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Metamath Proof Explorer

Theorem tendospcanN

Description: Cancellation law for trace-preserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014) (New usage is discouraged.)

Ref Expression
Hypotheses tendospcan.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
tendospcan.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
tendospcan.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
tendospcan.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
tendospcan.o ⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
Assertion tendospcanN ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ↔ 𝐹 = 𝐺 ) )

Proof

Step Hyp Ref Expression
1 tendospcan.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 tendospcan.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
3 tendospcan.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
4 tendospcan.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
5 tendospcan.o ⊢ 𝑂 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
6 2 3 4 tendocnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) → ◡ ( 𝑆 ‘ 𝐺 ) = ( 𝑆 ‘ ◡ 𝐺 ) )
7 6 3adant3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ◡ ( 𝑆 ‘ 𝐺 ) = ( 𝑆 ‘ ◡ 𝐺 ) )
8 7 coeq2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) = ( ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ ◡ 𝐺 ) ) )
9 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simp2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝑆 ∈ 𝐸 )
11 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐹 ∈ 𝑇 )
12 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → 𝐺 ∈ 𝑇 )
13 2 3 ltrncnv ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ◡ 𝐺 ∈ 𝑇 )
14 9 12 13 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ◡ 𝐺 ∈ 𝑇 )
15 2 3 4 tendospdi1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ ◡ 𝐺 ∈ 𝑇 ) ) → ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ ◡ 𝐺 ) ) )
16 9 10 11 14 15 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( ( 𝑆 ‘ 𝐹 ) ∘ ( 𝑆 ‘ ◡ 𝐺 ) ) )
17 8 16 eqtr4d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) = ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) )
18 17 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) = ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) )
19 18 eqeq1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) = ( I ↾ 𝐵 ) ↔ ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( I ↾ 𝐵 ) ) )
20 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
21 simpl2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → 𝑆 ∈ 𝐸 )
22 simpl3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → 𝐹 ∈ 𝑇 )
23 2 3 4 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ) → ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 )
24 20 21 22 23 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 )
25 simpl3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → 𝐺 ∈ 𝑇 )
26 2 3 4 tendocl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 )
27 20 21 25 26 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 )
28 1 2 3 ltrncoidN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ‘ 𝐹 ) ∈ 𝑇 ∧ ( 𝑆 ‘ 𝐺 ) ∈ 𝑇 ) → ( ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) = ( I ↾ 𝐵 ) ↔ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) )
29 20 24 27 28 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( ( 𝑆 ‘ 𝐹 ) ∘ ◡ ( 𝑆 ‘ 𝐺 ) ) = ( I ↾ 𝐵 ) ↔ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) )
30 20 25 13 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ◡ 𝐺 ∈ 𝑇 )
31 2 3 ltrnco ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ◡ 𝐺 ∈ 𝑇 ) → ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 )
32 20 22 30 31 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 )
33 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) )
34 1 2 3 4 5 tendoid0 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( ( 𝐹 ∘ ◡ 𝐺 ) ∈ 𝑇 ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) ) → ( ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( I ↾ 𝐵 ) ↔ 𝑆 = 𝑂 ) )
35 20 21 32 33 34 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑆 ‘ ( 𝐹 ∘ ◡ 𝐺 ) ) = ( I ↾ 𝐵 ) ↔ 𝑆 = 𝑂 ) )
36 19 29 35 3bitr3d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ↔ 𝑆 = 𝑂 ) )
37 36 biimpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) → 𝑆 = 𝑂 ) )
38 37 impancom ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) → ( ( 𝐹 ∘ ◡ 𝐺 ) ≠ ( I ↾ 𝐵 ) → 𝑆 = 𝑂 ) )
39 38 necon1d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) → ( 𝑆 ≠ 𝑂 → ( 𝐹 ∘ ◡ 𝐺 ) = ( I ↾ 𝐵 ) ) )
40 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
41 simpl3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) → 𝐹 ∈ 𝑇 )
42 simpl3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) → 𝐺 ∈ 𝑇 )
43 1 2 3 ltrncoidN ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝐹 ∘ ◡ 𝐺 ) = ( I ↾ 𝐵 ) ↔ 𝐹 = 𝐺 ) )
44 40 41 42 43 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) → ( ( 𝐹 ∘ ◡ 𝐺 ) = ( I ↾ 𝐵 ) ↔ 𝐹 = 𝐺 ) )
45 39 44 sylibd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐸 ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) ∧ ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ) → ( 𝑆 ≠ 𝑂 → 𝐹 = 𝐺 ) )
46 45 3exp1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑆 ∈ 𝐸 → ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) → ( 𝑆 ≠ 𝑂 → 𝐹 = 𝐺 ) ) ) ) )
47 46 com24 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) → ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( 𝑆 ∈ 𝐸 → ( 𝑆 ≠ 𝑂 → 𝐹 = 𝐺 ) ) ) ) )
48 47 imp5a ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) → ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂 ) → 𝐹 = 𝐺 ) ) ) )
49 48 com24 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂 ) → ( ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) → 𝐹 = 𝐺 ) ) ) )
50 49 3imp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) → 𝐹 = 𝐺 ) )
51 fveq2 ⊢ ( 𝐹 = 𝐺 → ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) )
52 50 51 impbid1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) → ( ( 𝑆 ‘ 𝐹 ) = ( 𝑆 ‘ 𝐺 ) ↔ 𝐹 = 𝐺 ) )