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Metamath Proof Explorer

Theorem dihord10

Description: Part of proof after Lemma N of Crawley p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014)

Ref Expression
Hypotheses dihjust.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
dihjust.l ⊢ ≤ = ( le ‘ 𝐾 )
dihjust.j ⊢ ∨ = ( join ‘ 𝐾 )
dihjust.m ⊢ ∧ = ( meet ‘ 𝐾 )
dihjust.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
dihjust.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
dihjust.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
dihjust.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
dihjust.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
dihjust.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
dihord2c.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
dihord2c.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
dihord2c.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
dihord2.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
dihord2.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
dihord2.d ⊢ + = ( +g ‘ 𝑈 )
dihord2.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 )
Assertion dihord10 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) )

Proof

Step Hyp Ref Expression
1 dihjust.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 dihjust.l ⊢ ≤ = ( le ‘ 𝐾 )
3 dihjust.j ⊢ ∨ = ( join ‘ 𝐾 )
4 dihjust.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 dihjust.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 dihjust.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 dihjust.i ⊢ 𝐼 = ( ( DIsoB ‘ 𝐾 ) ‘ 𝑊 )
8 dihjust.J ⊢ 𝐽 = ( ( DIsoC ‘ 𝐾 ) ‘ 𝑊 )
9 dihjust.u ⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
10 dihjust.s ⊢ ⊕ = ( LSSum ‘ 𝑈 )
11 dihord2c.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
12 dihord2c.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
13 dihord2c.o ⊢ 𝑂 = ( ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
14 dihord2.p ⊢ 𝑃 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
15 dihord2.e ⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
16 dihord2.d ⊢ + = ( +g ‘ 𝑈 )
17 dihord2.g ⊢ 𝐺 = ( ℩ ℎ ∈ 𝑇 ( ℎ ‘ 𝑃 ) = 𝑁 )
18 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
19 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
20 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) )
21 simp31l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑠 ∈ 𝐸 )
22 simp31r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑔 ∈ 𝑇 )
23 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) )
24 1 2 5 6 14 13 11 15 9 16 17 dihordlem7b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑓 = 𝑔 ∧ 𝑂 = 𝑠 ) )
25 24 simpld ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑓 = 𝑔 )
26 18 19 20 21 22 23 25 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → 𝑓 = 𝑔 )
27 26 fveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑅 ‘ 𝑓 ) = ( 𝑅 ‘ 𝑔 ) )
28 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) )
29 27 28 eqbrtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑁 ∈ 𝐴 ∧ ¬ 𝑁 ≤ 𝑊 ) ) ∧ ( 𝑓 ∈ 𝑇 ∧ ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑋 ∧ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝑔 ) ≤ ( 𝑌 ∧ 𝑊 ) ∧ ⟨ 𝑓 , 𝑂 ⟩ = ( ⟨ ( 𝑠 ‘ 𝐺 ) , 𝑠 ⟩ + ⟨ 𝑔 , 𝑂 ⟩ ) ) ) → ( 𝑅 ‘ 𝑓 ) ≤ ( 𝑌 ∧ 𝑊 ) )