cdleml6

Description: Part of proof of Lemma L of Crawley p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013)

	Ref	Expression
Hypotheses	cdleml6.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdleml6.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleml6.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleml6.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleml6.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
	cdleml6.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
	cdleml6.p	⊢ 𝑄 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
	cdleml6.z	⊢ 𝑍 = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( ℎ ‘ 𝑄 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ^◡ ( 𝑠 ‘ ℎ ) ) ) ) )
	cdleml6.y	⊢ 𝑌 = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ^◡ 𝑏 ) ) ) )
	cdleml6.x	⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ ( 𝑠 ‘ ℎ ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑄 ) = 𝑌 ) )
	cdleml6.u	⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( ( 𝑠 ‘ ℎ ) = ℎ , 𝑔 , 𝑋 ) )
	cdleml6.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
	cdleml6.o	⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
Assertion	cdleml6	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( 𝑈 ∈ 𝐸 ∧ ( 𝑈 ‘ ( 𝑠 ‘ ℎ ) ) = ℎ ) )

Proof

Step	Hyp	Ref	Expression
1		cdleml6.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdleml6.j	⊢ ∨ = ( join ‘ 𝐾 )
3		cdleml6.m	⊢ ∧ = ( meet ‘ 𝐾 )
4		cdleml6.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
5		cdleml6.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
6		cdleml6.r	⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
7		cdleml6.p	⊢ 𝑄 = ( ( oc ‘ 𝐾 ) ‘ 𝑊 )
8		cdleml6.z	⊢ 𝑍 = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝑏 ) ) ∧ ( ( ℎ ‘ 𝑄 ) ∨ ( 𝑅 ‘ ( 𝑏 ∘ ^◡ ( 𝑠 ‘ ℎ ) ) ) ) )
9		cdleml6.y	⊢ 𝑌 = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ^◡ 𝑏 ) ) ) )
10		cdleml6.x	⊢ 𝑋 = ( ℩ 𝑧 ∈ 𝑇 ∀ 𝑏 ∈ 𝑇 ( ( 𝑏 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ ( 𝑠 ‘ ℎ ) ) ∧ ( 𝑅 ‘ 𝑏 ) ≠ ( 𝑅 ‘ 𝑔 ) ) → ( 𝑧 ‘ 𝑄 ) = 𝑌 ) )
11		cdleml6.u	⊢ 𝑈 = ( 𝑔 ∈ 𝑇 ↦ if ( ( 𝑠 ‘ ℎ ) = ℎ , 𝑔 , 𝑋 ) )
12		cdleml6.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
13		cdleml6.o	⊢ 0 = ( 𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵 ) )
14		simp1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
15		simp3l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → 𝑠 ∈ 𝐸 )
16		simp2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ℎ ∈ 𝑇 )
17	4 5 12	tendocl	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑠 ∈ 𝐸 ∧ ℎ ∈ 𝑇 ) → ( 𝑠 ‘ ℎ ) ∈ 𝑇 )
18	14 15 16 17	syl3anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( 𝑠 ‘ ℎ ) ∈ 𝑇 )
19	1 4 5 6 12 13	tendotr	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ∧ ℎ ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝑠 ‘ ℎ ) ) = ( 𝑅 ‘ ℎ ) )
20	19	3com23	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( 𝑅 ‘ ( 𝑠 ‘ ℎ ) ) = ( 𝑅 ‘ ℎ ) )
21		eqid	⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
22		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
23	1 2 3 21 22 4 5 6 7 8 9 10 11 12	cdlemk56w	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑠 ‘ ℎ ) ∈ 𝑇 ∧ ℎ ∈ 𝑇 ) ∧ ( 𝑅 ‘ ( 𝑠 ‘ ℎ ) ) = ( 𝑅 ‘ ℎ ) ) → ( 𝑈 ∈ 𝐸 ∧ ( 𝑈 ‘ ( 𝑠 ‘ ℎ ) ) = ℎ ) )
24	14 18 16 20 23	syl121anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ℎ ∈ 𝑇 ∧ ( 𝑠 ∈ 𝐸 ∧ 𝑠 ≠ 0 ) ) → ( 𝑈 ∈ 𝐸 ∧ ( 𝑈 ‘ ( 𝑠 ‘ ℎ ) ) = ℎ ) )

Metamath Proof Explorer

Theorem cdleml6

Proof