Theorem cdlemk41

Description: Part of proof of Lemma K of Crawley p. 118. TODO: fix comment. (Contributed by NM, 19-Jul-2013)

	Ref	Expression
Hypothesis	cdlemk41.y	⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
Assertion	cdlemk41	⊢ ( 𝐺 ∈ 𝑇 → ⦋ 𝐺 / 𝑔 ⦌ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝑏 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		cdlemk41.y	⊢ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) )
2		nfcvd	⊢ ( 𝐺 ∈ 𝑇 → Ⅎ 𝑔 ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝑏 ) ) ) ) )
3		fveq2	⊢ ( 𝑔 = 𝐺 → ( 𝑅 ‘ 𝑔 ) = ( 𝑅 ‘ 𝐺 ) )
4	3	oveq2d	⊢ ( 𝑔 = 𝐺 → ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) = ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) )
5		coeq1	⊢ ( 𝑔 = 𝐺 → ( 𝑔 ∘ ◡ 𝑏 ) = ( 𝐺 ∘ ◡ 𝑏 ) )
6	5	fveq2d	⊢ ( 𝑔 = 𝐺 → ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝑏 ) ) )
7	6	oveq2d	⊢ ( 𝑔 = 𝐺 → ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) = ( 𝑍 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝑏 ) ) ) )
8	4 7	oveq12d	⊢ ( 𝑔 = 𝐺 → ( ( 𝑃 ∨ ( 𝑅 ‘ 𝑔 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝑔 ∘ ◡ 𝑏 ) ) ) ) = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝑏 ) ) ) ) )
9	1 8	syl5eq	⊢ ( 𝑔 = 𝐺 → 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝑏 ) ) ) ) )
10	2 9	csbiegf	⊢ ( 𝐺 ∈ 𝑇 → ⦋ 𝐺 / 𝑔 ⦌ 𝑌 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑍 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝑏 ) ) ) ) )