cdlemk41 - Mathbox for Norm Megill

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Theorem cdlemk41 40397

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

**Hypothesis**
Ref	Expression
cdlemk41.y	⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))

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

Distinct variable groups: ∧ ,𝑔 ∨ ,𝑔 𝑔,𝐺 𝑃,𝑔 𝑅,𝑔 𝑇,𝑔 𝑔,𝑍 𝑔,𝑏 Allowed substitution hints: 𝑃(𝑏) 𝑅(𝑏) 𝑇(𝑏) 𝐺(𝑏) ∨ (𝑏) ∧ (𝑏) 𝑌(𝑔,𝑏) 𝑍(𝑏)

**Proof of Theorem cdlemk41**
Step	Hyp	Ref	Expression
1		nfcvd 2899	. 2 ⊢ (𝐺 ∈ 𝑇 → Ⅎ𝑔((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ^◡𝑏)))))
2		cdlemk41.y	. . 3 ⊢ 𝑌 = ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))))
3		fveq2 6900	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘𝑔) = (𝑅‘𝐺))
4	3	oveq2d 7440	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑃 ∨ (𝑅‘𝑔)) = (𝑃 ∨ (𝑅‘𝐺)))
5		coeq1 5862	. . . . . 6 ⊢ (𝑔 = 𝐺 → (𝑔 ∘ ^◡𝑏) = (𝐺 ∘ ^◡𝑏))
6	5	fveq2d 6904	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑅‘(𝑔 ∘ ^◡𝑏)) = (𝑅‘(𝐺 ∘ ^◡𝑏)))
7	6	oveq2d 7440	. . . 4 ⊢ (𝑔 = 𝐺 → (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏))) = (𝑍 ∨ (𝑅‘(𝐺 ∘ ^◡𝑏))))
8	4, 7	oveq12d 7442	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑃 ∨ (𝑅‘𝑔)) ∧ (𝑍 ∨ (𝑅‘(𝑔 ∘ ^◡𝑏)))) = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ^◡𝑏)))))
9	2, 8	eqtrid 2779	. 2 ⊢ (𝑔 = 𝐺 → 𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ^◡𝑏)))))
10	1, 9	csbiegf 3926	1 ⊢ (𝐺 ∈ 𝑇 → ⦋𝐺 / 𝑔⦌𝑌 = ((𝑃 ∨ (𝑅‘𝐺)) ∧ (𝑍 ∨ (𝑅‘(𝐺 ∘ ^◡𝑏)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ⦋csb 3892 ^◡ccnv 5679 ∘ ccom 5684 ‘cfv 6551 (class class class)co 7424

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-co 5689 df-iota 6503 df-fv 6559 df-ov 7427

This theorem is referenced by: cdlemkid2 40401 cdlemkfid3N 40402 cdlemky 40403 cdlemk42yN 40421

	
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