Theorem mhmlem 19050

Description: Lemma for mhmmnd 19052 and ghmgrp 19054. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 12-May-2014.) (Revised by Thierry Arnoux, 25-Jan-2020.)

Hypotheses

Ref	Expression
ghmgrp.f	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
mhmlem.a	⊢ (𝜑 → 𝐴 ∈ 𝑋)
mhmlem.b	⊢ (𝜑 → 𝐵 ∈ 𝑋)

Assertion

Ref	Expression
mhmlem	⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵)))

Distinct variable groups:   𝑥,𝐹,𝑦   𝑥, + ,𝑦   𝑥,𝑋,𝑦   𝑥, ⨣ ,𝑦   𝜑,𝑥,𝑦   𝑥,𝐴,𝑦   𝑦,𝐵

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem mhmlem

Step	Hyp	Ref	Expression
1		id 22	. 2 ⊢ (𝜑 → 𝜑)
2		mhmlem.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
3		mhmlem.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
4		eleq1 2823	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
5	4	3anbi2d 1443	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
6		fvoveq1 7433	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝐴 + 𝑦)))
7		fveq2 6881	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
8	7	oveq1d 7425	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝑦)))
9	6, 8	eqeq12d 2752	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 + 𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝑦))))
10	5, 9	imbi12d 344	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦))) ↔ ((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝐴 + 𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝑦)))))
11		eleq1 2823	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
12	11	3anbi3d 1444	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
13		oveq2 7418	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
14	13	fveq2d 6885	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴 + 𝑦)) = (𝐹‘(𝐴 + 𝐵)))
15		fveq2 6881	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
16	15	oveq2d 7426	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵)))
17	14, 16	eqeq12d 2752	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴 + 𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))))
18	12, 17	imbi12d 344	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝐴 + 𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝑦))) ↔ ((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵)))))
19		ghmgrp.f	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)))
20	10, 18, 19	vtocl2g 3558	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))))
21	2, 3, 20	syl2anc 584	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))))
22	1, 2, 3, 21	mp3and 1466	1 ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ‘cfv 6536  (class class class)co 7410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-iota 6489  df-fv 6544  df-ov 7413

This theorem is referenced by:  mhmid  19051  mhmmnd  19052  ghmgrp  19054  ghmcmn  19817