Theorem mhmlem 12978

Description: Lemma for mhmmnd 12980 and ghmgrp 12982. (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 19	. 2 ⊢ (𝜑 → 𝜑)
2		mhmlem.a	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑋)
3		mhmlem.b	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑋)
4		eleq1 2240	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝑋 ↔ 𝐴 ∈ 𝑋))
5	4	3anbi2d 1317	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)))
6		fvoveq1 5898	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝐹‘(𝑥 + 𝑦)) = (𝐹‘(𝐴 + 𝑦)))
7		fveq2 5516	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
8	7	oveq1d 5890	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝑦)))
9	6, 8	eqeq12d 2192	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝐹‘(𝑥 + 𝑦)) = ((𝐹‘𝑥) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 + 𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝑦))))
10	5, 9	imbi12d 234	. . . 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 2240	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝑋 ↔ 𝐵 ∈ 𝑋))
12	11	3anbi3d 1318	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ↔ (𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)))
13		oveq2 5883	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐴 + 𝑦) = (𝐴 + 𝐵))
14	13	fveq2d 5520	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐹‘(𝐴 + 𝑦)) = (𝐹‘(𝐴 + 𝐵)))
15		fveq2 5516	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (𝐹‘𝑦) = (𝐹‘𝐵))
16	15	oveq2d 5891	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((𝐹‘𝐴) ⨣ (𝐹‘𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵)))
17	14, 16	eqeq12d 2192	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝐹‘(𝐴 + 𝑦)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝑦)) ↔ (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))))
18	12, 17	imbi12d 234	. . . 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 2802	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))))
21	2, 3, 20	syl2anc 411	. 2 ⊢ (𝜑 → ((𝜑 ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵))))
22	1, 2, 3, 21	mp3and 1340	1 ⊢ (𝜑 → (𝐹‘(𝐴 + 𝐵)) = ((𝐹‘𝐴) ⨣ (𝐹‘𝐵)))

Syntax hints:   → wi 4   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  ‘cfv 5217  (class class class)co 5875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225  df-ov 5878

This theorem is referenced by:  mhmid  12979  mhmmnd  12980  ghmgrp  12982