Theorem ghmpropd 19274

Description: Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
ghmpropd.a	⊢ (𝜑 → 𝐵 = (Base‘𝐽))
ghmpropd.b	⊢ (𝜑 → 𝐶 = (Base‘𝐾))
ghmpropd.c	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
ghmpropd.d	⊢ (𝜑 → 𝐶 = (Base‘𝑀))
ghmpropd.e	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
ghmpropd.f	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))

Assertion

Ref	Expression
ghmpropd	⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))

Distinct variable groups:   𝑥,𝑦,𝐽   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝑥,𝑀,𝑦   𝜑,𝑥,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem ghmpropd

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ghmpropd.a	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐽))
2		ghmpropd.c	. . . . . 6 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		ghmpropd.e	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐽)𝑦) = (𝑥(+g‘𝐿)𝑦))
4	1, 2, 3	grppropd 18969	. . . . 5 ⊢ (𝜑 → (𝐽 ∈ Grp ↔ 𝐿 ∈ Grp))
5		ghmpropd.b	. . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝐾))
6		ghmpropd.d	. . . . . 6 ⊢ (𝜑 → 𝐶 = (Base‘𝑀))
7		ghmpropd.f	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝑀)𝑦))
8	5, 6, 7	grppropd 18969	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝑀 ∈ Grp))
9	4, 8	anbi12d 632	. . . 4 ⊢ (𝜑 → ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ↔ (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp)))
10	1, 5, 2, 6, 3, 7	mhmpropd 18805	. . . . 5 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))
11	10	eleq2d 2827	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
12	9, 11	anbi12d 632	. . 3 ⊢ (𝜑 → (((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀))))
13		ghmgrp1 19236	. . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐽 ∈ Grp)
14		ghmgrp2 19237	. . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐾 ∈ Grp)
15	13, 14	jca 511	. . . 4 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp))
16		ghmmhmb 19245	. . . . 5 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾))
17	16	eleq2d 2827	. . . 4 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐽 MndHom 𝐾)))
18	15, 17	biadanii 822	. . 3 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)))
19		ghmgrp1 19236	. . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝐿 ∈ Grp)
20		ghmgrp2 19237	. . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝑀 ∈ Grp)
21	19, 20	jca 511	. . . 4 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp))
22		ghmmhmb 19245	. . . . 5 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀))
23	22	eleq2d 2827	. . . 4 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
24	21, 23	biadanii 822	. . 3 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀)))
25	12, 18, 24	3bitr4g 314	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
26	25	eqrdv 2735	1 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297   MndHom cmhm 18794  Grpcgrp 18951   GrpHom cghm 19230

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-grp 18954  df-ghm 19231

This theorem is referenced by:  rhmpropd  20609  lmhmpropd  21072  evls1maplmhm  22381