Theorem ghmpropd 19217

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 18915	. . . . 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 18915	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝑀 ∈ Grp))
9	4, 8	anbi12d 630	. . . 4 ⊢ (𝜑 → ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ↔ (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp)))
10	1, 5, 2, 6, 3, 7	mhmpropd 18756	. . . . 5 ⊢ (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))
11	10	eleq2d 2815	. . . 4 ⊢ (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
12	9, 11	anbi12d 630	. . 3 ⊢ (𝜑 → (((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀))))
13		ghmgrp1 19179	. . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐽 ∈ Grp)
14		ghmgrp2 19180	. . . . 5 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐾 ∈ Grp)
15	13, 14	jca 510	. . . 4 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp))
16		ghmmhmb 19188	. . . . 5 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾))
17	16	eleq2d 2815	. . . 4 ⊢ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐽 MndHom 𝐾)))
18	15, 17	biadanii 820	. . 3 ⊢ (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)))
19		ghmgrp1 19179	. . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝐿 ∈ Grp)
20		ghmgrp2 19180	. . . . 5 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝑀 ∈ Grp)
21	19, 20	jca 510	. . . 4 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp))
22		ghmmhmb 19188	. . . . 5 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀))
23	22	eleq2d 2815	. . . 4 ⊢ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
24	21, 23	biadanii 820	. . 3 ⊢ (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀)))
25	12, 18, 24	3bitr4g 313	. 2 ⊢ (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
26	25	eqrdv 2726	1 ⊢ (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ‘cfv 6553  (class class class)co 7426  Basecbs 17187  +_gcplusg 17240   MndHom cmhm 18745  Grpcgrp 18897   GrpHom cghm 19174

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 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

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-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-map 8853  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-mhm 18747  df-grp 18900  df-ghm 19175

This theorem is referenced by:  rhmpropd  20555  lmhmpropd  20965  evls1maplmhm  22303