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Theorem ghmpropd 18387
Description: Group homomorphism depends only on the group attributes of structures. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
ghmpropd.a (𝜑𝐵 = (Base‘𝐽))
ghmpropd.b (𝜑𝐶 = (Base‘𝐾))
ghmpropd.c (𝜑𝐵 = (Base‘𝐿))
ghmpropd.d (𝜑𝐶 = (Base‘𝑀))
ghmpropd.e ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
ghmpropd.f ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)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.
StepHypRef Expression
1 ghmpropd.a . . . . . 6 (𝜑𝐵 = (Base‘𝐽))
2 ghmpropd.c . . . . . 6 (𝜑𝐵 = (Base‘𝐿))
3 ghmpropd.e . . . . . 6 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))
41, 2, 3grppropd 18109 . . . . 5 (𝜑 → (𝐽 ∈ Grp ↔ 𝐿 ∈ Grp))
5 ghmpropd.b . . . . . 6 (𝜑𝐶 = (Base‘𝐾))
6 ghmpropd.d . . . . . 6 (𝜑𝐶 = (Base‘𝑀))
7 ghmpropd.f . . . . . 6 ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))
85, 6, 7grppropd 18109 . . . . 5 (𝜑 → (𝐾 ∈ Grp ↔ 𝑀 ∈ Grp))
94, 8anbi12d 633 . . . 4 (𝜑 → ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ↔ (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp)))
101, 5, 2, 6, 3, 7mhmpropd 17953 . . . . 5 (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))
1110eleq2d 2899 . . . 4 (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
129, 11anbi12d 633 . . 3 (𝜑 → (((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀))))
13 ghmgrp1 18351 . . . . 5 (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐽 ∈ Grp)
14 ghmgrp2 18352 . . . . 5 (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐾 ∈ Grp)
1513, 14jca 515 . . . 4 (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp))
16 ghmmhmb 18360 . . . . 5 ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾))
1716eleq2d 2899 . . . 4 ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐽 MndHom 𝐾)))
1815, 17biadanii 821 . . 3 (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)))
19 ghmgrp1 18351 . . . . 5 (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝐿 ∈ Grp)
20 ghmgrp2 18352 . . . . 5 (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝑀 ∈ Grp)
2119, 20jca 515 . . . 4 (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp))
22 ghmmhmb 18360 . . . . 5 ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀))
2322eleq2d 2899 . . . 4 ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
2421, 23biadanii 821 . . 3 (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀)))
2512, 18, 243bitr4g 317 . 2 (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
2625eqrdv 2820 1 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  cfv 6334  (class class class)co 7140  Basecbs 16474  +gcplusg 16556   MndHom cmhm 17945  Grpcgrp 18094   GrpHom cghm 18346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-map 8395  df-0g 16706  df-mgm 17843  df-sgrp 17892  df-mnd 17903  df-mhm 17947  df-grp 18097  df-ghm 18347
This theorem is referenced by:  rhmpropd  19562  lmhmpropd  19836
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