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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 (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 18969 . . . . 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 18969 . . . . 5 (𝜑 → (𝐾 ∈ Grp ↔ 𝑀 ∈ Grp))
94, 8anbi12d 632 . . . 4 (𝜑 → ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ↔ (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp)))
101, 5, 2, 6, 3, 7mhmpropd 18805 . . . . 5 (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))
1110eleq2d 2827 . . . 4 (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
129, 11anbi12d 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)
1513, 14jca 511 . . . 4 (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp))
16 ghmmhmb 19245 . . . . 5 ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾))
1716eleq2d 2827 . . . 4 ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐽 MndHom 𝐾)))
1815, 17biadanii 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)
2119, 20jca 511 . . . 4 (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp))
22 ghmmhmb 19245 . . . . 5 ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀))
2322eleq2d 2827 . . . 4 ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
2421, 23biadanii 822 . . 3 (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀)))
2512, 18, 243bitr4g 314 . 2 (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
2625eqrdv 2735 1 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
Colors of variables: wff setvar class
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
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