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Theorem ghmpropd 18049
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 17791 . . . . 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 17791 . . . . 5 (𝜑 → (𝐾 ∈ Grp ↔ 𝑀 ∈ Grp))
94, 8anbi12d 626 . . . 4 (𝜑 → ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ↔ (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp)))
101, 5, 2, 6, 3, 7mhmpropd 17694 . . . . 5 (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))
1110eleq2d 2892 . . . 4 (𝜑 → (𝑓 ∈ (𝐽 MndHom 𝐾) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
129, 11anbi12d 626 . . 3 (𝜑 → (((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀))))
13 ghmgrp1 18013 . . . . 5 (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐽 ∈ Grp)
14 ghmgrp2 18014 . . . . 5 (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐾 ∈ Grp)
1513, 14jca 509 . . . 4 (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp))
16 ghmmhmb 18022 . . . . 5 ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾))
1716eleq2d 2892 . . . 4 ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐽 MndHom 𝐾)))
1815, 17biadanii 859 . . 3 (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) ∧ 𝑓 ∈ (𝐽 MndHom 𝐾)))
19 ghmgrp1 18013 . . . . 5 (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝐿 ∈ Grp)
20 ghmgrp2 18014 . . . . 5 (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝑀 ∈ Grp)
2119, 20jca 509 . . . 4 (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp))
22 ghmmhmb 18022 . . . . 5 ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀))
2322eleq2d 2892 . . . 4 ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ 𝑓 ∈ (𝐿 MndHom 𝑀)))
2421, 23biadanii 859 . . 3 (𝑓 ∈ (𝐿 GrpHom 𝑀) ↔ ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) ∧ 𝑓 ∈ (𝐿 MndHom 𝑀)))
2512, 18, 243bitr4g 306 . 2 (𝜑 → (𝑓 ∈ (𝐽 GrpHom 𝐾) ↔ 𝑓 ∈ (𝐿 GrpHom 𝑀)))
2625eqrdv 2823 1 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1658  wcel 2166  cfv 6123  (class class class)co 6905  Basecbs 16222  +gcplusg 16305   MndHom cmhm 17686  Grpcgrp 17776   GrpHom cghm 18008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-map 8124  df-0g 16455  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-mhm 17688  df-grp 17779  df-ghm 18009
This theorem is referenced by:  rhmpropd  19171  lmhmpropd  19432
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