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Theorem ghmpropd 19287
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 18982 . . . . 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 18982 . . . . 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 18818 . . . . 5 (𝜑 → (𝐽 MndHom 𝐾) = (𝐿 MndHom 𝑀))
1110eleq2d 2825 . . . 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 19249 . . . . 5 (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐽 ∈ Grp)
14 ghmgrp2 19250 . . . . 5 (𝑓 ∈ (𝐽 GrpHom 𝐾) → 𝐾 ∈ Grp)
1513, 14jca 511 . . . 4 (𝑓 ∈ (𝐽 GrpHom 𝐾) → (𝐽 ∈ Grp ∧ 𝐾 ∈ Grp))
16 ghmmhmb 19258 . . . . 5 ((𝐽 ∈ Grp ∧ 𝐾 ∈ Grp) → (𝐽 GrpHom 𝐾) = (𝐽 MndHom 𝐾))
1716eleq2d 2825 . . . 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 19249 . . . . 5 (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝐿 ∈ Grp)
20 ghmgrp2 19250 . . . . 5 (𝑓 ∈ (𝐿 GrpHom 𝑀) → 𝑀 ∈ Grp)
2119, 20jca 511 . . . 4 (𝑓 ∈ (𝐿 GrpHom 𝑀) → (𝐿 ∈ Grp ∧ 𝑀 ∈ Grp))
22 ghmmhmb 19258 . . . . 5 ((𝐿 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐿 GrpHom 𝑀) = (𝐿 MndHom 𝑀))
2322eleq2d 2825 . . . 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 2733 1 (𝜑 → (𝐽 GrpHom 𝐾) = (𝐿 GrpHom 𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298   MndHom cmhm 18807  Grpcgrp 18964   GrpHom cghm 19243
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-map 8867  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-grp 18967  df-ghm 19244
This theorem is referenced by:  rhmpropd  20626  lmhmpropd  21090  evls1maplmhm  22397
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