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Theorem ismgmhm 42108
Description: Property of a magma homomorphism. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
ismgmhm.b 𝐵 = (Base‘𝑆)
ismgmhm.c 𝐶 = (Base‘𝑇)
ismgmhm.p + = (+g𝑆)
ismgmhm.q = (+g𝑇)
Assertion
Ref Expression
ismgmhm (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑆,𝑦   𝑥,𝑇,𝑦   𝑥,𝐹,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)   + (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem ismgmhm
Dummy variables 𝑓 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mgmhmrcl 42106 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2 fveq2 6229 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
3 ismgmhm.c . . . . . . . 8 𝐶 = (Base‘𝑇)
42, 3syl6eqr 2703 . . . . . . 7 (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
5 fveq2 6229 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
6 ismgmhm.b . . . . . . . 8 𝐵 = (Base‘𝑆)
75, 6syl6eqr 2703 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
84, 7oveqan12rd 6710 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) = (𝐶𝑚 𝐵))
97adantr 480 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
10 fveq2 6229 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
11 ismgmhm.p . . . . . . . . . . . 12 + = (+g𝑆)
1210, 11syl6eqr 2703 . . . . . . . . . . 11 (𝑠 = 𝑆 → (+g𝑠) = + )
1312oveqd 6707 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑥(+g𝑠)𝑦) = (𝑥 + 𝑦))
1413fveq2d 6233 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑓‘(𝑥(+g𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
15 fveq2 6229 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
16 ismgmhm.q . . . . . . . . . . 11 = (+g𝑇)
1715, 16syl6eqr 2703 . . . . . . . . . 10 (𝑡 = 𝑇 → (+g𝑡) = )
1817oveqd 6707 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
1914, 18eqeqan12d 2667 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
209, 19raleqbidv 3182 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
219, 20raleqbidv 3182 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
228, 21rabeqbidv 3226 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦))} = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))})
23 df-mgmhm 42104 . . . . 5 MgmHom = (𝑠 ∈ Mgm, 𝑡 ∈ Mgm ↦ {𝑓 ∈ ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) ∣ ∀𝑥 ∈ (Base‘𝑠)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦))})
24 ovex 6718 . . . . . 6 (𝐶𝑚 𝐵) ∈ V
2524rabex 4845 . . . . 5 {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ∈ V
2622, 23, 25ovmpt2a 6833 . . . 4 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝑆 MgmHom 𝑇) = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))})
2726eleq2d 2716 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))}))
28 fveq1 6228 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
29 fveq1 6228 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
30 fveq1 6228 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
3129, 30oveq12d 6708 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
3228, 31eqeq12d 2666 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
33322ralbidv 3018 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
3433elrab 3396 . . . 4 (𝐹 ∈ {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ↔ (𝐹 ∈ (𝐶𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
35 fvex 6239 . . . . . . 7 (Base‘𝑇) ∈ V
363, 35eqeltri 2726 . . . . . 6 𝐶 ∈ V
37 fvex 6239 . . . . . . 7 (Base‘𝑆) ∈ V
386, 37eqeltri 2726 . . . . . 6 𝐵 ∈ V
3936, 38elmap 7928 . . . . 5 (𝐹 ∈ (𝐶𝑚 𝐵) ↔ 𝐹:𝐵𝐶)
4039anbi1i 731 . . . 4 ((𝐹 ∈ (𝐶𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
4134, 40bitri 264 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
4227, 41syl6bb 276 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
431, 42biadan2 675 1 (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) ∧ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  Vcvv 3231  wf 5922  cfv 5926  (class class class)co 6690  𝑚 cmap 7899  Basecbs 15904  +gcplusg 15988  Mgmcmgm 17287   MgmHom cmgmhm 42102
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-map 7901  df-mgmhm 42104
This theorem is referenced by:  mgmhmf  42109  mgmhmpropd  42110  mgmhmlin  42111  mgmhmf1o  42112  idmgmhm  42113  resmgmhm  42123  resmgmhm2  42124  resmgmhm2b  42125  mgmhmco  42126  ismhm0  42130  mhmismgmhm  42131  isrnghmmul  42218  c0mgm  42234  c0snmgmhm  42239
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