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Theorem ismgmhm 43532
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 43530 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2 fveq2 6538 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
3 ismgmhm.c . . . . . . . 8 𝐶 = (Base‘𝑇)
42, 3syl6eqr 2849 . . . . . . 7 (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
5 fveq2 6538 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
6 ismgmhm.b . . . . . . . 8 𝐵 = (Base‘𝑆)
75, 6syl6eqr 2849 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
84, 7oveqan12rd 7036 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) = (𝐶𝑚 𝐵))
97adantr 481 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
10 fveq2 6538 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
11 ismgmhm.p . . . . . . . . . . . 12 + = (+g𝑆)
1210, 11syl6eqr 2849 . . . . . . . . . . 11 (𝑠 = 𝑆 → (+g𝑠) = + )
1312oveqd 7033 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑥(+g𝑠)𝑦) = (𝑥 + 𝑦))
1413fveq2d 6542 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑓‘(𝑥(+g𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
15 fveq2 6538 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
16 ismgmhm.q . . . . . . . . . . 11 = (+g𝑇)
1715, 16syl6eqr 2849 . . . . . . . . . 10 (𝑡 = 𝑇 → (+g𝑡) = )
1817oveqd 7033 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
1914, 18eqeqan12d 2811 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
209, 19raleqbidv 3361 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
219, 20raleqbidv 3361 . . . . . 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 3430 . . . . 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 43528 . . . . 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 7048 . . . . . 6 (𝐶𝑚 𝐵) ∈ V
2524rabex 5126 . . . . 5 {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ∈ V
2622, 23, 25ovmpoa 7161 . . . 4 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝑆 MgmHom 𝑇) = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))})
2726eleq2d 2868 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))}))
28 fveq1 6537 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
29 fveq1 6537 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
30 fveq1 6537 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
3129, 30oveq12d 7034 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
3228, 31eqeq12d 2810 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
33322ralbidv 3166 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
3433elrab 3618 . . . 4 (𝐹 ∈ {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ↔ (𝐹 ∈ (𝐶𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
353fvexi 6552 . . . . . 6 𝐶 ∈ V
366fvexi 6552 . . . . . 6 𝐵 ∈ V
3735, 36elmap 8285 . . . . 5 (𝐹 ∈ (𝐶𝑚 𝐵) ↔ 𝐹:𝐵𝐶)
3837anbi1i 623 . . . 4 ((𝐹 ∈ (𝐶𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
3934, 38bitri 276 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
4027, 39syl6bb 288 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
411, 40biadanii 819 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 207  wa 396   = wceq 1522  wcel 2081  wral 3105  {crab 3109  wf 6221  cfv 6225  (class class class)co 7016  𝑚 cmap 8256  Basecbs 16312  +gcplusg 16394  Mgmcmgm 17679   MgmHom cmgmhm 43526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-map 8258  df-mgmhm 43528
This theorem is referenced by:  mgmhmf  43533  mgmhmpropd  43534  mgmhmlin  43535  mgmhmf1o  43536  idmgmhm  43537  resmgmhm  43547  resmgmhm2  43548  resmgmhm2b  43549  mgmhmco  43550  ismhm0  43554  mhmismgmhm  43555  isrnghmmul  43642  c0mgm  43658  c0snmgmhm  43663
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