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Theorem ismgmhm 46163
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 46161 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2 fveq2 6843 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
3 ismgmhm.c . . . . . . . 8 𝐶 = (Base‘𝑇)
42, 3eqtr4di 2791 . . . . . . 7 (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
5 fveq2 6843 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
6 ismgmhm.b . . . . . . . 8 𝐵 = (Base‘𝑆)
75, 6eqtr4di 2791 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
84, 7oveqan12rd 7378 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ((Base‘𝑡) ↑m (Base‘𝑠)) = (𝐶m 𝐵))
97adantr 482 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
10 fveq2 6843 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
11 ismgmhm.p . . . . . . . . . . . 12 + = (+g𝑆)
1210, 11eqtr4di 2791 . . . . . . . . . . 11 (𝑠 = 𝑆 → (+g𝑠) = + )
1312oveqd 7375 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑥(+g𝑠)𝑦) = (𝑥 + 𝑦))
1413fveq2d 6847 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑓‘(𝑥(+g𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
15 fveq2 6843 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
16 ismgmhm.q . . . . . . . . . . 11 = (+g𝑇)
1715, 16eqtr4di 2791 . . . . . . . . . 10 (𝑡 = 𝑇 → (+g𝑡) = )
1817oveqd 7375 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
1914, 18eqeqan12d 2747 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
209, 19raleqbidv 3318 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
219, 20raleqbidv 3318 . . . . . 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 3423 . . . . 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 46159 . . . . 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 7391 . . . . . 6 (𝐶m 𝐵) ∈ V
2524rabex 5290 . . . . 5 {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ∈ V
2622, 23, 25ovmpoa 7511 . . . 4 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝑆 MgmHom 𝑇) = {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))})
2726eleq2d 2820 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))}))
28 fveq1 6842 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
29 fveq1 6842 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
30 fveq1 6842 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
3129, 30oveq12d 7376 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
3228, 31eqeq12d 2749 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
33322ralbidv 3209 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
3433elrab 3646 . . . 4 (𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ↔ (𝐹 ∈ (𝐶m 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
353fvexi 6857 . . . . . 6 𝐶 ∈ V
366fvexi 6857 . . . . . 6 𝐵 ∈ V
3735, 36elmap 8812 . . . . 5 (𝐹 ∈ (𝐶m 𝐵) ↔ 𝐹:𝐵𝐶)
3837anbi1i 625 . . . 4 ((𝐹 ∈ (𝐶m 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
3934, 38bitri 275 . . 3 (𝐹 ∈ {𝑓 ∈ (𝐶m 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
4027, 39bitrdi 287 . 2 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ (𝐹:𝐵𝐶 ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦)))))
411, 40biadanii 821 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 205  wa 397   = wceq 1542  wcel 2107  wral 3061  {crab 3406  wf 6493  cfv 6497  (class class class)co 7358  m cmap 8768  Basecbs 17088  +gcplusg 17138  Mgmcmgm 18500   MgmHom cmgmhm 46157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-mgmhm 46159
This theorem is referenced by:  mgmhmf  46164  mgmhmpropd  46165  mgmhmlin  46166  mgmhmf1o  46167  idmgmhm  46168  resmgmhm  46178  resmgmhm2  46179  resmgmhm2b  46180  mgmhmco  46181  ismhm0  46185  mhmismgmhm  46186  isrnghmmul  46277  c0mgm  46293  c0snmgmhm  46298
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