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Theorem ismgmhm 42301
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 42299 . 2 (𝐹 ∈ (𝑆 MgmHom 𝑇) → (𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm))
2 fveq2 6332 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
3 ismgmhm.c . . . . . . . 8 𝐶 = (Base‘𝑇)
42, 3syl6eqr 2822 . . . . . . 7 (𝑡 = 𝑇 → (Base‘𝑡) = 𝐶)
5 fveq2 6332 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
6 ismgmhm.b . . . . . . . 8 𝐵 = (Base‘𝑆)
75, 6syl6eqr 2822 . . . . . . 7 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
84, 7oveqan12rd 6812 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ((Base‘𝑡) ↑𝑚 (Base‘𝑠)) = (𝐶𝑚 𝐵))
97adantr 466 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝐵)
10 fveq2 6332 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
11 ismgmhm.p . . . . . . . . . . . 12 + = (+g𝑆)
1210, 11syl6eqr 2822 . . . . . . . . . . 11 (𝑠 = 𝑆 → (+g𝑠) = + )
1312oveqd 6809 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑥(+g𝑠)𝑦) = (𝑥 + 𝑦))
1413fveq2d 6336 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑓‘(𝑥(+g𝑠)𝑦)) = (𝑓‘(𝑥 + 𝑦)))
15 fveq2 6332 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
16 ismgmhm.q . . . . . . . . . . 11 = (+g𝑇)
1715, 16syl6eqr 2822 . . . . . . . . . 10 (𝑡 = 𝑇 → (+g𝑡) = )
1817oveqd 6809 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) = ((𝑓𝑥) (𝑓𝑦)))
1914, 18eqeqan12d 2786 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
209, 19raleqbidv 3300 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥(+g𝑠)𝑦)) = ((𝑓𝑥)(+g𝑡)(𝑓𝑦)) ↔ ∀𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))))
219, 20raleqbidv 3300 . . . . . 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 3344 . . . . 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 42297 . . . . 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 6822 . . . . . 6 (𝐶𝑚 𝐵) ∈ V
2524rabex 4943 . . . . 5 {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ∈ V
2622, 23, 25ovmpt2a 6937 . . . 4 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝑆 MgmHom 𝑇) = {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))})
2726eleq2d 2835 . . 3 ((𝑆 ∈ Mgm ∧ 𝑇 ∈ Mgm) → (𝐹 ∈ (𝑆 MgmHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))}))
28 fveq1 6331 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑦)) = (𝐹‘(𝑥 + 𝑦)))
29 fveq1 6331 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
30 fveq1 6331 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑦) = (𝐹𝑦))
3129, 30oveq12d 6810 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑥) (𝑓𝑦)) = ((𝐹𝑥) (𝐹𝑦)))
3228, 31eqeq12d 2785 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
33322ralbidv 3137 . . . . 5 (𝑓 = 𝐹 → (∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦)) ↔ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
3433elrab 3513 . . . 4 (𝐹 ∈ {𝑓 ∈ (𝐶𝑚 𝐵) ∣ ∀𝑥𝐵𝑦𝐵 (𝑓‘(𝑥 + 𝑦)) = ((𝑓𝑥) (𝑓𝑦))} ↔ (𝐹 ∈ (𝐶𝑚 𝐵) ∧ ∀𝑥𝐵𝑦𝐵 (𝐹‘(𝑥 + 𝑦)) = ((𝐹𝑥) (𝐹𝑦))))
35 fvex 6342 . . . . . . 7 (Base‘𝑇) ∈ V
363, 35eqeltri 2845 . . . . . 6 𝐶 ∈ V
37 fvex 6342 . . . . . . 7 (Base‘𝑆) ∈ V
386, 37eqeltri 2845 . . . . . 6 𝐵 ∈ V
3936, 38elmap 8037 . . . . 5 (𝐹 ∈ (𝐶𝑚 𝐵) ↔ 𝐹:𝐵𝐶)
4039anbi1i 602 . . . 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 802 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 382   = wceq 1630  wcel 2144  wral 3060  {crab 3064  Vcvv 3349  wf 6027  cfv 6031  (class class class)co 6792  𝑚 cmap 8008  Basecbs 16063  +gcplusg 16148  Mgmcmgm 17447   MgmHom cmgmhm 42295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-map 8010  df-mgmhm 42297
This theorem is referenced by:  mgmhmf  42302  mgmhmpropd  42303  mgmhmlin  42304  mgmhmf1o  42305  idmgmhm  42306  resmgmhm  42316  resmgmhm2  42317  resmgmhm2b  42318  mgmhmco  42319  ismhm0  42323  mhmismgmhm  42324  isrnghmmul  42411  c0mgm  42427  c0snmgmhm  42432
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