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Theorem isghm 19233
Description: Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.) (Proof shortened by SN, 5-Jun-2025.)
Hypotheses
Ref Expression
isghm.w 𝑋 = (Base‘𝑆)
isghm.x 𝑌 = (Base‘𝑇)
isghm.a + = (+g𝑆)
isghm.b = (+g𝑇)
Assertion
Ref Expression
isghm (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
Distinct variable groups:   𝑣,𝑢,𝑆   𝑢,𝑇,𝑣   𝑢,𝑋,𝑣   𝑢, + ,𝑣   𝑢,𝑌,𝑣   𝑢, ,𝑣   𝑢,𝐹,𝑣

Proof of Theorem isghm
Dummy variables 𝑡 𝑠 𝑤 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ghm 19231 . . 3 GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑓[(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))})
21elmpocl 7674 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
3 fvex 6919 . . . . . . . 8 (Base‘𝑠) ∈ V
4 feq2 6717 . . . . . . . . 9 (𝑤 = (Base‘𝑠) → (𝑓:𝑤⟶(Base‘𝑡) ↔ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)))
5 raleq 3323 . . . . . . . . . 10 (𝑤 = (Base‘𝑠) → (∀𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
65raleqbi1dv 3338 . . . . . . . . 9 (𝑤 = (Base‘𝑠) → (∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
74, 6anbi12d 632 . . . . . . . 8 (𝑤 = (Base‘𝑠) → ((𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))) ↔ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))))
83, 7sbcie 3830 . . . . . . 7 ([(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))) ↔ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
9 fveq2 6906 . . . . . . . . . . 11 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
10 isghm.w . . . . . . . . . . 11 𝑋 = (Base‘𝑆)
119, 10eqtr4di 2795 . . . . . . . . . 10 (𝑠 = 𝑆 → (Base‘𝑠) = 𝑋)
1211adantr 480 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝑋)
13 fveq2 6906 . . . . . . . . . . 11 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
14 isghm.x . . . . . . . . . . 11 𝑌 = (Base‘𝑇)
1513, 14eqtr4di 2795 . . . . . . . . . 10 (𝑡 = 𝑇 → (Base‘𝑡) = 𝑌)
1615adantl 481 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑡) = 𝑌)
1712, 16feq23d 6731 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ↔ 𝑓:𝑋𝑌))
18 fveq2 6906 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
19 isghm.a . . . . . . . . . . . . . 14 + = (+g𝑆)
2018, 19eqtr4di 2795 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (+g𝑠) = + )
2120oveqd 7448 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑢(+g𝑠)𝑣) = (𝑢 + 𝑣))
2221fveq2d 6910 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑓‘(𝑢(+g𝑠)𝑣)) = (𝑓‘(𝑢 + 𝑣)))
23 fveq2 6906 . . . . . . . . . . . . 13 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
24 isghm.b . . . . . . . . . . . . 13 = (+g𝑇)
2523, 24eqtr4di 2795 . . . . . . . . . . . 12 (𝑡 = 𝑇 → (+g𝑡) = )
2625oveqd 7448 . . . . . . . . . . 11 (𝑡 = 𝑇 → ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) = ((𝑓𝑢) (𝑓𝑣)))
2722, 26eqeqan12d 2751 . . . . . . . . . 10 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
2812, 27raleqbidv 3346 . . . . . . . . 9 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
2912, 28raleqbidv 3346 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
3017, 29anbi12d 632 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))))
318, 30bitrid 283 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → ([(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))))
3231abbidv 2808 . . . . 5 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑓[(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))} = {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))})
3314fvexi 6920 . . . . . . 7 𝑌 ∈ V
34 fsetex 8896 . . . . . . 7 (𝑌 ∈ V → {𝑓𝑓:𝑋𝑌} ∈ V)
3533, 34ax-mp 5 . . . . . 6 {𝑓𝑓:𝑋𝑌} ∈ V
36 abanssl 4311 . . . . . 6 {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ⊆ {𝑓𝑓:𝑋𝑌}
3735, 36ssexi 5322 . . . . 5 {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ∈ V
3832, 1, 37ovmpoa 7588 . . . 4 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))})
3938eleq2d 2827 . . 3 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))}))
4010fvexi 6920 . . . . . 6 𝑋 ∈ V
41 fex2 7958 . . . . . 6 ((𝐹:𝑋𝑌𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐹 ∈ V)
4240, 33, 41mp3an23 1455 . . . . 5 (𝐹:𝑋𝑌𝐹 ∈ V)
4342adantr 480 . . . 4 ((𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))) → 𝐹 ∈ V)
44 feq1 6716 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝑋𝑌𝐹:𝑋𝑌))
45 fveq1 6905 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝑣)))
46 fveq1 6905 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑢) = (𝐹𝑢))
47 fveq1 6905 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑣) = (𝐹𝑣))
4846, 47oveq12d 7449 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑢) (𝑓𝑣)) = ((𝐹𝑢) (𝐹𝑣)))
4945, 48eqeq12d 2753 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)) ↔ (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
50492ralbidv 3221 . . . . 5 (𝑓 = 𝐹 → (∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
5144, 50anbi12d 632 . . . 4 (𝑓 = 𝐹 → ((𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
5243, 51elab3 3686 . . 3 (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
5339, 52bitrdi 287 . 2 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
542, 53biadanii 822 1 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  wral 3061  Vcvv 3480  [wsbc 3788  wf 6557  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951   GrpHom cghm 19230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-map 8868  df-ghm 19231
This theorem is referenced by:  isghm3  19235  ghmgrp1  19236  ghmgrp2  19237  ghmf  19238  ghmlin  19239  isghmd  19243  idghm  19249  ghmf1o  19266  isrnghm  20441  rhmopp  20509  islmhm2  21037  expghm  21486  mulgghm2  21487  pi1xfr  25088  pi1coghm  25094  zringfrac  33582
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