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Theorem isghm 17707
Description: Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)
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 17705 . . 3 GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑓[(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))})
21elmpt2cl 6918 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
3 fvex 6239 . . . . . . . 8 (Base‘𝑠) ∈ V
4 feq2 6065 . . . . . . . . 9 (𝑤 = (Base‘𝑠) → (𝑓:𝑤⟶(Base‘𝑡) ↔ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)))
5 raleq 3168 . . . . . . . . . 10 (𝑤 = (Base‘𝑠) → (∀𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
65raleqbi1dv 3176 . . . . . . . . 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 747 . . . . . . . 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 3503 . . . . . . 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 6229 . . . . . . . . . 10 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
10 isghm.w . . . . . . . . . 10 𝑋 = (Base‘𝑆)
119, 10syl6eqr 2703 . . . . . . . . 9 (𝑠 = 𝑆 → (Base‘𝑠) = 𝑋)
1211feq2d 6069 . . . . . . . 8 (𝑠 = 𝑆 → (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶(Base‘𝑡)))
13 fveq2 6229 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
14 isghm.a . . . . . . . . . . . . . 14 + = (+g𝑆)
1513, 14syl6eqr 2703 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (+g𝑠) = + )
1615oveqd 6707 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑢(+g𝑠)𝑣) = (𝑢 + 𝑣))
1716fveq2d 6233 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑓‘(𝑢(+g𝑠)𝑣)) = (𝑓‘(𝑢 + 𝑣)))
1817eqeq1d 2653 . . . . . . . . . 10 (𝑠 = 𝑆 → ((𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
1911, 18raleqbidv 3182 . . . . . . . . 9 (𝑠 = 𝑆 → (∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
2011, 19raleqbidv 3182 . . . . . . . 8 (𝑠 = 𝑆 → (∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
2112, 20anbi12d 747 . . . . . . 7 (𝑠 = 𝑆 → ((𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))) ↔ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))))
228, 21syl5bb 272 . . . . . 6 (𝑠 = 𝑆 → ([(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))) ↔ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))))
2322abbidv 2770 . . . . 5 (𝑠 = 𝑆 → {𝑓[(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))} = {𝑓 ∣ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))})
24 fveq2 6229 . . . . . . . . 9 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
25 isghm.x . . . . . . . . 9 𝑌 = (Base‘𝑇)
2624, 25syl6eqr 2703 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = 𝑌)
2726feq3d 6070 . . . . . . 7 (𝑡 = 𝑇 → (𝑓:𝑋⟶(Base‘𝑡) ↔ 𝑓:𝑋𝑌))
28 fveq2 6229 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
29 isghm.b . . . . . . . . . . 11 = (+g𝑇)
3028, 29syl6eqr 2703 . . . . . . . . . 10 (𝑡 = 𝑇 → (+g𝑡) = )
3130oveqd 6707 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) = ((𝑓𝑢) (𝑓𝑣)))
3231eqeq2d 2661 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
33322ralbidv 3018 . . . . . . 7 (𝑡 = 𝑇 → (∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
3427, 33anbi12d 747 . . . . . 6 (𝑡 = 𝑇 → ((𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))))
3534abbidv 2770 . . . . 5 (𝑡 = 𝑇 → {𝑓 ∣ (𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))} = {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))})
36 fvex 6239 . . . . . . . 8 (Base‘𝑆) ∈ V
3710, 36eqeltri 2726 . . . . . . 7 𝑋 ∈ V
38 fvex 6239 . . . . . . . 8 (Base‘𝑇) ∈ V
3925, 38eqeltri 2726 . . . . . . 7 𝑌 ∈ V
40 mapex 7905 . . . . . . 7 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑓𝑓:𝑋𝑌} ∈ V)
4137, 39, 40mp2an 708 . . . . . 6 {𝑓𝑓:𝑋𝑌} ∈ V
42 simpl 472 . . . . . . 7 ((𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))) → 𝑓:𝑋𝑌)
4342ss2abi 3707 . . . . . 6 {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ⊆ {𝑓𝑓:𝑋𝑌}
4441, 43ssexi 4836 . . . . 5 {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ∈ V
4523, 35, 1, 44ovmpt2 6838 . . . 4 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))})
4645eleq2d 2716 . . 3 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))}))
47 fex 6530 . . . . . 6 ((𝐹:𝑋𝑌𝑋 ∈ V) → 𝐹 ∈ V)
4837, 47mpan2 707 . . . . 5 (𝐹:𝑋𝑌𝐹 ∈ V)
4948adantr 480 . . . 4 ((𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))) → 𝐹 ∈ V)
50 feq1 6064 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝑋𝑌𝐹:𝑋𝑌))
51 fveq1 6228 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝑣)))
52 fveq1 6228 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑢) = (𝐹𝑢))
53 fveq1 6228 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑣) = (𝐹𝑣))
5452, 53oveq12d 6708 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑢) (𝑓𝑣)) = ((𝐹𝑢) (𝐹𝑣)))
5551, 54eqeq12d 2666 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)) ↔ (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
56552ralbidv 3018 . . . . 5 (𝑓 = 𝐹 → (∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
5750, 56anbi12d 747 . . . 4 (𝑓 = 𝐹 → ((𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
5849, 57elab3 3390 . . 3 (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
5946, 58syl6bb 276 . 2 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
602, 59biadan2 675 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 196  wa 383   = wceq 1523  wcel 2030  {cab 2637  wral 2941  Vcvv 3231  [wsbc 3468  wf 5922  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Grpcgrp 17469   GrpHom cghm 17704
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-rep 4804  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-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-ghm 17705
This theorem is referenced by:  isghm3  17708  ghmgrp1  17709  ghmgrp2  17710  ghmf  17711  ghmlin  17712  isghmd  17716  idghm  17722  ghmf1o  17737  islmhm2  19086  expghm  19892  mulgghm2  19893  pi1xfr  22901  pi1coghm  22907  rhmopp  29947  isrnghm  42217
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