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Theorem isghm 17592
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 17590 . . 3 GrpHom = (𝑠 ∈ Grp, 𝑡 ∈ Grp ↦ {𝑓[(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))})
21elmpt2cl 6836 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) → (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp))
3 fvex 6163 . . . . . . . 8 (Base‘𝑠) ∈ V
4 feq2 5989 . . . . . . . . 9 (𝑤 = (Base‘𝑠) → (𝑓:𝑤⟶(Base‘𝑡) ↔ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)))
5 raleq 3130 . . . . . . . . . 10 (𝑤 = (Base‘𝑠) → (∀𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
65raleqbi1dv 3138 . . . . . . . . 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 746 . . . . . . . 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 3456 . . . . . . 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 6153 . . . . . . . . . 10 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
10 isghm.w . . . . . . . . . 10 𝑋 = (Base‘𝑆)
119, 10syl6eqr 2673 . . . . . . . . 9 (𝑠 = 𝑆 → (Base‘𝑠) = 𝑋)
1211feq2d 5993 . . . . . . . 8 (𝑠 = 𝑆 → (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ↔ 𝑓:𝑋⟶(Base‘𝑡)))
13 fveq2 6153 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
14 isghm.a . . . . . . . . . . . . . 14 + = (+g𝑆)
1513, 14syl6eqr 2673 . . . . . . . . . . . . 13 (𝑠 = 𝑆 → (+g𝑠) = + )
1615oveqd 6627 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (𝑢(+g𝑠)𝑣) = (𝑢 + 𝑣))
1716fveq2d 6157 . . . . . . . . . . 11 (𝑠 = 𝑆 → (𝑓‘(𝑢(+g𝑠)𝑣)) = (𝑓‘(𝑢 + 𝑣)))
1817eqeq1d 2623 . . . . . . . . . 10 (𝑠 = 𝑆 → ((𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
1911, 18raleqbidv 3144 . . . . . . . . 9 (𝑠 = 𝑆 → (∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
2011, 19raleqbidv 3144 . . . . . . . 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 746 . . . . . . 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 2738 . . . . 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 6153 . . . . . . . . 9 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
25 isghm.x . . . . . . . . 9 𝑌 = (Base‘𝑇)
2624, 25syl6eqr 2673 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = 𝑌)
2726feq3d 5994 . . . . . . 7 (𝑡 = 𝑇 → (𝑓:𝑋⟶(Base‘𝑡) ↔ 𝑓:𝑋𝑌))
28 fveq2 6153 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
29 isghm.b . . . . . . . . . . 11 = (+g𝑇)
3028, 29syl6eqr 2673 . . . . . . . . . 10 (𝑡 = 𝑇 → (+g𝑡) = )
3130oveqd 6627 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) = ((𝑓𝑢) (𝑓𝑣)))
3231eqeq2d 2631 . . . . . . . 8 (𝑡 = 𝑇 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
33322ralbidv 2984 . . . . . . 7 (𝑡 = 𝑇 → (∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
3427, 33anbi12d 746 . . . . . 6 (𝑡 = 𝑇 → ((𝑓:𝑋⟶(Base‘𝑡) ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))))
3534abbidv 2738 . . . . 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 6163 . . . . . . . 8 (Base‘𝑆) ∈ V
3710, 36eqeltri 2694 . . . . . . 7 𝑋 ∈ V
38 fvex 6163 . . . . . . . 8 (Base‘𝑇) ∈ V
3925, 38eqeltri 2694 . . . . . . 7 𝑌 ∈ V
40 mapex 7815 . . . . . . 7 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {𝑓𝑓:𝑋𝑌} ∈ V)
4137, 39, 40mp2an 707 . . . . . 6 {𝑓𝑓:𝑋𝑌} ∈ V
42 simpl 473 . . . . . . 7 ((𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))) → 𝑓:𝑋𝑌)
4342ss2abi 3658 . . . . . 6 {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ⊆ {𝑓𝑓:𝑋𝑌}
4441, 43ssexi 4768 . . . . 5 {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ∈ V
4523, 35, 1, 44ovmpt2 6756 . . . 4 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝑆 GrpHom 𝑇) = {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))})
4645eleq2d 2684 . . 3 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) → (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))}))
47 fex 6450 . . . . . 6 ((𝐹:𝑋𝑌𝑋 ∈ V) → 𝐹 ∈ V)
4837, 47mpan2 706 . . . . 5 (𝐹:𝑋𝑌𝐹 ∈ V)
4948adantr 481 . . . 4 ((𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))) → 𝐹 ∈ V)
50 feq1 5988 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝑋𝑌𝐹:𝑋𝑌))
51 fveq1 6152 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝑣)))
52 fveq1 6152 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑢) = (𝐹𝑢))
53 fveq1 6152 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑣) = (𝐹𝑣))
5452, 53oveq12d 6628 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑢) (𝑓𝑣)) = ((𝐹𝑢) (𝐹𝑣)))
5551, 54eqeq12d 2636 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)) ↔ (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
56552ralbidv 2984 . . . . 5 (𝑓 = 𝐹 → (∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
5750, 56anbi12d 746 . . . 4 (𝑓 = 𝐹 → ((𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
5849, 57elab3 3345 . . 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 673 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 384   = wceq 1480  wcel 1987  {cab 2607  wral 2907  Vcvv 3189  [wsbc 3421  wf 5848  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  Grpcgrp 17354   GrpHom cghm 17589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-ghm 17590
This theorem is referenced by:  isghm3  17593  ghmgrp1  17594  ghmgrp2  17595  ghmf  17596  ghmlin  17597  isghmd  17601  idghm  17607  ghmf1o  17622  islmhm2  18970  expghm  19776  mulgghm2  19777  pi1xfr  22778  pi1coghm  22784  rhmopp  29628  isrnghm  41206
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