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Theorem sn-isghm 42683
Description: Longer proof of isghm 19233, unsuccessfully attempting to simplify isghm 19233 using elovmpo 7678 according to an editorial note (now removed). (Contributed by SN, 7-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
sn-isghm.w 𝑋 = (Base‘𝑆)
sn-isghm.x 𝑌 = (Base‘𝑇)
sn-isghm.a + = (+g𝑆)
sn-isghm.b = (+g𝑇)
Assertion
Ref Expression
sn-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 sn-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)))})
2 fvex 6919 . . . . . 6 (Base‘𝑠) ∈ V
3 feq2 6717 . . . . . . 7 (𝑤 = (Base‘𝑠) → (𝑓:𝑤⟶(Base‘𝑡) ↔ 𝑓:(Base‘𝑠)⟶(Base‘𝑡)))
4 raleq 3323 . . . . . . . 8 (𝑤 = (Base‘𝑠) → (∀𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
54raleqbi1dv 3338 . . . . . . 7 (𝑤 = (Base‘𝑠) → (∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣))))
63, 5anbi12d 632 . . . . . 6 (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)))))
72, 6sbcie 3830 . . . . 5 ([(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))))
87abbii 2809 . . . 4 {𝑓[(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))} = {𝑓 ∣ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))}
9 fvex 6919 . . . . . 6 (Base‘𝑡) ∈ V
10 fsetex 8896 . . . . . 6 ((Base‘𝑡) ∈ V → {𝑓𝑓:(Base‘𝑠)⟶(Base‘𝑡)} ∈ V)
119, 10ax-mp 5 . . . . 5 {𝑓𝑓:(Base‘𝑠)⟶(Base‘𝑡)} ∈ V
12 abanssl 4311 . . . . 5 {𝑓 ∣ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))} ⊆ {𝑓𝑓:(Base‘𝑠)⟶(Base‘𝑡)}
1311, 12ssexi 5322 . . . 4 {𝑓 ∣ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))} ∈ V
148, 13eqeltri 2837 . . 3 {𝑓[(Base‘𝑠) / 𝑤](𝑓:𝑤⟶(Base‘𝑡) ∧ ∀𝑢𝑤𝑣𝑤 (𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))} ∈ V
15 fveq2 6906 . . . . . . . . 9 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
16 sn-isghm.w . . . . . . . . 9 𝑋 = (Base‘𝑆)
1715, 16eqtr4di 2795 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = 𝑋)
1817adantr 480 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑠) = 𝑋)
19 fveq2 6906 . . . . . . . . 9 (𝑡 = 𝑇 → (Base‘𝑡) = (Base‘𝑇))
20 sn-isghm.x . . . . . . . . 9 𝑌 = (Base‘𝑇)
2119, 20eqtr4di 2795 . . . . . . . 8 (𝑡 = 𝑇 → (Base‘𝑡) = 𝑌)
2221adantl 481 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (Base‘𝑡) = 𝑌)
2318, 22feq23d 6731 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ↔ 𝑓:𝑋𝑌))
24 fveq2 6906 . . . . . . . . . . . 12 (𝑠 = 𝑆 → (+g𝑠) = (+g𝑆))
25 sn-isghm.a . . . . . . . . . . . 12 + = (+g𝑆)
2624, 25eqtr4di 2795 . . . . . . . . . . 11 (𝑠 = 𝑆 → (+g𝑠) = + )
2726oveqd 7448 . . . . . . . . . 10 (𝑠 = 𝑆 → (𝑢(+g𝑠)𝑣) = (𝑢 + 𝑣))
2827fveq2d 6910 . . . . . . . . 9 (𝑠 = 𝑆 → (𝑓‘(𝑢(+g𝑠)𝑣)) = (𝑓‘(𝑢 + 𝑣)))
29 fveq2 6906 . . . . . . . . . . 11 (𝑡 = 𝑇 → (+g𝑡) = (+g𝑇))
30 sn-isghm.b . . . . . . . . . . 11 = (+g𝑇)
3129, 30eqtr4di 2795 . . . . . . . . . 10 (𝑡 = 𝑇 → (+g𝑡) = )
3231oveqd 7448 . . . . . . . . 9 (𝑡 = 𝑇 → ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) = ((𝑓𝑢) (𝑓𝑣)))
3328, 32eqeqan12d 2751 . . . . . . . 8 ((𝑠 = 𝑆𝑡 = 𝑇) → ((𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
3418, 33raleqbidv 3346 . . . . . . 7 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
3518, 34raleqbidv 3346 . . . . . 6 ((𝑠 = 𝑆𝑡 = 𝑇) → (∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))))
3623, 35anbi12d 632 . . . . 5 ((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)))))
3736abbidv 2808 . . . 4 ((𝑠 = 𝑆𝑡 = 𝑇) → {𝑓 ∣ (𝑓:(Base‘𝑠)⟶(Base‘𝑡) ∧ ∀𝑢 ∈ (Base‘𝑠)∀𝑣 ∈ (Base‘𝑠)(𝑓‘(𝑢(+g𝑠)𝑣)) = ((𝑓𝑢)(+g𝑡)(𝑓𝑣)))} = {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))})
388, 37eqtrid 2789 . . 3 ((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)))})
391, 14, 38elovmpo 7678 . 2 (𝐹 ∈ (𝑆 GrpHom 𝑇) ↔ (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))}))
4016fvexi 6920 . . . . . 6 𝑋 ∈ V
4120fvexi 6920 . . . . . 6 𝑌 ∈ V
42 fex2 7958 . . . . . 6 ((𝐹:𝑋𝑌𝑋 ∈ V ∧ 𝑌 ∈ V) → 𝐹 ∈ V)
4340, 41, 42mp3an23 1455 . . . . 5 (𝐹:𝑋𝑌𝐹 ∈ V)
4443adantr 480 . . . 4 ((𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))) → 𝐹 ∈ V)
45 feq1 6716 . . . . 5 (𝑓 = 𝐹 → (𝑓:𝑋𝑌𝐹:𝑋𝑌))
46 fveq1 6905 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑢 + 𝑣)) = (𝐹‘(𝑢 + 𝑣)))
47 fveq1 6905 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑢) = (𝐹𝑢))
48 fveq1 6905 . . . . . . . 8 (𝑓 = 𝐹 → (𝑓𝑣) = (𝐹𝑣))
4947, 48oveq12d 7449 . . . . . . 7 (𝑓 = 𝐹 → ((𝑓𝑢) (𝑓𝑣)) = ((𝐹𝑢) (𝐹𝑣)))
5046, 49eqeq12d 2753 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)) ↔ (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
51502ralbidv 3221 . . . . 5 (𝑓 = 𝐹 → (∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)) ↔ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
5245, 51anbi12d 632 . . . 4 (𝑓 = 𝐹 → ((𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣))) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
5344, 52elab3 3686 . . 3 (𝐹 ∈ {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))} ↔ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣))))
54533anbi3i 1160 . 2 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ 𝐹 ∈ {𝑓 ∣ (𝑓:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝑓‘(𝑢 + 𝑣)) = ((𝑓𝑢) (𝑓𝑣)))}) ↔ (𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
55 df-3an 1089 . 2 ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp ∧ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))) ↔ ((𝑆 ∈ Grp ∧ 𝑇 ∈ Grp) ∧ (𝐹:𝑋𝑌 ∧ ∀𝑢𝑋𝑣𝑋 (𝐹‘(𝑢 + 𝑣)) = ((𝐹𝑢) (𝐹𝑣)))))
5639, 54, 553bitri 297 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  w3a 1087   = 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: (None)
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