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Theorem symgvalstruct 18504
 Description: The value of the symmetric group function at 𝐴 represented as extensible structure with three slots. This corresponds to the former definition of SymGrp. (Contributed by Paul Chapman, 25-Feb-2008.) (Revised by Mario Carneiro, 12-Jan-2015.) (Revised by AV, 31-Mar-2024.)
Hypotheses
Ref Expression
symgvalstruct.g 𝐺 = (SymGrp‘𝐴)
symgvalstruct.b 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
symgvalstruct.m 𝑀 = (𝐴m 𝐴)
symgvalstruct.p + = (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔))
symgvalstruct.j 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
Assertion
Ref Expression
symgvalstruct (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Distinct variable groups:   𝐴,𝑓,𝑔   𝑥,𝐴   𝑥,𝐵   𝑥,𝐺   𝑥,𝐽   𝑓,𝑀,𝑔   𝑥,𝑉   𝑥, +
Allowed substitution hints:   𝐵(𝑓,𝑔)   + (𝑓,𝑔)   𝐺(𝑓,𝑔)   𝐽(𝑓,𝑔)   𝑀(𝑥)   𝑉(𝑓,𝑔)

Proof of Theorem symgvalstruct
StepHypRef Expression
1 hashv01gt1 13690 . 2 (𝐴𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
2 hasheq0 13709 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
3 0symgefmndeq 18501 . . . . . . . . 9 (EndoFMnd‘∅) = (SymGrp‘∅)
43eqcomi 2829 . . . . . . . 8 (SymGrp‘∅) = (EndoFMnd‘∅)
5 symgvalstruct.g . . . . . . . . 9 𝐺 = (SymGrp‘𝐴)
6 fveq2 6646 . . . . . . . . 9 (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
75, 6syl5eq 2867 . . . . . . . 8 (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8 fveq2 6646 . . . . . . . 8 (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
94, 7, 83eqtr4a 2881 . . . . . . 7 (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
109adantl 484 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11 eqid 2820 . . . . . . . 8 (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
12 symgvalstruct.m . . . . . . . 8 𝑀 = (𝐴m 𝐴)
13 symgvalstruct.p . . . . . . . 8 + = (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔))
14 symgvalstruct.j . . . . . . . 8 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
1511, 12, 13, 14efmnd 18014 . . . . . . 7 (𝐴𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
1615adantr 483 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
17 0map0sn0 8427 . . . . . . . . . . 11 (∅ ↑m ∅) = {∅}
18 id 22 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
1918, 18oveq12d 7151 . . . . . . . . . . 11 (𝐴 = ∅ → (𝐴m 𝐴) = (∅ ↑m ∅))
20 symgvalstruct.b . . . . . . . . . . . 12 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
217fveq2d 6650 . . . . . . . . . . . . 13 (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22 eqid 2820 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
235, 22symgbas 18478 . . . . . . . . . . . . 13 (Base‘𝐺) = {𝑥𝑥:𝐴1-1-onto𝐴}
24 symgbas0 18496 . . . . . . . . . . . . 13 (Base‘(SymGrp‘∅)) = {∅}
2521, 23, 243eqtr3g 2878 . . . . . . . . . . . 12 (𝐴 = ∅ → {𝑥𝑥:𝐴1-1-onto𝐴} = {∅})
2620, 25syl5eq 2867 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐵 = {∅})
2717, 19, 263eqtr4a 2881 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴m 𝐴) = 𝐵)
2827adantl 484 . . . . . . . . 9 ((𝐴𝑉𝐴 = ∅) → (𝐴m 𝐴) = 𝐵)
2912, 28syl5eq 2867 . . . . . . . 8 ((𝐴𝑉𝐴 = ∅) → 𝑀 = 𝐵)
3029opeq2d 4786 . . . . . . 7 ((𝐴𝑉𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
3130tpeq1d 4657 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3210, 16, 313eqtrd 2859 . . . . 5 ((𝐴𝑉𝐴 = ∅) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3332ex 415 . . . 4 (𝐴𝑉 → (𝐴 = ∅ → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
342, 33sylbid 242 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 0 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
35 hash1snb 13765 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 1 ↔ ∃𝑥 𝐴 = {𝑥}))
36 snex 5308 . . . . . . . 8 {𝑥} ∈ V
37 eleq1 2898 . . . . . . . 8 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
3836, 37mpbiri 260 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 ∈ V)
3911, 12, 13, 14efmnd 18014 . . . . . . 7 (𝐴 ∈ V → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
4038, 39syl 17 . . . . . 6 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
41 snsymgefmndeq 18502 . . . . . . 7 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
4241, 5syl6eqr 2873 . . . . . 6 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
4342fveq2d 6650 . . . . . . . . 9 (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44 eqid 2820 . . . . . . . . . . 11 (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
4511, 44efmndbas 18015 . . . . . . . . . 10 (Base‘(EndoFMnd‘𝐴)) = (𝐴m 𝐴)
4645, 12eqtr4i 2846 . . . . . . . . 9 (Base‘(EndoFMnd‘𝐴)) = 𝑀
4723, 20eqtr4i 2846 . . . . . . . . 9 (Base‘𝐺) = 𝐵
4843, 46, 473eqtr3g 2878 . . . . . . . 8 (𝐴 = {𝑥} → 𝑀 = 𝐵)
4948opeq2d 4786 . . . . . . 7 (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
5049tpeq1d 4657 . . . . . 6 (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5140, 42, 503eqtr3d 2863 . . . . 5 (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5251exlimiv 1931 . . . 4 (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5335, 52syl6bi 255 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54 ssnpss 4059 . . . . . . 7 ((𝐴m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴m 𝐴))
5511, 5symgpssefmnd 18503 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5620, 23eqtr4i 2846 . . . . . . . . 9 𝐵 = (Base‘𝐺)
5745eqcomi 2829 . . . . . . . . 9 (𝐴m 𝐴) = (Base‘(EndoFMnd‘𝐴))
5856, 57psseq12i 4047 . . . . . . . 8 (𝐵 ⊊ (𝐴m 𝐴) ↔ (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5955, 58sylibr 236 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊊ (𝐴m 𝐴))
6054, 59nsyl3 140 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ¬ (𝐴m 𝐴) ⊆ 𝐵)
61 fvexd 6661 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
625, 56symgbasex 18479 . . . . . . 7 (𝐴𝑉𝐵 ∈ V)
6362adantr 483 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
645, 20symgval 18476 . . . . . . 7 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
6564, 57ressval2 16532 . . . . . 6 ((¬ (𝐴m 𝐴) ⊆ 𝐵 ∧ (EndoFMnd‘𝐴) ∈ V ∧ 𝐵 ∈ V) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
6660, 61, 63, 65syl3anc 1367 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
67 ovex 7166 . . . . . . 7 (𝐴m 𝐴) ∈ V
6867inex2 5198 . . . . . 6 (𝐵 ∩ (𝐴m 𝐴)) ∈ V
69 setsval 16492 . . . . . 6 (((EndoFMnd‘𝐴) ∈ V ∧ (𝐵 ∩ (𝐴m 𝐴)) ∈ V) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
7061, 68, 69sylancl 588 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
7115adantr 483 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
7271reseq1d 5828 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
7372uneq1d 4117 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = (({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
74 eqidd 2821 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
75 fvexd 6661 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
76 fvexd 6661 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
7712, 67eqeltri 2907 . . . . . . . . . . 11 𝑀 ∈ V
7877, 77mpoex 7755 . . . . . . . . . 10 (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔)) ∈ V
7913, 78eqeltri 2907 . . . . . . . . 9 + ∈ V
8079a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
8114fvexi 6660 . . . . . . . . 9 𝐽 ∈ V
8281a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
83 basendxnplusgndx 16587 . . . . . . . . . 10 (Base‘ndx) ≠ (+g‘ndx)
8483necomi 3060 . . . . . . . . 9 (+g‘ndx) ≠ (Base‘ndx)
8584a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ≠ (Base‘ndx))
86 tsetndx 16638 . . . . . . . . . 10 (TopSet‘ndx) = 9
87 1re 10619 . . . . . . . . . . . 12 1 ∈ ℝ
88 1lt9 11822 . . . . . . . . . . . 12 1 < 9
8987, 88gtneii 10730 . . . . . . . . . . 11 9 ≠ 1
90 df-base 16468 . . . . . . . . . . . 12 Base = Slot 1
91 1nn 11627 . . . . . . . . . . . 12 1 ∈ ℕ
9290, 91ndxarg 16487 . . . . . . . . . . 11 (Base‘ndx) = 1
9389, 92neeqtrri 3079 . . . . . . . . . 10 9 ≠ (Base‘ndx)
9486, 93eqnetri 3076 . . . . . . . . 9 (TopSet‘ndx) ≠ (Base‘ndx)
9594a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ≠ (Base‘ndx))
9674, 75, 76, 80, 82, 85, 95tpres 6939 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
9796uneq1d 4117 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
98 uncom 4108 . . . . . . . 8 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
99 tpass 4664 . . . . . . . 8 {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
10098, 99eqtr4i 2846 . . . . . . 7 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}
1015, 56symgbasmap 18484 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ (𝐴m 𝐴))
102101a1i 11 . . . . . . . . . . 11 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝑥𝐵𝑥 ∈ (𝐴m 𝐴)))
103102ssrdv 3952 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴m 𝐴))
104 df-ss 3930 . . . . . . . . . 10 (𝐵 ⊆ (𝐴m 𝐴) ↔ (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
105103, 104sylib 220 . . . . . . . . 9 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
106105opeq2d 4786 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
107106tpeq1d 4657 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
108100, 107syl5eq 2867 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
10973, 97, 1083eqtrd 2859 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
11066, 70, 1093eqtrd 2859 . . . 4 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
111110ex 415 . . 3 (𝐴𝑉 → (1 < (♯‘𝐴) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
11234, 53, 1113jaod 1424 . 2 (𝐴𝑉 → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
1131, 112mpd 15 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   ∨ w3o 1082   = wceq 1537  ∃wex 1780   ∈ wcel 2114  {cab 2798   ≠ wne 3006  Vcvv 3473   ∖ cdif 3910   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913   ⊊ wpss 3914  ∅c0 4269  𝒫 cpw 4515  {csn 4543  {cpr 4545  {ctp 4547  ⟨cop 4549   class class class wbr 5042   × cxp 5529   ↾ cres 5533   ∘ ccom 5535  –1-1-onto→wf1o 6330  ‘cfv 6331  (class class class)co 7133   ∈ cmpo 7135   ↑m cmap 8384  0cc0 10515  1c1 10516   < clt 10653  9c9 11678  ♯chash 13675  ndxcnx 16459   sSet csts 16460  Basecbs 16462  +gcplusg 16544  TopSetcts 16550  ∏tcpt 16691  EndoFMndcefmnd 18012  SymGrpcsymg 18474 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-int 4853  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-1o 8080  df-oadd 8084  df-er 8267  df-map 8386  df-en 8488  df-dom 8489  df-sdom 8490  df-fin 8491  df-dju 9308  df-card 9346  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-nn 11617  df-2 11679  df-3 11680  df-4 11681  df-5 11682  df-6 11683  df-7 11684  df-8 11685  df-9 11686  df-n0 11877  df-xnn0 11947  df-z 11961  df-uz 12223  df-fz 12877  df-hash 13676  df-struct 16464  df-ndx 16465  df-slot 16466  df-base 16468  df-sets 16469  df-ress 16470  df-plusg 16557  df-tset 16563  df-efmnd 18013  df-symg 18475 This theorem is referenced by: (None)
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