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Theorem symgvalstruct 19301
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.) (Proof shortened by AV, 6-Nov-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 14301 . 2 (𝐴𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
2 hasheq0 14319 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
3 0symgefmndeq 19298 . . . . . . . . 9 (EndoFMnd‘∅) = (SymGrp‘∅)
43eqcomi 2733 . . . . . . . 8 (SymGrp‘∅) = (EndoFMnd‘∅)
5 symgvalstruct.g . . . . . . . . 9 𝐺 = (SymGrp‘𝐴)
6 fveq2 6881 . . . . . . . . 9 (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
75, 6eqtrid 2776 . . . . . . . 8 (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8 fveq2 6881 . . . . . . . 8 (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
94, 7, 83eqtr4a 2790 . . . . . . 7 (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
109adantl 481 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11 eqid 2724 . . . . . . . 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 18782 . . . . . . 7 (𝐴𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
1615adantr 480 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
17 0map0sn0 8874 . . . . . . . . . . 11 (∅ ↑m ∅) = {∅}
18 id 22 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
1918, 18oveq12d 7419 . . . . . . . . . . 11 (𝐴 = ∅ → (𝐴m 𝐴) = (∅ ↑m ∅))
20 symgvalstruct.b . . . . . . . . . . . 12 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
217fveq2d 6885 . . . . . . . . . . . . 13 (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22 eqid 2724 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
235, 22symgbas 19275 . . . . . . . . . . . . 13 (Base‘𝐺) = {𝑥𝑥:𝐴1-1-onto𝐴}
24 symgbas0 19293 . . . . . . . . . . . . 13 (Base‘(SymGrp‘∅)) = {∅}
2521, 23, 243eqtr3g 2787 . . . . . . . . . . . 12 (𝐴 = ∅ → {𝑥𝑥:𝐴1-1-onto𝐴} = {∅})
2620, 25eqtrid 2776 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐵 = {∅})
2717, 19, 263eqtr4a 2790 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴m 𝐴) = 𝐵)
2827adantl 481 . . . . . . . . 9 ((𝐴𝑉𝐴 = ∅) → (𝐴m 𝐴) = 𝐵)
2912, 28eqtrid 2776 . . . . . . . 8 ((𝐴𝑉𝐴 = ∅) → 𝑀 = 𝐵)
3029opeq2d 4872 . . . . . . 7 ((𝐴𝑉𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
3130tpeq1d 4741 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3210, 16, 313eqtrd 2768 . . . . 5 ((𝐴𝑉𝐴 = ∅) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3332ex 412 . . . 4 (𝐴𝑉 → (𝐴 = ∅ → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
342, 33sylbid 239 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 0 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
35 hash1snb 14375 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 1 ↔ ∃𝑥 𝐴 = {𝑥}))
36 vsnex 5419 . . . . . . . 8 {𝑥} ∈ V
37 eleq1 2813 . . . . . . . 8 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
3836, 37mpbiri 258 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 ∈ V)
3911, 12, 13, 14efmnd 18782 . . . . . . 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 19299 . . . . . . 7 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
4241, 5eqtr4di 2782 . . . . . 6 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
4342fveq2d 6885 . . . . . . . . 9 (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44 eqid 2724 . . . . . . . . . . 11 (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
4511, 44efmndbas 18783 . . . . . . . . . 10 (Base‘(EndoFMnd‘𝐴)) = (𝐴m 𝐴)
4645, 12eqtr4i 2755 . . . . . . . . 9 (Base‘(EndoFMnd‘𝐴)) = 𝑀
4723, 20eqtr4i 2755 . . . . . . . . 9 (Base‘𝐺) = 𝐵
4843, 46, 473eqtr3g 2787 . . . . . . . 8 (𝐴 = {𝑥} → 𝑀 = 𝐵)
4948opeq2d 4872 . . . . . . 7 (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
5049tpeq1d 4741 . . . . . 6 (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5140, 42, 503eqtr3d 2772 . . . . 5 (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5251exlimiv 1925 . . . 4 (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5335, 52syl6bi 253 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54 ssnpss 4095 . . . . . . 7 ((𝐴m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴m 𝐴))
5511, 5symgpssefmnd 19300 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5620, 23eqtr4i 2755 . . . . . . . . 9 𝐵 = (Base‘𝐺)
5745eqcomi 2733 . . . . . . . . 9 (𝐴m 𝐴) = (Base‘(EndoFMnd‘𝐴))
5856, 57psseq12i 4083 . . . . . . . 8 (𝐵 ⊊ (𝐴m 𝐴) ↔ (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5955, 58sylibr 233 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊊ (𝐴m 𝐴))
6054, 59nsyl3 138 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ¬ (𝐴m 𝐴) ⊆ 𝐵)
61 fvexd 6896 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
62 f1osetex 8848 . . . . . . . 8 {𝑥𝑥:𝐴1-1-onto𝐴} ∈ V
6320, 62eqeltri 2821 . . . . . . 7 𝐵 ∈ V
6463a1i 11 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
655, 20symgval 19273 . . . . . . 7 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
6665, 57ressval2 17174 . . . . . 6 ((¬ (𝐴m 𝐴) ⊆ 𝐵 ∧ (EndoFMnd‘𝐴) ∈ V ∧ 𝐵 ∈ V) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
6760, 61, 64, 66syl3anc 1368 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
68 ovex 7434 . . . . . . 7 (𝐴m 𝐴) ∈ V
6968inex2 5308 . . . . . 6 (𝐵 ∩ (𝐴m 𝐴)) ∈ V
70 setsval 17096 . . . . . 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))⟩}))
7161, 69, 70sylancl 585 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
7215adantr 480 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
7372reseq1d 5970 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
7473uneq1d 4154 . . . . . 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))⟩}))
75 eqidd 2725 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76 fvexd 6896 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
77 fvexd 6896 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
7812, 68eqeltri 2821 . . . . . . . . . . 11 𝑀 ∈ V
7978, 78mpoex 8059 . . . . . . . . . 10 (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔)) ∈ V
8013, 79eqeltri 2821 . . . . . . . . 9 + ∈ V
8180a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
8214fvexi 6895 . . . . . . . . 9 𝐽 ∈ V
8382a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
84 basendxnplusgndx 17223 . . . . . . . . . 10 (Base‘ndx) ≠ (+g‘ndx)
8584necomi 2987 . . . . . . . . 9 (+g‘ndx) ≠ (Base‘ndx)
8685a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ≠ (Base‘ndx))
87 tsetndxnbasendx 17297 . . . . . . . . 9 (TopSet‘ndx) ≠ (Base‘ndx)
8887a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ≠ (Base‘ndx))
8975, 76, 77, 81, 83, 86, 88tpres 7194 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
9089uneq1d 4154 . . . . . 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))⟩}))
91 uncom 4145 . . . . . . . 8 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
92 tpass 4748 . . . . . . . 8 {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
9391, 92eqtr4i 2755 . . . . . . 7 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}
945, 56symgbasmap 19281 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ (𝐴m 𝐴))
9594a1i 11 . . . . . . . . . . 11 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝑥𝐵𝑥 ∈ (𝐴m 𝐴)))
9695ssrdv 3980 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴m 𝐴))
97 df-ss 3957 . . . . . . . . . 10 (𝐵 ⊆ (𝐴m 𝐴) ↔ (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
9896, 97sylib 217 . . . . . . . . 9 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
9998opeq2d 4872 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
10099tpeq1d 4741 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
10193, 100eqtrid 2776 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
10274, 90, 1013eqtrd 2768 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
10367, 71, 1023eqtrd 2768 . . . 4 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
104103ex 412 . . 3 (𝐴𝑉 → (1 < (♯‘𝐴) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
10534, 53, 1043jaod 1425 . 2 (𝐴𝑉 → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
1061, 105mpd 15 1 (𝐴𝑉𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3o 1083   = wceq 1533  wex 1773  wcel 2098  {cab 2701  wne 2932  Vcvv 3466  cdif 3937  cun 3938  cin 3939  wss 3940  wpss 3941  c0 4314  𝒫 cpw 4594  {csn 4620  {cpr 4622  {ctp 4624  cop 4626   class class class wbr 5138   × cxp 5664  cres 5668  ccom 5670  1-1-ontowf1o 6532  cfv 6533  (class class class)co 7401  cmpo 7403  m cmap 8815  0cc0 11105  1c1 11106   < clt 11244  chash 14286   sSet csts 17092  ndxcnx 17122  Basecbs 17140  +gcplusg 17193  TopSetcts 17199  tcpt 17380  EndoFMndcefmnd 18780  SymGrpcsymg 19271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-tp 4625  df-op 4627  df-uni 4900  df-int 4941  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-oadd 8465  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-fin 8938  df-dju 9891  df-card 9929  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-3 12272  df-4 12273  df-5 12274  df-6 12275  df-7 12276  df-8 12277  df-9 12278  df-n0 12469  df-xnn0 12541  df-z 12555  df-uz 12819  df-fz 13481  df-hash 14287  df-struct 17076  df-sets 17093  df-slot 17111  df-ndx 17123  df-base 17141  df-ress 17170  df-plusg 17206  df-tset 17212  df-efmnd 18781  df-symg 19272
This theorem is referenced by: (None)
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