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Theorem symgvalstruct 19433
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 14390 . 2 (𝐴𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
2 hasheq0 14408 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
3 0symgefmndeq 19430 . . . . . . . . 9 (EndoFMnd‘∅) = (SymGrp‘∅)
43eqcomi 2743 . . . . . . . 8 (SymGrp‘∅) = (EndoFMnd‘∅)
5 symgvalstruct.g . . . . . . . . 9 𝐺 = (SymGrp‘𝐴)
6 fveq2 6919 . . . . . . . . 9 (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
75, 6eqtrid 2786 . . . . . . . 8 (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8 fveq2 6919 . . . . . . . 8 (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
94, 7, 83eqtr4a 2800 . . . . . . 7 (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
109adantl 481 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11 eqid 2734 . . . . . . . 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 18900 . . . . . . 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 8939 . . . . . . . . . . 11 (∅ ↑m ∅) = {∅}
18 id 22 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
1918, 18oveq12d 7463 . . . . . . . . . . 11 (𝐴 = ∅ → (𝐴m 𝐴) = (∅ ↑m ∅))
20 symgvalstruct.b . . . . . . . . . . . 12 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
217fveq2d 6923 . . . . . . . . . . . . 13 (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22 eqid 2734 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
235, 22symgbas 19408 . . . . . . . . . . . . 13 (Base‘𝐺) = {𝑥𝑥:𝐴1-1-onto𝐴}
24 symgbas0 19425 . . . . . . . . . . . . 13 (Base‘(SymGrp‘∅)) = {∅}
2521, 23, 243eqtr3g 2797 . . . . . . . . . . . 12 (𝐴 = ∅ → {𝑥𝑥:𝐴1-1-onto𝐴} = {∅})
2620, 25eqtrid 2786 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐵 = {∅})
2717, 19, 263eqtr4a 2800 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴m 𝐴) = 𝐵)
2827adantl 481 . . . . . . . . 9 ((𝐴𝑉𝐴 = ∅) → (𝐴m 𝐴) = 𝐵)
2912, 28eqtrid 2786 . . . . . . . 8 ((𝐴𝑉𝐴 = ∅) → 𝑀 = 𝐵)
3029opeq2d 4904 . . . . . . 7 ((𝐴𝑉𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
3130tpeq1d 4770 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3210, 16, 313eqtrd 2778 . . . . 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 240 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 0 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
35 hash1snb 14464 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 1 ↔ ∃𝑥 𝐴 = {𝑥}))
36 vsnex 5452 . . . . . . . 8 {𝑥} ∈ V
37 eleq1 2826 . . . . . . . 8 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
3836, 37mpbiri 258 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 ∈ V)
3911, 12, 13, 14efmnd 18900 . . . . . . 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 19431 . . . . . . 7 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
4241, 5eqtr4di 2792 . . . . . 6 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
4342fveq2d 6923 . . . . . . . . 9 (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44 eqid 2734 . . . . . . . . . . 11 (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
4511, 44efmndbas 18901 . . . . . . . . . 10 (Base‘(EndoFMnd‘𝐴)) = (𝐴m 𝐴)
4645, 12eqtr4i 2765 . . . . . . . . 9 (Base‘(EndoFMnd‘𝐴)) = 𝑀
4723, 20eqtr4i 2765 . . . . . . . . 9 (Base‘𝐺) = 𝐵
4843, 46, 473eqtr3g 2797 . . . . . . . 8 (𝐴 = {𝑥} → 𝑀 = 𝐵)
4948opeq2d 4904 . . . . . . 7 (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
5049tpeq1d 4770 . . . . . 6 (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5140, 42, 503eqtr3d 2782 . . . . 5 (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5251exlimiv 1929 . . . 4 (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5335, 52biimtrdi 253 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54 ssnpss 4123 . . . . . . 7 ((𝐴m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴m 𝐴))
5511, 5symgpssefmnd 19432 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5620, 23eqtr4i 2765 . . . . . . . . 9 𝐵 = (Base‘𝐺)
5745eqcomi 2743 . . . . . . . . 9 (𝐴m 𝐴) = (Base‘(EndoFMnd‘𝐴))
5856, 57psseq12i 4111 . . . . . . . 8 (𝐵 ⊊ (𝐴m 𝐴) ↔ (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5955, 58sylibr 234 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊊ (𝐴m 𝐴))
6054, 59nsyl3 138 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ¬ (𝐴m 𝐴) ⊆ 𝐵)
61 fvexd 6934 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
62 f1osetex 8913 . . . . . . . 8 {𝑥𝑥:𝐴1-1-onto𝐴} ∈ V
6320, 62eqeltri 2834 . . . . . . 7 𝐵 ∈ V
6463a1i 11 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
655, 20symgval 19407 . . . . . . 7 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
6665, 57ressval2 17287 . . . . . 6 ((¬ (𝐴m 𝐴) ⊆ 𝐵 ∧ (EndoFMnd‘𝐴) ∈ V ∧ 𝐵 ∈ V) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
6760, 61, 64, 66syl3anc 1371 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
68 ovex 7478 . . . . . . 7 (𝐴m 𝐴) ∈ V
6968inex2 5339 . . . . . 6 (𝐵 ∩ (𝐴m 𝐴)) ∈ V
70 setsval 17209 . . . . . 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 6007 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
7473uneq1d 4184 . . . . . 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 2735 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76 fvexd 6934 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
77 fvexd 6934 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
7812, 68eqeltri 2834 . . . . . . . . . . 11 𝑀 ∈ V
7978, 78mpoex 8116 . . . . . . . . . 10 (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔)) ∈ V
8013, 79eqeltri 2834 . . . . . . . . 9 + ∈ V
8180a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
8214fvexi 6933 . . . . . . . . 9 𝐽 ∈ V
8382a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
84 basendxnplusgndx 17336 . . . . . . . . . 10 (Base‘ndx) ≠ (+g‘ndx)
8584necomi 2997 . . . . . . . . 9 (+g‘ndx) ≠ (Base‘ndx)
8685a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ≠ (Base‘ndx))
87 tsetndxnbasendx 17410 . . . . . . . . 9 (TopSet‘ndx) ≠ (Base‘ndx)
8887a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ≠ (Base‘ndx))
8975, 76, 77, 81, 83, 86, 88tpres 7236 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
9089uneq1d 4184 . . . . . 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 4175 . . . . . . . 8 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
92 tpass 4777 . . . . . . . 8 {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
9391, 92eqtr4i 2765 . . . . . . 7 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}
945, 56symgbasmap 19413 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ (𝐴m 𝐴))
9594a1i 11 . . . . . . . . . . 11 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝑥𝐵𝑥 ∈ (𝐴m 𝐴)))
9695ssrdv 4008 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴m 𝐴))
97 dfss2 3988 . . . . . . . . . 10 (𝐵 ⊆ (𝐴m 𝐴) ↔ (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
9896, 97sylib 218 . . . . . . . . 9 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
9998opeq2d 4904 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
10099tpeq1d 4770 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
10193, 100eqtrid 2786 . . . . . 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 2778 . . . . 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 2778 . . . 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 1429 . 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 1086   = wceq 1537  wex 1777  wcel 2103  {cab 2711  wne 2942  Vcvv 3482  cdif 3967  cun 3968  cin 3969  wss 3970  wpss 3971  c0 4347  𝒫 cpw 4622  {csn 4648  {cpr 4650  {ctp 4652  cop 4654   class class class wbr 5169   × cxp 5697  cres 5701  ccom 5703  1-1-ontowf1o 6571  cfv 6572  (class class class)co 7445  cmpo 7447  m cmap 8880  0cc0 11180  1c1 11181   < clt 11320  chash 14375   sSet csts 17205  ndxcnx 17235  Basecbs 17253  +gcplusg 17306  TopSetcts 17312  tcpt 17493  EndoFMndcefmnd 18898  SymGrpcsymg 19405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-rep 5306  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766  ax-cnex 11236  ax-resscn 11237  ax-1cn 11238  ax-icn 11239  ax-addcl 11240  ax-addrcl 11241  ax-mulcl 11242  ax-mulrcl 11243  ax-mulcom 11244  ax-addass 11245  ax-mulass 11246  ax-distr 11247  ax-i2m1 11248  ax-1ne0 11249  ax-1rid 11250  ax-rnegex 11251  ax-rrecex 11252  ax-cnre 11253  ax-pre-lttri 11254  ax-pre-lttrn 11255  ax-pre-ltadd 11256  ax-pre-mulgt0 11257
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3384  df-rab 3439  df-v 3484  df-sbc 3799  df-csb 3916  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-pss 3990  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-tp 4653  df-op 4655  df-uni 4932  df-int 4973  df-iun 5021  df-br 5170  df-opab 5232  df-mpt 5253  df-tr 5287  df-id 5597  df-eprel 5603  df-po 5611  df-so 5612  df-fr 5654  df-we 5656  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-pred 6331  df-ord 6397  df-on 6398  df-lim 6399  df-suc 6400  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-f1 6577  df-fo 6578  df-f1o 6579  df-fv 6580  df-riota 7401  df-ov 7448  df-oprab 7449  df-mpo 7450  df-om 7900  df-1st 8026  df-2nd 8027  df-frecs 8318  df-wrecs 8349  df-recs 8423  df-rdg 8462  df-1o 8518  df-oadd 8522  df-er 8759  df-map 8882  df-en 9000  df-dom 9001  df-sdom 9002  df-fin 9003  df-dju 9966  df-card 10004  df-pnf 11322  df-mnf 11323  df-xr 11324  df-ltxr 11325  df-le 11326  df-sub 11518  df-neg 11519  df-nn 12290  df-2 12352  df-3 12353  df-4 12354  df-5 12355  df-6 12356  df-7 12357  df-8 12358  df-9 12359  df-n0 12550  df-xnn0 12622  df-z 12636  df-uz 12900  df-fz 13564  df-hash 14376  df-struct 17189  df-sets 17206  df-slot 17224  df-ndx 17236  df-base 17254  df-ress 17283  df-plusg 17319  df-tset 17325  df-efmnd 18899  df-symg 19406
This theorem is referenced by: (None)
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