MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  symgvalstruct Structured version   Visualization version   GIF version

Theorem symgvalstruct 19002
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 14057 . 2 (𝐴𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
2 hasheq0 14076 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
3 0symgefmndeq 18999 . . . . . . . . 9 (EndoFMnd‘∅) = (SymGrp‘∅)
43eqcomi 2749 . . . . . . . 8 (SymGrp‘∅) = (EndoFMnd‘∅)
5 symgvalstruct.g . . . . . . . . 9 𝐺 = (SymGrp‘𝐴)
6 fveq2 6771 . . . . . . . . 9 (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
75, 6eqtrid 2792 . . . . . . . 8 (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8 fveq2 6771 . . . . . . . 8 (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
94, 7, 83eqtr4a 2806 . . . . . . 7 (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
109adantl 482 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11 eqid 2740 . . . . . . . 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 18507 . . . . . . 7 (𝐴𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
1615adantr 481 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
17 0map0sn0 8656 . . . . . . . . . . 11 (∅ ↑m ∅) = {∅}
18 id 22 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
1918, 18oveq12d 7289 . . . . . . . . . . 11 (𝐴 = ∅ → (𝐴m 𝐴) = (∅ ↑m ∅))
20 symgvalstruct.b . . . . . . . . . . . 12 𝐵 = {𝑥𝑥:𝐴1-1-onto𝐴}
217fveq2d 6775 . . . . . . . . . . . . 13 (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22 eqid 2740 . . . . . . . . . . . . . 14 (Base‘𝐺) = (Base‘𝐺)
235, 22symgbas 18976 . . . . . . . . . . . . 13 (Base‘𝐺) = {𝑥𝑥:𝐴1-1-onto𝐴}
24 symgbas0 18994 . . . . . . . . . . . . 13 (Base‘(SymGrp‘∅)) = {∅}
2521, 23, 243eqtr3g 2803 . . . . . . . . . . . 12 (𝐴 = ∅ → {𝑥𝑥:𝐴1-1-onto𝐴} = {∅})
2620, 25eqtrid 2792 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐵 = {∅})
2717, 19, 263eqtr4a 2806 . . . . . . . . . 10 (𝐴 = ∅ → (𝐴m 𝐴) = 𝐵)
2827adantl 482 . . . . . . . . 9 ((𝐴𝑉𝐴 = ∅) → (𝐴m 𝐴) = 𝐵)
2912, 28eqtrid 2792 . . . . . . . 8 ((𝐴𝑉𝐴 = ∅) → 𝑀 = 𝐵)
3029opeq2d 4817 . . . . . . 7 ((𝐴𝑉𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
3130tpeq1d 4687 . . . . . 6 ((𝐴𝑉𝐴 = ∅) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3210, 16, 313eqtrd 2784 . . . . 5 ((𝐴𝑉𝐴 = ∅) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
3332ex 413 . . . 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 14132 . . . 4 (𝐴𝑉 → ((♯‘𝐴) = 1 ↔ ∃𝑥 𝐴 = {𝑥}))
36 snex 5358 . . . . . . . 8 {𝑥} ∈ V
37 eleq1 2828 . . . . . . . 8 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
3836, 37mpbiri 257 . . . . . . 7 (𝐴 = {𝑥} → 𝐴 ∈ V)
3911, 12, 13, 14efmnd 18507 . . . . . . 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 19000 . . . . . . 7 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
4241, 5eqtr4di 2798 . . . . . 6 (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
4342fveq2d 6775 . . . . . . . . 9 (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44 eqid 2740 . . . . . . . . . . 11 (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
4511, 44efmndbas 18508 . . . . . . . . . 10 (Base‘(EndoFMnd‘𝐴)) = (𝐴m 𝐴)
4645, 12eqtr4i 2771 . . . . . . . . 9 (Base‘(EndoFMnd‘𝐴)) = 𝑀
4723, 20eqtr4i 2771 . . . . . . . . 9 (Base‘𝐺) = 𝐵
4843, 46, 473eqtr3g 2803 . . . . . . . 8 (𝐴 = {𝑥} → 𝑀 = 𝐵)
4948opeq2d 4817 . . . . . . 7 (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
5049tpeq1d 4687 . . . . . 6 (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5140, 42, 503eqtr3d 2788 . . . . 5 (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5251exlimiv 1937 . . . 4 (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
5335, 52syl6bi 252 . . 3 (𝐴𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54 ssnpss 4043 . . . . . . 7 ((𝐴m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴m 𝐴))
5511, 5symgpssefmnd 19001 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
5620, 23eqtr4i 2771 . . . . . . . . 9 𝐵 = (Base‘𝐺)
5745eqcomi 2749 . . . . . . . . 9 (𝐴m 𝐴) = (Base‘(EndoFMnd‘𝐴))
5856, 57psseq12i 4031 . . . . . . . 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 6786 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
62 f1osetex 8630 . . . . . . . 8 {𝑥𝑥:𝐴1-1-onto𝐴} ∈ V
6320, 62eqeltri 2837 . . . . . . 7 𝐵 ∈ V
6463a1i 11 . . . . . 6 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
655, 20symgval 18974 . . . . . . 7 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
6665, 57ressval2 16944 . . . . . 6 ((¬ (𝐴m 𝐴) ⊆ 𝐵 ∧ (EndoFMnd‘𝐴) ∈ V ∧ 𝐵 ∈ V) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
6760, 61, 64, 66syl3anc 1370 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩))
68 ovex 7304 . . . . . . 7 (𝐴m 𝐴) ∈ V
6968inex2 5246 . . . . . 6 (𝐵 ∩ (𝐴m 𝐴)) ∈ V
70 setsval 16866 . . . . . 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 586 . . . . 5 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}))
7215adantr 481 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
7372reseq1d 5889 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
7473uneq1d 4101 . . . . . 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 2741 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76 fvexd 6786 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
77 fvexd 6786 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
7812, 68eqeltri 2837 . . . . . . . . . . 11 𝑀 ∈ V
7978, 78mpoex 7913 . . . . . . . . . 10 (𝑓𝑀, 𝑔𝑀 ↦ (𝑓𝑔)) ∈ V
8013, 79eqeltri 2837 . . . . . . . . 9 + ∈ V
8180a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
8214fvexi 6785 . . . . . . . . 9 𝐽 ∈ V
8382a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
84 basendxnplusgndx 16990 . . . . . . . . . 10 (Base‘ndx) ≠ (+g‘ndx)
8584necomi 3000 . . . . . . . . 9 (+g‘ndx) ≠ (Base‘ndx)
8685a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ≠ (Base‘ndx))
87 tsetndxnbasendx 17064 . . . . . . . . 9 (TopSet‘ndx) ≠ (Base‘ndx)
8887a1i 11 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ≠ (Base‘ndx))
8975, 76, 77, 81, 83, 86, 88tpres 7073 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
9089uneq1d 4101 . . . . . 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 4092 . . . . . . . 8 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
92 tpass 4694 . . . . . . . 8 {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
9391, 92eqtr4i 2771 . . . . . . 7 ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩}) = {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}
945, 56symgbasmap 18982 . . . . . . . . . . . 12 (𝑥𝐵𝑥 ∈ (𝐴m 𝐴))
9594a1i 11 . . . . . . . . . . 11 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝑥𝐵𝑥 ∈ (𝐴m 𝐴)))
9695ssrdv 3932 . . . . . . . . . 10 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴m 𝐴))
97 df-ss 3909 . . . . . . . . . 10 (𝐵 ⊆ (𝐴m 𝐴) ↔ (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
9896, 97sylib 217 . . . . . . . . 9 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴m 𝐴)) = 𝐵)
9998opeq2d 4817 . . . . . . . 8 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
10099tpeq1d 4687 . . . . . . 7 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), (𝐵 ∩ (𝐴m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
10193, 100eqtrid 2792 . . . . . 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 2784 . . . . 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 2784 . . . 4 ((𝐴𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
104103ex 413 . . 3 (𝐴𝑉 → (1 < (♯‘𝐴) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
10534, 53, 1043jaod 1427 . 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 396  w3o 1085   = wceq 1542  wex 1786  wcel 2110  {cab 2717  wne 2945  Vcvv 3431  cdif 3889  cun 3890  cin 3891  wss 3892  wpss 3893  c0 4262  𝒫 cpw 4539  {csn 4567  {cpr 4569  {ctp 4571  cop 4573   class class class wbr 5079   × cxp 5588  cres 5592  ccom 5594  1-1-ontowf1o 6431  cfv 6432  (class class class)co 7271  cmpo 7273  m cmap 8598  0cc0 10872  1c1 10873   < clt 11010  chash 14042   sSet csts 16862  ndxcnx 16892  Basecbs 16910  +gcplusg 16960  TopSetcts 16966  tcpt 17147  EndoFMndcefmnd 18505  SymGrpcsymg 18972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-oadd 8292  df-er 8481  df-map 8600  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-dju 9660  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12582  df-fz 13239  df-hash 14043  df-struct 16846  df-sets 16863  df-slot 16881  df-ndx 16893  df-base 16911  df-ress 16940  df-plusg 16973  df-tset 16979  df-efmnd 18506  df-symg 18973
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator