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 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
symgvalstruct.m	⊢ 𝑀 = (𝐴 ↑m 𝐴)
symgvalstruct.p	⊢ + = (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔))
symgvalstruct.j	⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))

Assertion

Ref	Expression
symgvalstruct	⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(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

Step	Hyp	Ref	Expression
1		hashv01gt1 14301	. 2 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)))
2		hasheq0 14319	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 ↔ 𝐴 = ∅))
3		0symgefmndeq 19298	. . . . . . . . 9 ⊢ (EndoFMnd‘∅) = (SymGrp‘∅)
4	3	eqcomi 2733	. . . . . . . 8 ⊢ (SymGrp‘∅) = (EndoFMnd‘∅)
5		symgvalstruct.g	. . . . . . . . 9 ⊢ 𝐺 = (SymGrp‘𝐴)
6		fveq2 6881	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
7	5, 6	eqtrid 2776	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8		fveq2 6881	. . . . . . . 8 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
9	4, 7, 8	3eqtr4a 2790	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
10	9	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11		eqid 2724	. . . . . . . 8 ⊢ (EndoFMnd‘𝐴) = (EndoFMnd‘𝐴)
12		symgvalstruct.m	. . . . . . . 8 ⊢ 𝑀 = (𝐴 ↑m 𝐴)
13		symgvalstruct.p	. . . . . . . 8 ⊢ + = (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔))
14		symgvalstruct.j	. . . . . . . 8 ⊢ 𝐽 = (∏t‘(𝐴 × {𝒫 𝐴}))
15	11, 12, 13, 14	efmnd 18782	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
16	15	adantr 480	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
17		0map0sn0 8874	. . . . . . . . . . 11 ⊢ (∅ ↑m ∅) = {∅}
18		id 22	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
19	18, 18	oveq12d 7419	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐴) = (∅ ↑m ∅))
20		symgvalstruct.b	. . . . . . . . . . . 12 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
21	7	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22		eqid 2724	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
23	5, 22	symgbas 19275	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
24		symgbas0 19293	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘∅)) = {∅}
25	21, 23, 24	3eqtr3g 2787	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = {∅})
26	20, 25	eqtrid 2776	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → 𝐵 = {∅})
27	17, 19, 26	3eqtr4a 2790	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐴) = 𝐵)
28	27	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → (𝐴 ↑m 𝐴) = 𝐵)
29	12, 28	eqtrid 2776	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝑀 = 𝐵)
30	29	opeq2d 4872	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
31	30	tpeq1d 4741	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
32	10, 16, 31	3eqtrd 2768	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
33	32	ex 412	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝐴 = ∅ → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
34	2, 33	sylbid 239	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 0 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
35		hash1snb 14375	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 ↔ ∃𝑥 𝐴 = {𝑥}))
36		vsnex 5419	. . . . . . . 8 ⊢ {𝑥} ∈ V
37		eleq1 2813	. . . . . . . 8 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
38	36, 37	mpbiri 258	. . . . . . 7 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
39	11, 12, 13, 14	efmnd 18782	. . . . . . 7 ⊢ (𝐴 ∈ V → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
40	38, 39	syl 17	. . . . . 6 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
41		snsymgefmndeq 19299	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
42	41, 5	eqtr4di 2782	. . . . . 6 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
43	42	fveq2d 6885	. . . . . . . . 9 ⊢ (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44		eqid 2724	. . . . . . . . . . 11 ⊢ (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
45	11, 44	efmndbas 18783	. . . . . . . . . 10 ⊢ (Base‘(EndoFMnd‘𝐴)) = (𝐴 ↑m 𝐴)
46	45, 12	eqtr4i 2755	. . . . . . . . 9 ⊢ (Base‘(EndoFMnd‘𝐴)) = 𝑀
47	23, 20	eqtr4i 2755	. . . . . . . . 9 ⊢ (Base‘𝐺) = 𝐵
48	43, 46, 47	3eqtr3g 2787	. . . . . . . 8 ⊢ (𝐴 = {𝑥} → 𝑀 = 𝐵)
49	48	opeq2d 4872	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
50	49	tpeq1d 4741	. . . . . 6 ⊢ (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
51	40, 42, 50	3eqtr3d 2772	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
52	51	exlimiv 1925	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
53	35, 52	syl6bi 253	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54		ssnpss 4095	. . . . . . 7 ⊢ ((𝐴 ↑m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴 ↑m 𝐴))
55	11, 5	symgpssefmnd 19300	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
56	20, 23	eqtr4i 2755	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
57	45	eqcomi 2733	. . . . . . . . 9 ⊢ (𝐴 ↑m 𝐴) = (Base‘(EndoFMnd‘𝐴))
58	56, 57	psseq12i 4083	. . . . . . . 8 ⊢ (𝐵 ⊊ (𝐴 ↑m 𝐴) ↔ (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
59	55, 58	sylibr 233	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊊ (𝐴 ↑m 𝐴))
60	54, 59	nsyl3 138	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ¬ (𝐴 ↑m 𝐴) ⊆ 𝐵)
61		fvexd 6896	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
62		f1osetex 8848	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ∈ V
63	20, 62	eqeltri 2821	. . . . . . 7 ⊢ 𝐵 ∈ V
64	63	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
65	5, 20	symgval 19273	. . . . . . 7 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
66	65, 57	ressval2 17174	. . . . . 6 ⊢ ((¬ (𝐴 ↑m 𝐴) ⊆ 𝐵 ∧ (EndoFMnd‘𝐴) ∈ V ∧ 𝐵 ∈ V) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩))
67	60, 61, 64, 66	syl3anc 1368	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩))
68		ovex 7434	. . . . . . 7 ⊢ (𝐴 ↑m 𝐴) ∈ V
69	68	inex2 5308	. . . . . 6 ⊢ (𝐵 ∩ (𝐴 ↑m 𝐴)) ∈ V
70		setsval 17096	. . . . . 6 ⊢ (((EndoFMnd‘𝐴) ∈ V ∧ (𝐵 ∩ (𝐴 ↑m 𝐴)) ∈ V) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}))
71	61, 69, 70	sylancl 585	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩) = (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}))
72	15	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
73	72	reseq1d 5970	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
74	73	uneq1d 4154	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = (({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}))
75		eqidd 2725	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76		fvexd 6896	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
77		fvexd 6896	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
78	12, 68	eqeltri 2821	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
79	78, 78	mpoex 8059	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔)) ∈ V
80	13, 79	eqeltri 2821	. . . . . . . . 9 ⊢ + ∈ V
81	80	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
82	14	fvexi 6895	. . . . . . . . 9 ⊢ 𝐽 ∈ V
83	82	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
84		basendxnplusgndx 17223	. . . . . . . . . 10 ⊢ (Base‘ndx) ≠ (+g‘ndx)
85	84	necomi 2987	. . . . . . . . 9 ⊢ (+g‘ndx) ≠ (Base‘ndx)
86	85	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ≠ (Base‘ndx))
87		tsetndxnbasendx 17297	. . . . . . . . 9 ⊢ (TopSet‘ndx) ≠ (Base‘ndx)
88	87	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ≠ (Base‘ndx))
89	75, 76, 77, 81, 83, 86, 88	tpres 7194	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
90	89	uneq1d 4154	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}))
91		uncom 4145	. . . . . . . 8 ⊢ ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
92		tpass 4748	. . . . . . . 8 ⊢ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = ({⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩} ∪ {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
93	91, 92	eqtr4i 2755	. . . . . . 7 ⊢ ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}
94	5, 56	symgbasmap 19281	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐴 ↑m 𝐴))
95	94	a1i 11	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (𝑥 ∈ 𝐵 → 𝑥 ∈ (𝐴 ↑m 𝐴)))
96	95	ssrdv 3980	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴 ↑m 𝐴))
97		df-ss 3957	. . . . . . . . . 10 ⊢ (𝐵 ⊆ (𝐴 ↑m 𝐴) ↔ (𝐵 ∩ (𝐴 ↑m 𝐴)) = 𝐵)
98	96, 97	sylib 217	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴 ↑m 𝐴)) = 𝐵)
99	98	opeq2d 4872	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
100	99	tpeq1d 4741	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
101	93, 100	eqtrid 2776	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
102	74, 90, 101	3eqtrd 2768	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) ∪ {⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩}) = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
103	67, 71, 102	3eqtrd 2768	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
104	103	ex 412	. . 3 ⊢ (𝐴 ∈ 𝑉 → (1 < (♯‘𝐴) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
105	34, 53, 104	3jaod 1425	. 2 ⊢ (𝐴 ∈ 𝑉 → (((♯‘𝐴) = 0 ∨ (♯‘𝐴) = 1 ∨ 1 < (♯‘𝐴)) → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
106	1, 105	mpd 15	1 ⊢ (𝐴 ∈ 𝑉 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})

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-onto→wf1o 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)