Theorem symgvalstruct 19366

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 19363	. . . . . . . . 9 ⊢ (EndoFMnd‘∅) = (SymGrp‘∅)
4	3	eqcomi 2746	. . . . . . . 8 ⊢ (SymGrp‘∅) = (EndoFMnd‘∅)
5		symgvalstruct.g	. . . . . . . . 9 ⊢ 𝐺 = (SymGrp‘𝐴)
6		fveq2 6835	. . . . . . . . 9 ⊢ (𝐴 = ∅ → (SymGrp‘𝐴) = (SymGrp‘∅))
7	5, 6	eqtrid 2784	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐺 = (SymGrp‘∅))
8		fveq2 6835	. . . . . . . 8 ⊢ (𝐴 = ∅ → (EndoFMnd‘𝐴) = (EndoFMnd‘∅))
9	4, 7, 8	3eqtr4a 2798	. . . . . . 7 ⊢ (𝐴 = ∅ → 𝐺 = (EndoFMnd‘𝐴))
10	9	adantl 481	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝐺 = (EndoFMnd‘𝐴))
11		eqid 2737	. . . . . . . 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 18832	. . . . . . 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 8827	. . . . . . . . . . 11 ⊢ (∅ ↑m ∅) = {∅}
18		id 22	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
19	18, 18	oveq12d 7379	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐴) = (∅ ↑m ∅))
20		symgvalstruct.b	. . . . . . . . . . . 12 ⊢ 𝐵 = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
21	7	fveq2d 6839	. . . . . . . . . . . . 13 ⊢ (𝐴 = ∅ → (Base‘𝐺) = (Base‘(SymGrp‘∅)))
22		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (Base‘𝐺) = (Base‘𝐺)
23	5, 22	symgbas 19341	. . . . . . . . . . . . 13 ⊢ (Base‘𝐺) = {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴}
24		symgbas0 19358	. . . . . . . . . . . . 13 ⊢ (Base‘(SymGrp‘∅)) = {∅}
25	21, 23, 24	3eqtr3g 2795	. . . . . . . . . . . 12 ⊢ (𝐴 = ∅ → {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} = {∅})
26	20, 25	eqtrid 2784	. . . . . . . . . . 11 ⊢ (𝐴 = ∅ → 𝐵 = {∅})
27	17, 19, 26	3eqtr4a 2798	. . . . . . . . . 10 ⊢ (𝐴 = ∅ → (𝐴 ↑m 𝐴) = 𝐵)
28	27	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → (𝐴 ↑m 𝐴) = 𝐵)
29	12, 28	eqtrid 2784	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → 𝑀 = 𝐵)
30	29	opeq2d 4824	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 = ∅) → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
31	30	tpeq1d 4690	. . . . . 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 2776	. . . . 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 240	. . 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 5373	. . . . . . . 8 ⊢ {𝑥} ∈ V
37		eleq1 2825	. . . . . . . 8 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
38	36, 37	mpbiri 258	. . . . . . 7 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
39	11, 12, 13, 14	efmnd 18832	. . . . . . 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 19364	. . . . . . 7 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = (SymGrp‘𝐴))
42	41, 5	eqtr4di 2790	. . . . . 6 ⊢ (𝐴 = {𝑥} → (EndoFMnd‘𝐴) = 𝐺)
43	42	fveq2d 6839	. . . . . . . . 9 ⊢ (𝐴 = {𝑥} → (Base‘(EndoFMnd‘𝐴)) = (Base‘𝐺))
44		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘(EndoFMnd‘𝐴)) = (Base‘(EndoFMnd‘𝐴))
45	11, 44	efmndbas 18833	. . . . . . . . . 10 ⊢ (Base‘(EndoFMnd‘𝐴)) = (𝐴 ↑m 𝐴)
46	45, 12	eqtr4i 2763	. . . . . . . . 9 ⊢ (Base‘(EndoFMnd‘𝐴)) = 𝑀
47	23, 20	eqtr4i 2763	. . . . . . . . 9 ⊢ (Base‘𝐺) = 𝐵
48	43, 46, 47	3eqtr3g 2795	. . . . . . . 8 ⊢ (𝐴 = {𝑥} → 𝑀 = 𝐵)
49	48	opeq2d 4824	. . . . . . 7 ⊢ (𝐴 = {𝑥} → ⟨(Base‘ndx), 𝑀⟩ = ⟨(Base‘ndx), 𝐵⟩)
50	49	tpeq1d 4690	. . . . . 6 ⊢ (𝐴 = {𝑥} → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
51	40, 42, 50	3eqtr3d 2780	. . . . 5 ⊢ (𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
52	51	exlimiv 1932	. . . 4 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
53	35, 52	biimtrdi 253	. . 3 ⊢ (𝐴 ∈ 𝑉 → ((♯‘𝐴) = 1 → 𝐺 = {⟨(Base‘ndx), 𝐵⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩}))
54		ssnpss 4047	. . . . . . 7 ⊢ ((𝐴 ↑m 𝐴) ⊆ 𝐵 → ¬ 𝐵 ⊊ (𝐴 ↑m 𝐴))
55	11, 5	symgpssefmnd 19365	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
56	20, 23	eqtr4i 2763	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
57	45	eqcomi 2746	. . . . . . . . 9 ⊢ (𝐴 ↑m 𝐴) = (Base‘(EndoFMnd‘𝐴))
58	56, 57	psseq12i 4035	. . . . . . . 8 ⊢ (𝐵 ⊊ (𝐴 ↑m 𝐴) ↔ (Base‘𝐺) ⊊ (Base‘(EndoFMnd‘𝐴)))
59	55, 58	sylibr 234	. . . . . . 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 6850	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (EndoFMnd‘𝐴) ∈ V)
62		f1osetex 8800	. . . . . . . 8 ⊢ {𝑥 ∣ 𝑥:𝐴–1-1-onto→𝐴} ∈ V
63	20, 62	eqeltri 2833	. . . . . . 7 ⊢ 𝐵 ∈ V
64	63	a1i 11	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ∈ V)
65	5, 20	symgval 19340	. . . . . . 7 ⊢ 𝐺 = ((EndoFMnd‘𝐴) ↾s 𝐵)
66	65, 57	ressval2 17199	. . . . . 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 1374	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐺 = ((EndoFMnd‘𝐴) sSet ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩))
68		ovex 7394	. . . . . . 7 ⊢ (𝐴 ↑m 𝐴) ∈ V
69	68	inex2 5256	. . . . . 6 ⊢ (𝐵 ∩ (𝐴 ↑m 𝐴)) ∈ V
70		setsval 17131	. . . . . 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 587	. . . . 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 5938	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ((EndoFMnd‘𝐴) ↾ (V ∖ {(Base‘ndx)})) = ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})))
74	73	uneq1d 4108	. . . . . 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 2738	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} = {⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
76		fvexd 6850	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (+g‘ndx) ∈ V)
77		fvexd 6850	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (TopSet‘ndx) ∈ V)
78	12, 68	eqeltri 2833	. . . . . . . . . . 11 ⊢ 𝑀 ∈ V
79	78, 78	mpoex 8026	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝑀, 𝑔 ∈ 𝑀 ↦ (𝑓 ∘ 𝑔)) ∈ V
80	13, 79	eqeltri 2833	. . . . . . . . 9 ⊢ + ∈ V
81	80	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → + ∈ V)
82	14	fvexi 6849	. . . . . . . . 9 ⊢ 𝐽 ∈ V
83	82	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐽 ∈ V)
84		basendxnplusgndx 17244	. . . . . . . . . 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 17313	. . . . . . . . 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 7150	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ({⟨(Base‘ndx), 𝑀⟩, ⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩} ↾ (V ∖ {(Base‘ndx)})) = {⟨(+g‘ndx), + ⟩, ⟨(TopSet‘ndx), 𝐽⟩})
90	89	uneq1d 4108	. . . . . 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 4099	. . . . . . . 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 4697	. . . . . . . 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 2763	. . . . . . 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 19346	. . . . . . . . . . . 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 3928	. . . . . . . . . 10 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → 𝐵 ⊆ (𝐴 ↑m 𝐴))
97		dfss2 3908	. . . . . . . . . 10 ⊢ (𝐵 ⊆ (𝐴 ↑m 𝐴) ↔ (𝐵 ∩ (𝐴 ↑m 𝐴)) = 𝐵)
98	96, 97	sylib 218	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → (𝐵 ∩ (𝐴 ↑m 𝐴)) = 𝐵)
99	98	opeq2d 4824	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 1 < (♯‘𝐴)) → ⟨(Base‘ndx), (𝐵 ∩ (𝐴 ↑m 𝐴))⟩ = ⟨(Base‘ndx), 𝐵⟩)
100	99	tpeq1d 4690	. . . . . . 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 2784	. . . . . 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 2776	. . . . 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 2776	. . . 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 1432	. 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 1086   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715   ≠ wne 2933  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888   ∩ cin 3889   ⊆ wss 3890   ⊊ wpss 3891  ∅c0 4274  𝒫 cpw 4542  {csn 4568  {cpr 4570  {ctp 4572  ⟨cop 4574   class class class wbr 5086   × cxp 5623   ↾ cres 5627   ∘ ccom 5629  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363   ↑_m cmap 8767  0cc0 11032  1c1 11033   < clt 11173  ♯chash 14286   sSet csts 17127  ndxcnx 17157  Basecbs 17173  +_gcplusg 17214  TopSetcts 17220  ∏_tcpt 17395  EndoFMndcefmnd 18830  SymGrpcsymg 19338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683  ax-cnex 11088  ax-resscn 11089  ax-1cn 11090  ax-icn 11091  ax-addcl 11092  ax-addrcl 11093  ax-mulcl 11094  ax-mulrcl 11095  ax-mulcom 11096  ax-addass 11097  ax-mulass 11098  ax-distr 11099  ax-i2m1 11100  ax-1ne0 11101  ax-1rid 11102  ax-rnegex 11103  ax-rrecex 11104  ax-cnre 11105  ax-pre-lttri 11106  ax-pre-lttrn 11107  ax-pre-ltadd 11108  ax-pre-mulgt0 11109

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7318  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7812  df-1st 7936  df-2nd 7937  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-1o 8399  df-oadd 8403  df-er 8637  df-map 8769  df-en 8888  df-dom 8889  df-sdom 8890  df-fin 8891  df-dju 9819  df-card 9857  df-pnf 11175  df-mnf 11176  df-xr 11177  df-ltxr 11178  df-le 11179  df-sub 11373  df-neg 11374  df-nn 12169  df-2 12238  df-3 12239  df-4 12240  df-5 12241  df-6 12242  df-7 12243  df-8 12244  df-9 12245  df-n0 12432  df-xnn0 12505  df-z 12519  df-uz 12783  df-fz 13456  df-hash 14287  df-struct 17111  df-sets 17128  df-slot 17146  df-ndx 17158  df-base 17174  df-ress 17195  df-plusg 17227  df-tset 17233  df-efmnd 18831  df-symg 19339

This theorem is referenced by: (None)