Theorem smndex1mgm 18932

Description: The monoid of endofunctions on ℕ₀ restricted to the modulo function 𝐼 and the constant functions (𝐺‘𝐾) is a magma. (Contributed by AV, 14-Feb-2024.)

Hypotheses

Ref	Expression
smndex1ibas.m	⊢ 𝑀 = (EndoFMnd‘ℕ0)
smndex1ibas.n	⊢ 𝑁 ∈ ℕ
smndex1ibas.i	⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))
smndex1ibas.g	⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))
smndex1mgm.b	⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})
smndex1mgm.s	⊢ 𝑆 = (𝑀 ↾s 𝐵)

Assertion

Ref	Expression
smndex1mgm	⊢ 𝑆 ∈ Mgm

Distinct variable groups:   𝑥,𝑁,𝑛   𝑥,𝑀   𝑛,𝐺   𝑛,𝑀   𝑥,𝐺   𝑛,𝐼,𝑥

Allowed substitution hints:   𝐵(𝑥,𝑛)   𝑆(𝑥,𝑛)

Proof of Theorem smndex1mgm

Dummy variables 𝑏 𝑘 𝑎 𝑚 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smndex1ibas.m	. . . . . . 7 ⊢ 𝑀 = (EndoFMnd‘ℕ0)
2		smndex1ibas.n	. . . . . . 7 ⊢ 𝑁 ∈ ℕ
3		smndex1ibas.i	. . . . . . 7 ⊢ 𝐼 = (𝑥 ∈ ℕ0 ↦ (𝑥 mod 𝑁))
4		smndex1ibas.g	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ (0..^𝑁) ↦ (𝑥 ∈ ℕ0 ↦ 𝑛))
5		smndex1mgm.b	. . . . . . 7 ⊢ 𝐵 = ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)})
6	1, 2, 3, 4, 5	smndex1basss 18930	. . . . . 6 ⊢ 𝐵 ⊆ (Base‘𝑀)
7		ssel 3988	. . . . . . 7 ⊢ (𝐵 ⊆ (Base‘𝑀) → (𝑎 ∈ 𝐵 → 𝑎 ∈ (Base‘𝑀)))
8		ssel 3988	. . . . . . 7 ⊢ (𝐵 ⊆ (Base‘𝑀) → (𝑏 ∈ 𝐵 → 𝑏 ∈ (Base‘𝑀)))
9	7, 8	anim12d 609	. . . . . 6 ⊢ (𝐵 ⊆ (Base‘𝑀) → ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀))))
10	6, 9	ax-mp 5	. . . . 5 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)))
11		eqid 2734	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
12		eqid 2734	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	1, 11, 12	efmndov 18906	. . . . 5 ⊢ ((𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(+g‘𝑀)𝑏) = (𝑎 ∘ 𝑏))
14	10, 13	syl 17	. . . 4 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) = (𝑎 ∘ 𝑏))
15		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → 𝑎 = 𝐼)
16		simpr 484	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → 𝑏 = 𝐼)
17	15, 16	coeq12d 5877	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → (𝑎 ∘ 𝑏) = (𝐼 ∘ 𝐼))
18	1, 2, 3	smndex1iidm 18926	. . . . . . . . . . 11 ⊢ (𝐼 ∘ 𝐼) = 𝐼
19	17, 18	eqtrdi 2790	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → (𝑎 ∘ 𝑏) = 𝐼)
20	19	orcd 873	. . . . . . . . 9 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
21	20	ex 412	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑏 = 𝐼 → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
22		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → 𝑎 = 𝐼)
23		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → 𝑏 = (𝐺‘𝑘))
24	22, 23	coeq12d 5877	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐼 ∘ (𝐺‘𝑘)))
25	1, 2, 3, 4	smndex1igid 18929	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘))
26	25	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘))
27	24, 26	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
28	27	ex 412	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑏 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
29	28	reximdva 3165	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
30	29	imp 406	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
31	30	olcd 874	. . . . . . . . 9 ⊢ ((𝑎 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
32	31	ex 412	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
33	21, 32	jaod 859	. . . . . . 7 ⊢ (𝑎 = 𝐼 → ((𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
34		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑎 = (𝐺‘𝑘))
35		simpll 767	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑏 = 𝐼)
36	34, 35	coeq12d 5877	. . . . . . . . . . . . . 14 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = ((𝐺‘𝑘) ∘ 𝐼))
37	1, 2, 3	smndex1ibas 18925	. . . . . . . . . . . . . . . 16 ⊢ 𝐼 ∈ (Base‘𝑀)
38	1, 2, 3, 4	smndex1gid 18928	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ (Base‘𝑀) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
39	37, 38	mpan 690	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
40	39	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
41	36, 40	eqtrd 2774	. . . . . . . . . . . . 13 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
42	41	ex 412	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑎 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
43	42	reximdva 3165	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
44	43	imp 406	. . . . . . . . . 10 ⊢ ((𝑏 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
45	44	olcd 874	. . . . . . . . 9 ⊢ ((𝑏 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
46	45	expcom 413	. . . . . . . 8 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → (𝑏 = 𝐼 → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
47		fveq2 6906	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝐺‘𝑘) = (𝐺‘𝑚))
48	47	eqeq2d 2745	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝑏 = (𝐺‘𝑘) ↔ 𝑏 = (𝐺‘𝑚)))
49	48	cbvrexvw 3235	. . . . . . . . 9 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) ↔ ∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚))
50		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑎 = (𝐺‘𝑘))
51		simpllr 776	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑏 = (𝐺‘𝑚))
52	50, 51	coeq12d 5877	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = ((𝐺‘𝑘) ∘ (𝐺‘𝑚)))
53	1, 2, 3, 4	smndex1gbas 18927	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0..^𝑁) → (𝐺‘𝑚) ∈ (Base‘𝑀))
54	1, 2, 3, 4	smndex1gid 18928	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑚) ∈ (Base‘𝑀) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
55	53, 54	sylan 580	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
56	55	ad4ant13 751	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
57	52, 56	eqtrd 2774	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
58	57	ex 412	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) → (𝑎 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
59	58	reximdva 3165	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
60	59	rexlimiva 3144	. . . . . . . . . . . 12 ⊢ (∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚) → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
61	60	imp 406	. . . . . . . . . . 11 ⊢ ((∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚) ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
62	61	olcd 874	. . . . . . . . . 10 ⊢ ((∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚) ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
63	62	expcom 413	. . . . . . . . 9 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → (∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
64	49, 63	biimtrid 242	. . . . . . . 8 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
65	46, 64	jaod 859	. . . . . . 7 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ((𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
66	33, 65	jaoi 857	. . . . . 6 ⊢ ((𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ((𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
67	66	imp 406	. . . . 5 ⊢ (((𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) ∧ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘))) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
68	5	eleq2i 2830	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
69		fveq2 6906	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
70	69	sneqd 4642	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → {(𝐺‘𝑛)} = {(𝐺‘𝑘)})
71	70	cbviunv 5044	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} = ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}
72	71	uneq2i 4174	. . . . . . . . 9 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) = ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)})
73	72	eleq2i 2830	. . . . . . . 8 ⊢ (𝑎 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ 𝑎 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
74	68, 73	bitri 275	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
75		elun 4162	. . . . . . 7 ⊢ (𝑎 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑎 ∈ {𝐼} ∨ 𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
76		velsn 4646	. . . . . . . 8 ⊢ (𝑎 ∈ {𝐼} ↔ 𝑎 = 𝐼)
77		eliun 4999	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑎 ∈ {(𝐺‘𝑘)})
78		velsn 4646	. . . . . . . . . 10 ⊢ (𝑎 ∈ {(𝐺‘𝑘)} ↔ 𝑎 = (𝐺‘𝑘))
79	78	rexbii 3091	. . . . . . . . 9 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑎 ∈ {(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘))
80	77, 79	bitri 275	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘))
81	76, 80	orbi12i 914	. . . . . . 7 ⊢ ((𝑎 ∈ {𝐼} ∨ 𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)))
82	74, 75, 81	3bitri 297	. . . . . 6 ⊢ (𝑎 ∈ 𝐵 ↔ (𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)))
83	5	eleq2i 2830	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
84	72	eleq2i 2830	. . . . . . . 8 ⊢ (𝑏 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
85	83, 84	bitri 275	. . . . . . 7 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
86		elun 4162	. . . . . . 7 ⊢ (𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑏 ∈ {𝐼} ∨ 𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
87		velsn 4646	. . . . . . . 8 ⊢ (𝑏 ∈ {𝐼} ↔ 𝑏 = 𝐼)
88		eliun 4999	. . . . . . . . 9 ⊢ (𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)})
89		velsn 4646	. . . . . . . . . 10 ⊢ (𝑏 ∈ {(𝐺‘𝑘)} ↔ 𝑏 = (𝐺‘𝑘))
90	89	rexbii 3091	. . . . . . . . 9 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘))
91	88, 90	bitri 275	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘))
92	87, 91	orbi12i 914	. . . . . . 7 ⊢ ((𝑏 ∈ {𝐼} ∨ 𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)))
93	85, 86, 92	3bitri 297	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)))
94	82, 93	anbi12i 628	. . . . 5 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ↔ ((𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) ∧ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘))))
95	5	eleq2i 2830	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ 𝐵 ↔ (𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
96	72	eleq2i 2830	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
97	95, 96	bitri 275	. . . . . 6 ⊢ ((𝑎 ∘ 𝑏) ∈ 𝐵 ↔ (𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
98		elun 4162	. . . . . 6 ⊢ ((𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ ((𝑎 ∘ 𝑏) ∈ {𝐼} ∨ (𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
99		vex 3481	. . . . . . . . 9 ⊢ 𝑎 ∈ V
100		vex 3481	. . . . . . . . 9 ⊢ 𝑏 ∈ V
101	99, 100	coex 7952	. . . . . . . 8 ⊢ (𝑎 ∘ 𝑏) ∈ V
102	101	elsn 4645	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ {𝐼} ↔ (𝑎 ∘ 𝑏) = 𝐼)
103		eliun 4999	. . . . . . . 8 ⊢ ((𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) ∈ {(𝐺‘𝑘)})
104	101	elsn 4645	. . . . . . . . 9 ⊢ ((𝑎 ∘ 𝑏) ∈ {(𝐺‘𝑘)} ↔ (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
105	104	rexbii 3091	. . . . . . . 8 ⊢ (∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) ∈ {(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
106	103, 105	bitri 275	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
107	102, 106	orbi12i 914	. . . . . 6 ⊢ (((𝑎 ∘ 𝑏) ∈ {𝐼} ∨ (𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
108	97, 98, 107	3bitri 297	. . . . 5 ⊢ ((𝑎 ∘ 𝑏) ∈ 𝐵 ↔ ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
109	67, 94, 108	3imtr4i 292	. . . 4 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 ∘ 𝑏) ∈ 𝐵)
110	14, 109	eqeltrd 2838	. . 3 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
111	110	rgen2 3196	. 2 ⊢ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵
112		smndex1mgm.s	. . . 4 ⊢ 𝑆 = (𝑀 ↾s 𝐵)
113	112	ovexi 7464	. . 3 ⊢ 𝑆 ∈ V
114	1, 2, 3, 4, 5, 112	smndex1bas 18931	. . . . 5 ⊢ (Base‘𝑆) = 𝐵
115	114	eqcomi 2743	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
116	115	fvexi 6920	. . . . 5 ⊢ 𝐵 ∈ V
117	112, 12	ressplusg 17335	. . . . 5 ⊢ (𝐵 ∈ V → (+g‘𝑀) = (+g‘𝑆))
118	116, 117	ax-mp 5	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑆)
119	115, 118	ismgm 18666	. . 3 ⊢ (𝑆 ∈ V → (𝑆 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵))
120	113, 119	ax-mp 5	. 2 ⊢ (𝑆 ∈ Mgm ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
121	111, 120	mpbir 231	1 ⊢ 𝑆 ∈ Mgm

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  Vcvv 3477   ∪ cun 3960   ⊆ wss 3962  {csn 4630  ∪ ciun 4995   ↦ cmpt 5230   ∘ ccom 5692  ‘cfv 6562  (class class class)co 7430  0cc0 11152  ℕcn 12263  ℕ₀cn0 12523  ..^cfzo 13690   mod cmo 13905  Basecbs 17244   ↾_s cress 17273  +_gcplusg 17297  Mgmcmgm 18663  EndoFMndcefmnd 18893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229  ax-pre-sup 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-er 8743  df-map 8866  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-sup 9479  df-inf 9480  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-div 11918  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-uz 12876  df-rp 13032  df-fz 13544  df-fzo 13691  df-fl 13828  df-mod 13906  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-tset 17316  df-mgm 18665  df-efmnd 18894

This theorem is referenced by:  smndex1sgrp  18933