Theorem smndex1mgm 18964

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 18962	. . . . . 6 ⊢ 𝐵 ⊆ (Base‘𝑀)
7		ssel 3931	. . . . . . 7 ⊢ (𝐵 ⊆ (Base‘𝑀) → (𝑎 ∈ 𝐵 → 𝑎 ∈ (Base‘𝑀)))
8		ssel 3931	. . . . . . 7 ⊢ (𝐵 ⊆ (Base‘𝑀) → (𝑏 ∈ 𝐵 → 𝑏 ∈ (Base‘𝑀)))
9	7, 8	anim12d 620	. . . . . 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 2763	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
12		eqid 2763	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	1, 11, 12	efmndov 18935	. . . . 5 ⊢ ((𝑎 ∈ (Base‘𝑀) ∧ 𝑏 ∈ (Base‘𝑀)) → (𝑎(+g‘𝑀)𝑏) = (𝑎 ∘ 𝑏))
14	10, 13	syl 18	. . . 4 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) = (𝑎 ∘ 𝑏))
15		simpl 487	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → 𝑎 = 𝐼)
16		simpr 489	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → 𝑏 = 𝐼)
17	15, 16	coeq12d 5850	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → (𝑎 ∘ 𝑏) = (𝐼 ∘ 𝐼))
18	1, 2, 3	smndex1iidm 18955	. . . . . . . . . . 11 ⊢ (𝐼 ∘ 𝐼) = 𝐼
19	17, 18	eqtrdi 2814	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → (𝑎 ∘ 𝑏) = 𝐼)
20	19	orcd 886	. . . . . . . . 9 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
21	20	ex 417	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑏 = 𝐼 → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
22		simpll 778	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → 𝑎 = 𝐼)
23		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → 𝑏 = (𝐺‘𝑘))
24	22, 23	coeq12d 5850	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐼 ∘ (𝐺‘𝑘)))
25	1, 2, 3, 4	smndex1igid 18960	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘))
26	25	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘))
27	24, 26	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
28	27	ex 417	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑏 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
29	28	reximdva 3178	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
30	29	imp 411	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
31	30	olcd 887	. . . . . . . . 9 ⊢ ((𝑎 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
32	31	ex 417	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
33	21, 32	jaod 872	. . . . . . 7 ⊢ (𝑎 = 𝐼 → ((𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
34		simpr 489	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑎 = (𝐺‘𝑘))
35		simpll 778	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑏 = 𝐼)
36	34, 35	coeq12d 5850	. . . . . . . . . . . . . 14 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = ((𝐺‘𝑘) ∘ 𝐼))
37	1, 2, 3	smndex1ibas 18954	. . . . . . . . . . . . . . . 16 ⊢ 𝐼 ∈ (Base‘𝑀)
38	1, 2, 3, 4	smndex1gid 18958	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ (Base‘𝑀) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
39	37, 38	mpan 702	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
40	39	ad2antlr 739	. . . . . . . . . . . . . 14 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
41	36, 40	eqtrd 2798	. . . . . . . . . . . . 13 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
42	41	ex 417	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑎 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
43	42	reximdva 3178	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
44	43	imp 411	. . . . . . . . . 10 ⊢ ((𝑏 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
45	44	olcd 887	. . . . . . . . 9 ⊢ ((𝑏 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
46	45	expcom 418	. . . . . . . 8 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → (𝑏 = 𝐼 → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
47		fveq2 6881	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝐺‘𝑘) = (𝐺‘𝑚))
48	47	eqeq2d 2774	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝑏 = (𝐺‘𝑘) ↔ 𝑏 = (𝐺‘𝑚)))
49	48	cbvrexvw 3244	. . . . . . . . 9 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) ↔ ∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚))
50		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑎 = (𝐺‘𝑘))
51		simpllr 787	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑏 = (𝐺‘𝑚))
52	50, 51	coeq12d 5850	. . . . . . . . . . . . . . . 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 18956	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0..^𝑁) → (𝐺‘𝑚) ∈ (Base‘𝑀))
54	1, 2, 3, 4	smndex1gid 18958	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑚) ∈ (Base‘𝑀) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
55	53, 54	sylan 591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
56	55	ad4ant13 763	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
57	52, 56	eqtrd 2798	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
58	57	ex 417	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) → (𝑎 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
59	58	reximdva 3178	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
60	59	rexlimiva 3158	. . . . . . . . . . . 12 ⊢ (∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚) → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
61	60	imp 411	. . . . . . . . . . 11 ⊢ ((∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚) ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
62	61	olcd 887	. . . . . . . . . 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 418	. . . . . . . . 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 245	. . . . . . . 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 872	. . . . . . 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 870	. . . . . 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 411	. . . . 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 2855	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
69		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
70	69	sneqd 4601	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → {(𝐺‘𝑛)} = {(𝐺‘𝑘)})
71	70	cbviunv 5003	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} = ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}
72	71	uneq2i 4119	. . . . . . . . 9 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) = ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)})
73	72	eleq2i 2855	. . . . . . . 8 ⊢ (𝑎 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ 𝑎 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
74	68, 73	bitri 278	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
75		elun 4107	. . . . . . 7 ⊢ (𝑎 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑎 ∈ {𝐼} ∨ 𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
76		velsn 4605	. . . . . . . 8 ⊢ (𝑎 ∈ {𝐼} ↔ 𝑎 = 𝐼)
77		eliun 4960	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑎 ∈ {(𝐺‘𝑘)})
78		velsn 4605	. . . . . . . . . 10 ⊢ (𝑎 ∈ {(𝐺‘𝑘)} ↔ 𝑎 = (𝐺‘𝑘))
79	78	rexbii 3112	. . . . . . . . 9 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑎 ∈ {(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘))
80	77, 79	bitri 278	. . . . . . . 8 ⊢ (𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘))
81	76, 80	orbi12i 927	. . . . . . 7 ⊢ ((𝑎 ∈ {𝐼} ∨ 𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)))
82	74, 75, 81	3bitri 300	. . . . . 6 ⊢ (𝑎 ∈ 𝐵 ↔ (𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)))
83	5	eleq2i 2855	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
84	72	eleq2i 2855	. . . . . . . 8 ⊢ (𝑏 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
85	83, 84	bitri 278	. . . . . . 7 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
86		elun 4107	. . . . . . 7 ⊢ (𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑏 ∈ {𝐼} ∨ 𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
87		velsn 4605	. . . . . . . 8 ⊢ (𝑏 ∈ {𝐼} ↔ 𝑏 = 𝐼)
88		eliun 4960	. . . . . . . . 9 ⊢ (𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)})
89		velsn 4605	. . . . . . . . . 10 ⊢ (𝑏 ∈ {(𝐺‘𝑘)} ↔ 𝑏 = (𝐺‘𝑘))
90	89	rexbii 3112	. . . . . . . . 9 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘))
91	88, 90	bitri 278	. . . . . . . 8 ⊢ (𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘))
92	87, 91	orbi12i 927	. . . . . . 7 ⊢ ((𝑏 ∈ {𝐼} ∨ 𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)))
93	85, 86, 92	3bitri 300	. . . . . 6 ⊢ (𝑏 ∈ 𝐵 ↔ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)))
94	82, 93	anbi12i 639	. . . . 5 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ↔ ((𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) ∧ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘))))
95	5	eleq2i 2855	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ 𝐵 ↔ (𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
96	72	eleq2i 2855	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) ↔ (𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
97	95, 96	bitri 278	. . . . . 6 ⊢ ((𝑎 ∘ 𝑏) ∈ 𝐵 ↔ (𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
98		elun 4107	. . . . . 6 ⊢ ((𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ ((𝑎 ∘ 𝑏) ∈ {𝐼} ∨ (𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
99		vex 3459	. . . . . . . . 9 ⊢ 𝑎 ∈ V
100		vex 3459	. . . . . . . . 9 ⊢ 𝑏 ∈ V
101	99, 100	coex 7923	. . . . . . . 8 ⊢ (𝑎 ∘ 𝑏) ∈ V
102	101	elsn 4604	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ {𝐼} ↔ (𝑎 ∘ 𝑏) = 𝐼)
103		eliun 4960	. . . . . . . 8 ⊢ ((𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) ∈ {(𝐺‘𝑘)})
104	101	elsn 4604	. . . . . . . . 9 ⊢ ((𝑎 ∘ 𝑏) ∈ {(𝐺‘𝑘)} ↔ (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
105	104	rexbii 3112	. . . . . . . 8 ⊢ (∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) ∈ {(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
106	103, 105	bitri 278	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
107	102, 106	orbi12i 927	. . . . . 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 300	. . . . 5 ⊢ ((𝑎 ∘ 𝑏) ∈ 𝐵 ↔ ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
109	67, 94, 108	3imtr4i 295	. . . 4 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 ∘ 𝑏) ∈ 𝐵)
110	14, 109	eqeltrd 2863	. . 3 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
111	110	rgen2 3205	. 2 ⊢ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵
112		smndex1mgm.s	. . . 4 ⊢ 𝑆 = (𝑀 ↾s 𝐵)
113	112	ovexi 7444	. . 3 ⊢ 𝑆 ∈ V
114	1, 2, 3, 4, 5, 112	smndex1bas 18963	. . . . 5 ⊢ (Base‘𝑆) = 𝐵
115	114	eqcomi 2772	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
116	115	fvexi 6895	. . . . 5 ⊢ 𝐵 ∈ V
117	112, 12	ressplusg 17339	. . . . 5 ⊢ (𝐵 ∈ V → (+g‘𝑀) = (+g‘𝑆))
118	116, 117	ax-mp 5	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑆)
119	115, 118	ismgm 18694	. . 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 234	1 ⊢ 𝑆 ∈ Mgm

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  {csn 4589  ∪ ciun 4956   ↦ cmpt 5192   ∘ ccom 5665  ‘cfv 6536  (class class class)co 7410  0cc0 11095  ℕcn 12228  ℕ₀cn0 12499  ..^cfzo 13678   mod cmo 13898  Basecbs 17264   ↾_s cress 17285  +_gcplusg 17305  Mgmcmgm 18691  EndoFMndcefmnd 18922

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172  ax-pre-sup 11173

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-sup 9398  df-inf 9399  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-div 11867  df-nn 12229  df-2 12298  df-3 12299  df-4 12300  df-5 12301  df-6 12302  df-7 12303  df-8 12304  df-9 12305  df-n0 12500  df-z 12587  df-uz 12858  df-rp 13012  df-fz 13531  df-fzo 13679  df-fl 13821  df-mod 13899  df-struct 17202  df-sets 17219  df-slot 17237  df-ndx 17249  df-base 17265  df-ress 17286  df-plusg 17318  df-tset 17324  df-mgm 18693  df-efmnd 18923

This theorem is referenced by:  smndex1sgrp  18965