Theorem smndex1mgm 19106

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 19104	. . . . . 6 ⊢ 𝐵 ⊆ (Base‘𝑀)
7		ssel 3925	. . . . . . 7 ⊢ (𝐵 ⊆ (Base‘𝑀) → (𝑎 ∈ 𝐵 → 𝑎 ∈ (Base‘𝑀)))
8		ssel 3925	. . . . . . 7 ⊢ (𝐵 ⊆ (Base‘𝑀) → (𝑏 ∈ 𝐵 → 𝑏 ∈ (Base‘𝑀)))
9	7, 8	anim12d 621	. . . . . 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 2761	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
12		eqid 2761	. . . . . 6 ⊢ (+g‘𝑀) = (+g‘𝑀)
13	1, 11, 12	efmndov 19077	. . . . 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 488	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → 𝑎 = 𝐼)
16		simpr 490	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → 𝑏 = 𝐼)
17	15, 16	coeq12d 5842	. . . . . . . . . . 11 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → (𝑎 ∘ 𝑏) = (𝐼 ∘ 𝐼))
18	1, 2, 3	smndex1iidm 19097	. . . . . . . . . . 11 ⊢ (𝐼 ∘ 𝐼) = 𝐼
19	17, 18	eqtrdi 2812	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → (𝑎 ∘ 𝑏) = 𝐼)
20	19	orcd 887	. . . . . . . . 9 ⊢ ((𝑎 = 𝐼 ∧ 𝑏 = 𝐼) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
21	20	ex 418	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (𝑏 = 𝐼 → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
22		simpll 779	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → 𝑎 = 𝐼)
23		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → 𝑏 = (𝐺‘𝑘))
24	22, 23	coeq12d 5842	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐼 ∘ (𝐺‘𝑘)))
25	1, 2, 3, 4	smndex1igid 19102	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) → (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘))
26	25	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝐼 ∘ (𝐺‘𝑘)) = (𝐺‘𝑘))
27	24, 26	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑏 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
28	27	ex 418	. . . . . . . . . . . 12 ⊢ ((𝑎 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑏 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
29	28	reximdva 3176	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
30	29	imp 412	. . . . . . . . . 10 ⊢ ((𝑎 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
31	30	olcd 888	. . . . . . . . 9 ⊢ ((𝑎 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
32	31	ex 418	. . . . . . . 8 ⊢ (𝑎 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
33	21, 32	jaod 873	. . . . . . 7 ⊢ (𝑎 = 𝐼 → ((𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
34		simpr 490	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑎 = (𝐺‘𝑘))
35		simpll 779	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑏 = 𝐼)
36	34, 35	coeq12d 5842	. . . . . . . . . . . . . 14 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = ((𝐺‘𝑘) ∘ 𝐼))
37	1, 2, 3	smndex1ibas 19096	. . . . . . . . . . . . . . . 16 ⊢ 𝐼 ∈ (Base‘𝑀)
38	1, 2, 3, 4	smndex1gid 19100	. . . . . . . . . . . . . . . 16 ⊢ ((𝐼 ∈ (Base‘𝑀) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
39	37, 38	mpan 703	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0..^𝑁) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
40	39	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → ((𝐺‘𝑘) ∘ 𝐼) = (𝐺‘𝑘))
41	36, 40	eqtrd 2796	. . . . . . . . . . . . 13 ⊢ (((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
42	41	ex 418	. . . . . . . . . . . 12 ⊢ ((𝑏 = 𝐼 ∧ 𝑘 ∈ (0..^𝑁)) → (𝑎 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
43	42	reximdva 3176	. . . . . . . . . . 11 ⊢ (𝑏 = 𝐼 → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
44	43	imp 412	. . . . . . . . . 10 ⊢ ((𝑏 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
45	44	olcd 888	. . . . . . . . 9 ⊢ ((𝑏 = 𝐼 ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
46	45	expcom 419	. . . . . . . 8 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → (𝑏 = 𝐼 → ((𝑎 ∘ 𝑏) = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))))
47		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑘 = 𝑚 → (𝐺‘𝑘) = (𝐺‘𝑚))
48	47	eqeq2d 2772	. . . . . . . . . 10 ⊢ (𝑘 = 𝑚 → (𝑏 = (𝐺‘𝑘) ↔ 𝑏 = (𝐺‘𝑚)))
49	48	cbvrexvw 3242	. . . . . . . . 9 ⊢ (∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘) ↔ ∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚))
50		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑎 = (𝐺‘𝑘))
51		simpllr 788	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → 𝑏 = (𝐺‘𝑚))
52	50, 51	coeq12d 5842	. . . . . . . . . . . . . . . 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 19098	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (0..^𝑁) → (𝐺‘𝑚) ∈ (Base‘𝑀))
54	1, 2, 3, 4	smndex1gid 19100	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐺‘𝑚) ∈ (Base‘𝑀) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
55	53, 54	sylan 592	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑚 ∈ (0..^𝑁) ∧ 𝑘 ∈ (0..^𝑁)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
56	55	ad4ant13 764	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → ((𝐺‘𝑘) ∘ (𝐺‘𝑚)) = (𝐺‘𝑘))
57	52, 56	eqtrd 2796	. . . . . . . . . . . . . . 15 ⊢ ((((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) ∧ 𝑎 = (𝐺‘𝑘)) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
58	57	ex 418	. . . . . . . . . . . . . 14 ⊢ (((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) ∧ 𝑘 ∈ (0..^𝑁)) → (𝑎 = (𝐺‘𝑘) → (𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
59	58	reximdva 3176	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ (0..^𝑁) ∧ 𝑏 = (𝐺‘𝑚)) → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
60	59	rexlimiva 3156	. . . . . . . . . . . 12 ⊢ (∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚) → (∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘)))
61	60	imp 412	. . . . . . . . . . 11 ⊢ ((∃𝑚 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑚) ∧ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) → ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) = (𝐺‘𝑘))
62	61	olcd 888	. . . . . . . . . 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 419	. . . . . . . . 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 873	. . . . . . 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 871	. . . . . 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 412	. . . . 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 2853	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐵 ↔ 𝑎 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
69		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑘 → (𝐺‘𝑛) = (𝐺‘𝑘))
70	69	sneqd 4596	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑘 → {(𝐺‘𝑛)} = {(𝐺‘𝑘)})
71	70	cbviunv 4997	. . . . . . . . . 10 ⊢ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)} = ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}
72	71	uneq2i 4112	. . . . . . . . 9 ⊢ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}) = ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)})
73	72	eleq2i 2853	. . . . . . . 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 4100	. . . . . . 7 ⊢ (𝑎 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑎 ∈ {𝐼} ∨ 𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
76		velsn 4600	. . . . . . . 8 ⊢ (𝑎 ∈ {𝐼} ↔ 𝑎 = 𝐼)
77		eliun 4955	. . . . . . . . 9 ⊢ (𝑎 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑎 ∈ {(𝐺‘𝑘)})
78		velsn 4600	. . . . . . . . . 10 ⊢ (𝑎 ∈ {(𝐺‘𝑘)} ↔ 𝑎 = (𝐺‘𝑘))
79	78	rexbii 3110	. . . . . . . . 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 928	. . . . . . 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 2853	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 ↔ 𝑏 ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
84	72	eleq2i 2853	. . . . . . . 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 4100	. . . . . . 7 ⊢ (𝑏 ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ (𝑏 ∈ {𝐼} ∨ 𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
87		velsn 4600	. . . . . . . 8 ⊢ (𝑏 ∈ {𝐼} ↔ 𝑏 = 𝐼)
88		eliun 4955	. . . . . . . . 9 ⊢ (𝑏 ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)𝑏 ∈ {(𝐺‘𝑘)})
89		velsn 4600	. . . . . . . . . 10 ⊢ (𝑏 ∈ {(𝐺‘𝑘)} ↔ 𝑏 = (𝐺‘𝑘))
90	89	rexbii 3110	. . . . . . . . 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 928	. . . . . . 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 640	. . . . 5 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) ↔ ((𝑎 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑎 = (𝐺‘𝑘)) ∧ (𝑏 = 𝐼 ∨ ∃𝑘 ∈ (0..^𝑁)𝑏 = (𝐺‘𝑘))))
95	5	eleq2i 2853	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ 𝐵 ↔ (𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑛 ∈ (0..^𝑁){(𝐺‘𝑛)}))
96	72	eleq2i 2853	. . . . . . 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 4100	. . . . . 6 ⊢ ((𝑎 ∘ 𝑏) ∈ ({𝐼} ∪ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}) ↔ ((𝑎 ∘ 𝑏) ∈ {𝐼} ∨ (𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)}))
99		vex 3455	. . . . . . . . 9 ⊢ 𝑎 ∈ V
100		vex 3455	. . . . . . . . 9 ⊢ 𝑏 ∈ V
101	99, 100	coex 7942	. . . . . . . 8 ⊢ (𝑎 ∘ 𝑏) ∈ V
102	101	elsn 4599	. . . . . . 7 ⊢ ((𝑎 ∘ 𝑏) ∈ {𝐼} ↔ (𝑎 ∘ 𝑏) = 𝐼)
103		eliun 4955	. . . . . . . 8 ⊢ ((𝑎 ∘ 𝑏) ∈ ∪ 𝑘 ∈ (0..^𝑁){(𝐺‘𝑘)} ↔ ∃𝑘 ∈ (0..^𝑁)(𝑎 ∘ 𝑏) ∈ {(𝐺‘𝑘)})
104	101	elsn 4599	. . . . . . . . 9 ⊢ ((𝑎 ∘ 𝑏) ∈ {(𝐺‘𝑘)} ↔ (𝑎 ∘ 𝑏) = (𝐺‘𝑘))
105	104	rexbii 3110	. . . . . . . 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 928	. . . . . 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 2861	. . 3 ⊢ ((𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎(+g‘𝑀)𝑏) ∈ 𝐵)
111	110	rgen2 3203	. 2 ⊢ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 (𝑎(+g‘𝑀)𝑏) ∈ 𝐵
112		smndex1mgm.s	. . . 4 ⊢ 𝑆 = (𝑀 ↾s 𝐵)
113	112	ovexi 7454	. . 3 ⊢ 𝑆 ∈ V
114	1, 2, 3, 4, 5, 112	smndex1bas 19105	. . . . 5 ⊢ (Base‘𝑆) = 𝐵
115	114	eqcomi 2770	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
116	115	fvexi 6899	. . . . 5 ⊢ 𝐵 ∈ V
117	112, 12	ressplusg 17462	. . . . 5 ⊢ (𝐵 ∈ V → (+g‘𝑀) = (+g‘𝑆))
118	116, 117	ax-mp 5	. . . 4 ⊢ (+g‘𝑀) = (+g‘𝑆)
119	115, 118	ismgm 18817	. . 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

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899  {csn 4584  ∪ ciun 4951   ↦ cmpt 5186   ∘ ccom 5655  ‘cfv 6538  (class class class)co 7420  0cc0 11200  ℕcn 12335  ℕ₀cn0 12606  ..^cfzo 13788   mod cmo 14009  Basecbs 17387   ↾_s cress 17408  +_gcplusg 17428  Mgmcmgm 18814  EndoFMndcefmnd 19064

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277  ax-pre-sup 11278

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-er 8717  df-map 8849  df-en 8974  df-dom 8975  df-sdom 8976  df-fin 8977  df-sup 9434  df-inf 9435  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-div 11974  df-nn 12336  df-2 12405  df-3 12406  df-4 12407  df-5 12408  df-6 12409  df-7 12410  df-8 12411  df-9 12412  df-n0 12607  df-z 12694  df-uz 12966  df-rp 13121  df-fz 13640  df-fzo 13789  df-fl 13932  df-mod 14010  df-struct 17325  df-sets 17342  df-slot 17360  df-ndx 17372  df-base 17388  df-ress 17409  df-plusg 17441  df-tset 17447  df-mgm 18816  df-efmnd 19065

This theorem is used by:  smndex1sgrp  19107