Theorem omndmul2 20099

Description: In an ordered monoid, the ordering is compatible with group power. This version does not require the monoid to be commutative. (Contributed by Thierry Arnoux, 23-Mar-2018.)

Hypotheses

Ref	Expression
omndmul.0	⊢ 𝐵 = (Base‘𝑀)
omndmul.1	⊢ ≤ = (le‘𝑀)
omndmul2.2	⊢ · = (.g‘𝑀)
omndmul2.3	⊢ 0 = (0g‘𝑀)

Assertion

Ref	Expression
omndmul2	⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))

Proof of Theorem omndmul2

Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-3an 1089	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0)) ∧ 0 ≤ 𝑋))
2		anass 468	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ↔ (𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0)))
3	2	anbi1i 625	. . 3 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0)) ∧ 0 ≤ 𝑋))
4	1, 3	bitr4i 278	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ↔ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋))
5		simplr 769	. . 3 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 𝑁 ∈ ℕ0)
6		oveq1 7367	. . . . 5 ⊢ (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
7	6	breq2d 5098	. . . 4 ⊢ (𝑚 = 0 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (0 · 𝑋)))
8		oveq1 7367	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
9	8	breq2d 5098	. . . 4 ⊢ (𝑚 = 𝑛 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑛 · 𝑋)))
10		oveq1 7367	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
11	10	breq2d 5098	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ ((𝑛 + 1) · 𝑋)))
12		oveq1 7367	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
13	12	breq2d 5098	. . . 4 ⊢ (𝑚 = 𝑁 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑁 · 𝑋)))
14		omndtos 20093	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15		tospos 18375	. . . . . . . 8 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17		omndmnd 20092	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18		omndmul.0	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
19		omndmul2.3	. . . . . . . . 9 ⊢ 0 = (0g‘𝑀)
20	18, 19	mndidcl 18708	. . . . . . . 8 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵)
21	17, 20	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ oMnd → 0 ∈ 𝐵)
22		omndmul.1	. . . . . . . 8 ⊢ ≤ = (le‘𝑀)
23	18, 22	posref 18275	. . . . . . 7 ⊢ ((𝑀 ∈ Poset ∧ 0 ∈ 𝐵) → 0 ≤ 0 )
24	16, 21, 23	syl2anc 585	. . . . . 6 ⊢ (𝑀 ∈ oMnd → 0 ≤ 0 )
25	24	ad3antrrr 731	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ 0 )
26		omndmul2.2	. . . . . . 7 ⊢ · = (.g‘𝑀)
27	18, 19, 26	mulg0 19041	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 )
28	27	ad3antlr 732	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → (0 · 𝑋) = 0 )
29	25, 28	breqtrrd 5114	. . . 4 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (0 · 𝑋))
30	16	ad5antr 735	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑀 ∈ Poset)
31	17	ad5antr 735	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑀 ∈ Mnd)
32	31, 20	syl 17	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 0 ∈ 𝐵)
33		simplr 769	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑛 ∈ ℕ0)
34		simp-5r 786	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑋 ∈ 𝐵)
35	18, 26, 31, 33, 34	mulgnn0cld 19062	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
36		simpr32 1266	. . . . . . . . . 10 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0 ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ0)
37		1nn0 12444	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
38	37	a1i 11	. . . . . . . . . 10 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0 ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋)))) → 1 ∈ ℕ0)
39	36, 38	nn0addcld 12493	. . . . . . . . 9 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0 ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ0)
40	39	3anassrs 1362	. . . . . . . 8 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ0)
41	40	3anassrs 1362	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ0)
42	18, 26, 31, 41, 34	mulgnn0cld 19062	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
43	32, 35, 42	3jca 1129	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → ( 0 ∈ 𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵))
44		simpr 484	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 0 ≤ (𝑛 · 𝑋))
45		simp-4l 783	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ oMnd)
46	17	ad4antr 733	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ Mnd)
47	46, 20	syl 17	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 ∈ 𝐵)
48		simp-4r 784	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋 ∈ 𝐵)
49		simpr 484	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
50	18, 26, 46, 49, 48	mulgnn0cld 19062	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ∈ 𝐵)
51		simplr 769	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 ≤ 𝑋)
52		eqid 2737	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
53	18, 22, 52	omndadd 20094	. . . . . . . 8 ⊢ ((𝑀 ∈ oMnd ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) ≤ (𝑋(+g‘𝑀)(𝑛 · 𝑋)))
54	45, 47, 48, 50, 51, 53	syl131anc 1386	. . . . . . 7 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) ≤ (𝑋(+g‘𝑀)(𝑛 · 𝑋)))
55	18, 52, 19	mndlid 18713	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ (𝑛 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
56	46, 50, 55	syl2anc 585	. . . . . . 7 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
57	37	a1i 11	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 1 ∈ ℕ0)
58	18, 26, 52	mulgnn0dir 19071	. . . . . . . . 9 ⊢ ((𝑀 ∈ Mnd ∧ (1 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g‘𝑀)(𝑛 · 𝑋)))
59	46, 57, 49, 48, 58	syl13anc 1375	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g‘𝑀)(𝑛 · 𝑋)))
60		1cnd 11130	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 1 ∈ ℂ)
61		simpr3 1198	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0)
62	61	nn0cnd 12491	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝑛 ∈ ℂ)
63	60, 62	addcomd 11339	. . . . . . . . . 10 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → (1 + 𝑛) = (𝑛 + 1))
64	63	3anassrs 1362	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 + 𝑛) = (𝑛 + 1))
65	64	oveq1d 7375	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
66	18, 26	mulg1 19048	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋)
67	48, 66	syl 17	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
68	67	oveq1d 7375	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 · 𝑋)(+g‘𝑀)(𝑛 · 𝑋)) = (𝑋(+g‘𝑀)(𝑛 · 𝑋)))
69	59, 65, 68	3eqtr3rd 2781	. . . . . . 7 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑋(+g‘𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
70	54, 56, 69	3brtr3d 5117	. . . . . 6 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))
71	70	adantr 480	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))
72	18, 22	postr 18277	. . . . . 6 ⊢ ((𝑀 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) → (( 0 ≤ (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋)) → 0 ≤ ((𝑛 + 1) · 𝑋)))
73	72	imp 406	. . . . 5 ⊢ (((𝑀 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) ∧ ( 0 ≤ (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))) → 0 ≤ ((𝑛 + 1) · 𝑋))
74	30, 43, 44, 71, 73	syl22anc 839	. . . 4 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 0 ≤ ((𝑛 + 1) · 𝑋))
75	7, 9, 11, 13, 29, 74	nn0indd 12617	. . 3 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑁 ∈ ℕ0) → 0 ≤ (𝑁 · 𝑋))
76	5, 75	mpdan 688	. 2 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))
77	4, 76	sylbi 217	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  0cc0 11029  1c1 11030   + caddc 11032  ℕ₀cn0 12428  Basecbs 17170  +_gcplusg 17211  lecple 17218  0_gc0g 17393  Posetcpo 18264  Tosetctos 18371  Mndcmnd 18693  ._gcmg 19034  oMndcomnd 20085

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-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

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-rmo 3343  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-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-n0 12429  df-z 12516  df-uz 12780  df-fz 13453  df-seq 13955  df-0g 17395  df-proset 18251  df-poset 18270  df-toset 18372  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-mulg 19035  df-omnd 20087

This theorem is referenced by:  omndmul3  20100