Theorem omndmul2 33028

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 1088	. . 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 624	. . 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 768	. . 3 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 𝑁 ∈ ℕ0)
6		oveq1 7420	. . . . 5 ⊢ (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
7	6	breq2d 5135	. . . 4 ⊢ (𝑚 = 0 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (0 · 𝑋)))
8		oveq1 7420	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
9	8	breq2d 5135	. . . 4 ⊢ (𝑚 = 𝑛 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑛 · 𝑋)))
10		oveq1 7420	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
11	10	breq2d 5135	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ ((𝑛 + 1) · 𝑋)))
12		oveq1 7420	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
13	12	breq2d 5135	. . . 4 ⊢ (𝑚 = 𝑁 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑁 · 𝑋)))
14		omndtos 33021	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15		tospos 18434	. . . . . . . 8 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17		omndmnd 33020	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18		omndmul.0	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
19		omndmul2.3	. . . . . . . . 9 ⊢ 0 = (0g‘𝑀)
20	18, 19	mndidcl 18731	. . . . . . . 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 18334	. . . . . . 7 ⊢ ((𝑀 ∈ Poset ∧ 0 ∈ 𝐵) → 0 ≤ 0 )
24	16, 21, 23	syl2anc 584	. . . . . 6 ⊢ (𝑀 ∈ oMnd → 0 ≤ 0 )
25	24	ad3antrrr 730	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ 0 )
26		omndmul2.2	. . . . . . 7 ⊢ · = (.g‘𝑀)
27	18, 19, 26	mulg0 19061	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 )
28	27	ad3antlr 731	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → (0 · 𝑋) = 0 )
29	25, 28	breqtrrd 5151	. . . 4 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (0 · 𝑋))
30	16	ad5antr 734	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑀 ∈ Poset)
31	17	ad5antr 734	. . . . . . 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 768	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑛 ∈ ℕ0)
34		simp-5r 785	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑋 ∈ 𝐵)
35	18, 26, 31, 33, 34	mulgnn0cld 19082	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
36		simpr32 1264	. . . . . . . . . 10 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0 ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ0)
37		1nn0 12525	. . . . . . . . . . 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 12574	. . . . . . . . 9 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0 ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ0)
40	39	3anassrs 1360	. . . . . . . 8 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ0)
41	40	3anassrs 1360	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ0)
42	18, 26, 31, 41, 34	mulgnn0cld 19082	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
43	32, 35, 42	3jca 1128	. . . . 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 782	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ oMnd)
46	17	ad4antr 732	. . . . . . . . 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 783	. . . . . . . 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 19082	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ∈ 𝐵)
51		simplr 768	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 ≤ 𝑋)
52		eqid 2734	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
53	18, 22, 52	omndadd 33022	. . . . . . . 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 1384	. . . . . . 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 18736	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ (𝑛 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
56	46, 50, 55	syl2anc 584	. . . . . . 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 19091	. . . . . . . . 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 1373	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g‘𝑀)(𝑛 · 𝑋)))
60		1cnd 11238	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 1 ∈ ℂ)
61		simpr3 1196	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0)
62	61	nn0cnd 12572	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝑛 ∈ ℂ)
63	60, 62	addcomd 11445	. . . . . . . . . 10 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → (1 + 𝑛) = (𝑛 + 1))
64	63	3anassrs 1360	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 + 𝑛) = (𝑛 + 1))
65	64	oveq1d 7428	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
66	18, 26	mulg1 19068	. . . . . . . . . 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 7428	. . . . . . . 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 2778	. . . . . . 7 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑋(+g‘𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
70	54, 56, 69	3brtr3d 5154	. . . . . 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 18336	. . . . . 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 838	. . . 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 12698	. . 3 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑁 ∈ ℕ0) → 0 ≤ (𝑁 · 𝑋))
76	5, 75	mpdan 687	. 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 1086   = wceq 1539   ∈ wcel 2107   class class class wbr 5123  ‘cfv 6541  (class class class)co 7413  0cc0 11137  1c1 11138   + caddc 11140  ℕ₀cn0 12509  Basecbs 17229  +_gcplusg 17273  lecple 17280  0_gc0g 17455  Posetcpo 18323  Tosetctos 18430  Mndcmnd 18716  ._gcmg 19054  oMndcomnd 33013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pow 5345  ax-pr 5412  ax-un 7737  ax-cnex 11193  ax-resscn 11194  ax-1cn 11195  ax-icn 11196  ax-addcl 11197  ax-addrcl 11198  ax-mulcl 11199  ax-mulrcl 11200  ax-mulcom 11201  ax-addass 11202  ax-mulass 11203  ax-distr 11204  ax-i2m1 11205  ax-1ne0 11206  ax-1rid 11207  ax-rnegex 11208  ax-rrecex 11209  ax-cnre 11210  ax-pre-lttri 11211  ax-pre-lttrn 11212  ax-pre-ltadd 11213  ax-pre-mulgt0 11214

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-iun 4973  df-br 5124  df-opab 5186  df-mpt 5206  df-tr 5240  df-id 5558  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-we 5619  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-pred 6301  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7870  df-1st 7996  df-2nd 7997  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432  df-er 8727  df-en 8968  df-dom 8969  df-sdom 8970  df-pnf 11279  df-mnf 11280  df-xr 11281  df-ltxr 11282  df-le 11283  df-sub 11476  df-neg 11477  df-nn 12249  df-n0 12510  df-z 12597  df-uz 12861  df-fz 13530  df-seq 14025  df-0g 17457  df-proset 18310  df-poset 18329  df-toset 18431  df-mgm 18622  df-sgrp 18701  df-mnd 18717  df-mulg 19055  df-omnd 33015

This theorem is referenced by:  omndmul3  33029