Theorem omndmul2 29043

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 1033	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0)) ∧ 0 ≤ 𝑋))
2		anass 679	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ↔ (𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0)))
3	2	anbi1i 727	. . 3 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0)) ∧ 0 ≤ 𝑋))
4	1, 3	bitr4i 266	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ↔ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋))
5		simplr 788	. . 3 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 𝑁 ∈ ℕ0)
6		oveq1 6556	. . . . 5 ⊢ (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
7	6	breq2d 4595	. . . 4 ⊢ (𝑚 = 0 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (0 · 𝑋)))
8		oveq1 6556	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
9	8	breq2d 4595	. . . 4 ⊢ (𝑚 = 𝑛 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑛 · 𝑋)))
10		oveq1 6556	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
11	10	breq2d 4595	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ ((𝑛 + 1) · 𝑋)))
12		oveq1 6556	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
13	12	breq2d 4595	. . . 4 ⊢ (𝑚 = 𝑁 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑁 · 𝑋)))
14		omndtos 29036	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15		tospos 28989	. . . . . . . 8 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17		omndmnd 29035	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18		omndmul.0	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
19		omndmul2.3	. . . . . . . . 9 ⊢ 0 = (0g‘𝑀)
20	18, 19	mndidcl 17131	. . . . . . . 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 16774	. . . . . . 7 ⊢ ((𝑀 ∈ Poset ∧ 0 ∈ 𝐵) → 0 ≤ 0 )
24	16, 21, 23	syl2anc 691	. . . . . 6 ⊢ (𝑀 ∈ oMnd → 0 ≤ 0 )
25	24	ad3antrrr 762	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ 0 )
26		omndmul2.2	. . . . . . 7 ⊢ · = (.g‘𝑀)
27	18, 19, 26	mulg0 17369	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 )
28	27	ad3antlr 763	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → (0 · 𝑋) = 0 )
29	25, 28	breqtrrd 4611	. . . 4 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (0 · 𝑋))
30	16	ad5antr 766	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑀 ∈ Poset)
31	17	ad5antr 766	. . . . . . 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 788	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑛 ∈ ℕ0)
34		simp-5r 805	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑋 ∈ 𝐵)
35	18, 26	mulgnn0cl 17381	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → (𝑛 · 𝑋) ∈ 𝐵)
36	31, 33, 34, 35	syl3anc 1318	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
37		simpr32 1145	. . . . . . . . . 10 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0 ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ0)
38		1nn0 11185	. . . . . . . . . . 11 ⊢ 1 ∈ ℕ0
39	38	a1i 11	. . . . . . . . . 10 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0 ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋)))) → 1 ∈ ℕ0)
40	37, 39	nn0addcld 11232	. . . . . . . . 9 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0 ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ0)
41	40	3anassrs 1282	. . . . . . . 8 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0 ∧ 0 ≤ (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ0)
42	41	3anassrs 1282	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ0)
43	18, 26	mulgnn0cl 17381	. . . . . . 7 ⊢ ((𝑀 ∈ Mnd ∧ (𝑛 + 1) ∈ ℕ0 ∧ 𝑋 ∈ 𝐵) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
44	31, 42, 34, 43	syl3anc 1318	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
45	32, 36, 44	3jca 1235	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → ( 0 ∈ 𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵))
46		simpr 476	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 0 ≤ (𝑛 · 𝑋))
47		simp-4l 802	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ oMnd)
48	17	ad4antr 764	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ Mnd)
49	48, 20	syl 17	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 ∈ 𝐵)
50		simp-4r 803	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋 ∈ 𝐵)
51		simpr 476	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
52	48, 51, 50, 35	syl3anc 1318	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ∈ 𝐵)
53		simplr 788	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 ≤ 𝑋)
54		eqid 2610	. . . . . . . . 9 ⊢ (+g‘𝑀) = (+g‘𝑀)
55	18, 22, 54	omndadd 29037	. . . . . . . 8 ⊢ ((𝑀 ∈ oMnd ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) ≤ (𝑋(+g‘𝑀)(𝑛 · 𝑋)))
56	47, 49, 50, 52, 53, 55	syl131anc 1331	. . . . . . 7 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) ≤ (𝑋(+g‘𝑀)(𝑛 · 𝑋)))
57	18, 54, 19	mndlid 17134	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ (𝑛 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
58	48, 52, 57	syl2anc 691	. . . . . . 7 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g‘𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
59	38	a1i 11	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → 1 ∈ ℕ0)
60	18, 26, 54	mulgnn0dir 17394	. . . . . . . . 9 ⊢ ((𝑀 ∈ Mnd ∧ (1 ∈ ℕ0 ∧ 𝑛 ∈ ℕ0 ∧ 𝑋 ∈ 𝐵)) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g‘𝑀)(𝑛 · 𝑋)))
61	48, 59, 51, 50, 60	syl13anc 1320	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g‘𝑀)(𝑛 · 𝑋)))
62		1cnd 9935	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 1 ∈ ℂ)
63		simpr3 1062	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0)
64	63	nn0cnd 11230	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → 𝑛 ∈ ℂ)
65	62, 64	addcomd 10117	. . . . . . . . . 10 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ0 ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ0)) → (1 + 𝑛) = (𝑛 + 1))
66	65	3anassrs 1282	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 + 𝑛) = (𝑛 + 1))
67	66	oveq1d 6564	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
68	18, 26	mulg1 17371	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝐵 → (1 · 𝑋) = 𝑋)
69	50, 68	syl 17	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
70	69	oveq1d 6564	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 · 𝑋)(+g‘𝑀)(𝑛 · 𝑋)) = (𝑋(+g‘𝑀)(𝑛 · 𝑋)))
71	61, 67, 70	3eqtr3rd 2653	. . . . . . 7 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑋(+g‘𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
72	56, 58, 71	3brtr3d 4614	. . . . . 6 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))
73	72	adantr 480	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))
74	18, 22	postr 16776	. . . . . 6 ⊢ ((𝑀 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) → (( 0 ≤ (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋)) → 0 ≤ ((𝑛 + 1) · 𝑋)))
75	74	imp 444	. . . . 5 ⊢ (((𝑀 ∈ Poset ∧ ( 0 ∈ 𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) ∧ ( 0 ≤ (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))) → 0 ≤ ((𝑛 + 1) · 𝑋))
76	30, 45, 46, 73, 75	syl22anc 1319	. . . 4 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 ≤ (𝑛 · 𝑋)) → 0 ≤ ((𝑛 + 1) · 𝑋))
77	7, 9, 11, 13, 29, 76	nn0indd 11350	. . 3 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) ∧ 𝑁 ∈ ℕ0) → 0 ≤ (𝑁 · 𝑋))
78	5, 77	mpdan 699	. 2 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))
79	4, 78	sylbi 206	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ0) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  ℕ₀cn0 11169  Basecbs 15695  +_gcplusg 15768  lecple 15775  0_gc0g 15923  Posetcpo 16763  Tosetctos 16856  Mndcmnd 17117  ._gcmg 17363  oMndcomnd 29028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-0g 15925  df-preset 16751  df-poset 16769  df-toset 16857  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mulg 17364  df-omnd 29030

This theorem is referenced by:  omndmul3  29044