omndmul2 - Mathbox for Thierry Arnoux

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Theorem omndmul2 32809

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 = (0_g‘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 1086	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀)) ∧ 0 ≤ 𝑋))
2		anass 467	. . . 4 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ↔ (𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀)))
3	2	anbi1i 622	. . 3 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀)) ∧ 0 ≤ 𝑋))
4	1, 3	bitr4i 277	. 2 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ↔ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋))
5		simplr 767	. . 3 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) → 𝑁 ∈ ℕ₀)
6		oveq1 7421	. . . . 5 ⊢ (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
7	6	breq2d 5153	. . . 4 ⊢ (𝑚 = 0 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (0 · 𝑋)))
8		oveq1 7421	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
9	8	breq2d 5153	. . . 4 ⊢ (𝑚 = 𝑛 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑛 · 𝑋)))
10		oveq1 7421	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
11	10	breq2d 5153	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ ((𝑛 + 1) · 𝑋)))
12		oveq1 7421	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
13	12	breq2d 5153	. . . 4 ⊢ (𝑚 = 𝑁 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑁 · 𝑋)))
14		omndtos 32802	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15		tospos 18409	. . . . . . . 8 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17		omndmnd 32801	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18		omndmul.0	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
19		omndmul2.3	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑀)
20	18, 19	mndidcl 18706	. . . . . . . 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 18307	. . . . . . 7 ⊢ ((𝑀 ∈ Poset ∧ 0 ∈ 𝐵) → 0 ≤ 0 )
24	16, 21, 23	syl2anc 582	. . . . . 6 ⊢ (𝑀 ∈ oMnd → 0 ≤ 0 )
25	24	ad3antrrr 728	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) → 0 ≤ 0 )
26		omndmul2.2	. . . . . . 7 ⊢ · = (._g‘𝑀)
27	18, 19, 26	mulg0 19032	. . . . . 6 ⊢ (𝑋 ∈ 𝐵 → (0 · 𝑋) = 0 )
28	27	ad3antlr 729	. . . . 5 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) → (0 · 𝑋) = 0 )
29	25, 28	breqtrrd 5169	. . . 4 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) → 0 ≤ (0 · 𝑋))
30	16	ad5antr 732	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑀 ∈ Poset)
31	17	ad5antr 732	. . . . . . 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 767	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑛 ∈ ℕ₀)
34		simp-5r 784	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → 𝑋 ∈ 𝐵)
35	18, 26, 31, 33, 34	mulgnn0cld 19052	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
36		simpr32 1261	. . . . . . . . . 10 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀ ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ₀ ∧ 0 ≤ (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ₀)
37		1nn0 12516	. . . . . . . . . . 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 12564	. . . . . . . . 9 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀ ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ₀ ∧ 0 ≤ (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ₀)
40	39	3anassrs 1357	. . . . . . . 8 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ ( 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ₀ ∧ 0 ≤ (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ₀)
41	40	3anassrs 1357	. . . . . . 7 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ₀)
42	18, 26, 31, 41, 34	mulgnn0cld 19052	. . . . . 6 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
43	32, 35, 42	3jca 1125	. . . . 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 483	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → 0 ≤ (𝑛 · 𝑋))
45		simp-4l 781	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → 𝑀 ∈ oMnd)
46	17	ad4antr 730	. . . . . . . . 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 782	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → 𝑋 ∈ 𝐵)
49		simpr 483	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
50	18, 26, 46, 49, 48	mulgnn0cld 19052	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · 𝑋) ∈ 𝐵)
51		simplr 767	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → 0 ≤ 𝑋)
52		eqid 2725	. . . . . . . . 9 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
53	18, 22, 52	omndadd 32803	. . . . . . . 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 1380	. . . . . . 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 18711	. . . . . . . 8 ⊢ ((𝑀 ∈ Mnd ∧ (𝑛 · 𝑋) ∈ 𝐵) → ( 0 (+_g‘𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
56	46, 50, 55	syl2anc 582	. . . . . . 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 19061	. . . . . . . . 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 1369	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+_g‘𝑀)(𝑛 · 𝑋)))
60		1cnd 11237	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ₀)) → 1 ∈ ℂ)
61		simpr3 1193	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ₀)) → 𝑛 ∈ ℕ₀)
62	61	nn0cnd 12562	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ₀)) → 𝑛 ∈ ℂ)
63	60, 62	addcomd 11444	. . . . . . . . . 10 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ₀)) → (1 + 𝑛) = (𝑛 + 1))
64	63	3anassrs 1357	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → (1 + 𝑛) = (𝑛 + 1))
65	64	oveq1d 7429	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
66	18, 26	mulg1 19038	. . . . . . . . . 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 7429	. . . . . . . 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 2774	. . . . . . 7 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → (𝑋(+_g‘𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
70	54, 56, 69	3brtr3d 5172	. . . . . 6 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))
71	70	adantr 479	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))
72	18, 22	postr 18309	. . . . . 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 405	. . . . 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 837	. . . 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 12687	. . 3 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑁 ∈ ℕ₀) → 0 ≤ (𝑁 · 𝑋))
76	5, 75	mpdan 685	. 2 ⊢ ((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))
77	4, 76	sylbi 216	1 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) → 0 ≤ (𝑁 · 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 class class class wbr 5141 ‘cfv 6541 (class class class)co 7414 0cc0 11136 1c1 11137 + caddc 11139 ℕ₀cn0 12500 Basecbs 17177 +_gcplusg 17230 lecple 17237 0_gc0g 17418 Posetcpo 18296 Tosetctos 18405 Mndcmnd 18691 ._gcmg 19025 oMndcomnd 32794

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5292 ax-nul 5299 ax-pow 5357 ax-pr 5421 ax-un 7736 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3958 df-nul 4317 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 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 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-nn 12241 df-n0 12501 df-z 12587 df-uz 12851 df-fz 13515 df-seq 13997 df-0g 17420 df-proset 18284 df-poset 18302 df-toset 18406 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mulg 19026 df-omnd 32796

This theorem is referenced by: omndmul3 32810

W3C validator