omndmul2 - Mathbox for Thierry Arnoux

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Theorem omndmul2 32217

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 1089	. . 3 ⊢ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑁 ∈ ℕ₀)) ∧ 0 ≤ 𝑋))
2		anass 469	. . . 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 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 7412	. . . . 5 ⊢ (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
7	6	breq2d 5159	. . . 4 ⊢ (𝑚 = 0 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (0 · 𝑋)))
8		oveq1 7412	. . . . 5 ⊢ (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
9	8	breq2d 5159	. . . 4 ⊢ (𝑚 = 𝑛 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑛 · 𝑋)))
10		oveq1 7412	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
11	10	breq2d 5159	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ ((𝑛 + 1) · 𝑋)))
12		oveq1 7412	. . . . 5 ⊢ (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
13	12	breq2d 5159	. . . 4 ⊢ (𝑚 = 𝑁 → ( 0 ≤ (𝑚 · 𝑋) ↔ 0 ≤ (𝑁 · 𝑋)))
14		omndtos 32210	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15		tospos 18369	. . . . . . . 8 ⊢ (𝑀 ∈ Toset → 𝑀 ∈ Poset)
16	14, 15	syl 17	. . . . . . 7 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17		omndmnd 32209	. . . . . . . 8 ⊢ (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18		omndmul.0	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑀)
19		omndmul2.3	. . . . . . . . 9 ⊢ 0 = (0_g‘𝑀)
20	18, 19	mndidcl 18636	. . . . . . . 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 18267	. . . . . . 7 ⊢ ((𝑀 ∈ Poset ∧ 0 ∈ 𝐵) → 0 ≤ 0 )
24	16, 21, 23	syl2anc 584	. . . . . 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 18951	. . . . . 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 5175	. . . 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 18969	. . . . . 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 12484	. . . . . . . . . . 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 12532	. . . . . . . . 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 18969	. . . . . 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 485	. . . . 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 485	. . . . . . . . 9 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → 𝑛 ∈ ℕ₀)
50	18, 26, 46, 49, 48	mulgnn0cld 18969	. . . . . . . 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 2732	. . . . . . . . 9 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
53	18, 22, 52	omndadd 32211	. . . . . . . 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 1383	. . . . . . 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 18641	. . . . . . . 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 18978	. . . . . . . . 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 1372	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+_g‘𝑀)(𝑛 · 𝑋)))
60		1cnd 11205	. . . . . . . . . . 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 12530	. . . . . . . . . . 11 ⊢ (((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ (𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝑋 ∧ 𝑛 ∈ ℕ₀)) → 𝑛 ∈ ℂ)
63	60, 62	addcomd 11412	. . . . . . . . . 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 7420	. . . . . . . 8 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
66	18, 26	mulg1 18955	. . . . . . . . . 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 7420	. . . . . . . 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 5178	. . . . . 6 ⊢ (((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) → (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))
71	70	adantr 481	. . . . 5 ⊢ ((((((𝑀 ∈ oMnd ∧ 𝑋 ∈ 𝐵) ∧ 𝑁 ∈ ℕ₀) ∧ 0 ≤ 𝑋) ∧ 𝑛 ∈ ℕ₀) ∧ 0 ≤ (𝑛 · 𝑋)) → (𝑛 · 𝑋) ≤ ((𝑛 + 1) · 𝑋))
72	18, 22	postr 18269	. . . . . 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 407	. . . . 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 12655	. . 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 + caddc 11109 ℕ₀cn0 12468 Basecbs 17140 +_gcplusg 17193 lecple 17200 0_gc0g 17381 Posetcpo 18256 Tosetctos 18365 Mndcmnd 18621 ._gcmg 18944 oMndcomnd 32202

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-seq 13963 df-0g 17383 df-proset 18244 df-poset 18262 df-toset 18366 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mulg 18945 df-omnd 32204

This theorem is referenced by: omndmul3 32218

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