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Theorem omndmul2 30640
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 · = (.g𝑀)
omndmul2.3 0 = (0g𝑀)
Assertion
Ref Expression
omndmul2 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))

Proof of Theorem omndmul2
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-3an 1081 . . 3 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)) ∧ 0 𝑋))
2 anass 469 . . . 4 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ↔ (𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)))
32anbi1i 623 . . 3 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)) ∧ 0 𝑋))
41, 3bitr4i 279 . 2 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋))
5 simplr 765 . . 3 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 𝑁 ∈ ℕ0)
6 oveq1 7152 . . . . 5 (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
76breq2d 5069 . . . 4 (𝑚 = 0 → ( 0 (𝑚 · 𝑋) ↔ 0 (0 · 𝑋)))
8 oveq1 7152 . . . . 5 (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
98breq2d 5069 . . . 4 (𝑚 = 𝑛 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑛 · 𝑋)))
10 oveq1 7152 . . . . 5 (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
1110breq2d 5069 . . . 4 (𝑚 = (𝑛 + 1) → ( 0 (𝑚 · 𝑋) ↔ 0 ((𝑛 + 1) · 𝑋)))
12 oveq1 7152 . . . . 5 (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
1312breq2d 5069 . . . 4 (𝑚 = 𝑁 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑁 · 𝑋)))
14 omndtos 30633 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15 tospos 30572 . . . . . . . 8 (𝑀 ∈ Toset → 𝑀 ∈ Poset)
1614, 15syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17 omndmnd 30632 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18 omndmul.0 . . . . . . . . 9 𝐵 = (Base‘𝑀)
19 omndmul2.3 . . . . . . . . 9 0 = (0g𝑀)
2018, 19mndidcl 17914 . . . . . . . 8 (𝑀 ∈ Mnd → 0𝐵)
2117, 20syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 0𝐵)
22 omndmul.1 . . . . . . . 8 = (le‘𝑀)
2318, 22posref 17549 . . . . . . 7 ((𝑀 ∈ Poset ∧ 0𝐵) → 0 0 )
2416, 21, 23syl2anc 584 . . . . . 6 (𝑀 ∈ oMnd → 0 0 )
2524ad3antrrr 726 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 0 )
26 omndmul2.2 . . . . . . 7 · = (.g𝑀)
2718, 19, 26mulg0 18169 . . . . . 6 (𝑋𝐵 → (0 · 𝑋) = 0 )
2827ad3antlr 727 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → (0 · 𝑋) = 0 )
2925, 28breqtrrd 5085 . . . 4 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (0 · 𝑋))
3016ad5antr 730 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑀 ∈ Poset)
3117ad5antr 730 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑀 ∈ Mnd)
3231, 20syl 17 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0𝐵)
33 simplr 765 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑛 ∈ ℕ0)
34 simp-5r 782 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑋𝐵)
3518, 26mulgnn0cl 18182 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑋𝐵) → (𝑛 · 𝑋) ∈ 𝐵)
3631, 33, 34, 35syl3anc 1363 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
37 simpr32 1256 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ0)
38 1nn0 11901 . . . . . . . . . . 11 1 ∈ ℕ0
3938a1i 11 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 1 ∈ ℕ0)
4037, 39nn0addcld 11947 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ0)
41403anassrs 1352 . . . . . . . 8 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ0)
42413anassrs 1352 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ0)
4318, 26mulgnn0cl 18182 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑛 + 1) ∈ ℕ0𝑋𝐵) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4431, 42, 34, 43syl3anc 1363 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4532, 36, 443jca 1120 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵))
46 simpr 485 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 (𝑛 · 𝑋))
47 simp-4l 779 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ oMnd)
4817ad4antr 728 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ Mnd)
4948, 20syl 17 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 0𝐵)
50 simp-4r 780 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋𝐵)
51 simpr 485 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
5248, 51, 50, 35syl3anc 1363 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ∈ 𝐵)
53 simplr 765 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 𝑋)
54 eqid 2818 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
5518, 22, 54omndadd 30634 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5647, 49, 50, 52, 53, 55syl131anc 1375 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5718, 54, 19mndlid 17919 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑛 · 𝑋) ∈ 𝐵) → ( 0 (+g𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
5848, 52, 57syl2anc 584 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
5938a1i 11 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 1 ∈ ℕ0)
6018, 26, 54mulgnn0dir 18195 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (1 ∈ ℕ0𝑛 ∈ ℕ0𝑋𝐵)) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
6148, 59, 51, 50, 60syl13anc 1364 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
62 1cnd 10624 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 1 ∈ ℂ)
63 simpr3 1188 . . . . . . . . . . . 12 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0)
6463nn0cnd 11945 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℂ)
6562, 64addcomd 10830 . . . . . . . . . 10 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → (1 + 𝑛) = (𝑛 + 1))
66653anassrs 1352 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 + 𝑛) = (𝑛 + 1))
6766oveq1d 7160 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
6818, 26mulg1 18173 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
6950, 68syl 17 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
7069oveq1d 7160 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)) = (𝑋(+g𝑀)(𝑛 · 𝑋)))
7161, 67, 703eqtr3rd 2862 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑋(+g𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
7256, 58, 713brtr3d 5088 . . . . . 6 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))
7372adantr 481 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))
7418, 22postr 17551 . . . . . 6 ((𝑀 ∈ Poset ∧ ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) → (( 0 (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋)) → 0 ((𝑛 + 1) · 𝑋)))
7574imp 407 . . . . 5 (((𝑀 ∈ Poset ∧ ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) ∧ ( 0 (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))) → 0 ((𝑛 + 1) · 𝑋))
7630, 45, 46, 73, 75syl22anc 834 . . . 4 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 ((𝑛 + 1) · 𝑋))
777, 9, 11, 13, 29, 76nn0indd 12067 . . 3 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑁 ∈ ℕ0) → 0 (𝑁 · 𝑋))
785, 77mpdan 683 . 2 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))
794, 78sylbi 218 1 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105   class class class wbr 5057  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526   + caddc 10528  0cn0 11885  Basecbs 16471  +gcplusg 16553  lecple 16560  0gc0g 16701  Posetcpo 17538  Tosetctos 17631  Mndcmnd 17899  .gcmg 18162  oMndcomnd 30625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-seq 13358  df-0g 16703  df-proset 17526  df-poset 17544  df-toset 17632  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-mulg 18163  df-omnd 30627
This theorem is referenced by:  omndmul3  30641
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