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Theorem omndmul2 30042
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 1074 . . 3 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)) ∧ 0 𝑋))
2 anass 684 . . . 4 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ↔ (𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)))
32anbi1i 733 . . 3 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)) ∧ 0 𝑋))
41, 3bitr4i 267 . 2 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋))
5 simplr 809 . . 3 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 𝑁 ∈ ℕ0)
6 oveq1 6821 . . . . 5 (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
76breq2d 4816 . . . 4 (𝑚 = 0 → ( 0 (𝑚 · 𝑋) ↔ 0 (0 · 𝑋)))
8 oveq1 6821 . . . . 5 (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
98breq2d 4816 . . . 4 (𝑚 = 𝑛 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑛 · 𝑋)))
10 oveq1 6821 . . . . 5 (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
1110breq2d 4816 . . . 4 (𝑚 = (𝑛 + 1) → ( 0 (𝑚 · 𝑋) ↔ 0 ((𝑛 + 1) · 𝑋)))
12 oveq1 6821 . . . . 5 (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
1312breq2d 4816 . . . 4 (𝑚 = 𝑁 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑁 · 𝑋)))
14 omndtos 30035 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15 tospos 29988 . . . . . . . 8 (𝑀 ∈ Toset → 𝑀 ∈ Poset)
1614, 15syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17 omndmnd 30034 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18 omndmul.0 . . . . . . . . 9 𝐵 = (Base‘𝑀)
19 omndmul2.3 . . . . . . . . 9 0 = (0g𝑀)
2018, 19mndidcl 17529 . . . . . . . 8 (𝑀 ∈ Mnd → 0𝐵)
2117, 20syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 0𝐵)
22 omndmul.1 . . . . . . . 8 = (le‘𝑀)
2318, 22posref 17172 . . . . . . 7 ((𝑀 ∈ Poset ∧ 0𝐵) → 0 0 )
2416, 21, 23syl2anc 696 . . . . . 6 (𝑀 ∈ oMnd → 0 0 )
2524ad3antrrr 768 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 0 )
26 omndmul2.2 . . . . . . 7 · = (.g𝑀)
2718, 19, 26mulg0 17767 . . . . . 6 (𝑋𝐵 → (0 · 𝑋) = 0 )
2827ad3antlr 769 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → (0 · 𝑋) = 0 )
2925, 28breqtrrd 4832 . . . 4 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (0 · 𝑋))
3016ad5antr 775 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑀 ∈ Poset)
3117ad5antr 775 . . . . . . 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 809 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑛 ∈ ℕ0)
34 simp-5r 831 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑋𝐵)
3518, 26mulgnn0cl 17779 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑋𝐵) → (𝑛 · 𝑋) ∈ 𝐵)
3631, 33, 34, 35syl3anc 1477 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
37 simpr32 1347 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ0)
38 1nn0 11520 . . . . . . . . . . 11 1 ∈ ℕ0
3938a1i 11 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 1 ∈ ℕ0)
4037, 39nn0addcld 11567 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ0)
41403anassrs 1454 . . . . . . . 8 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ0)
42413anassrs 1454 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ0)
4318, 26mulgnn0cl 17779 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑛 + 1) ∈ ℕ0𝑋𝐵) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4431, 42, 34, 43syl3anc 1477 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4532, 36, 443jca 1123 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵))
46 simpr 479 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 (𝑛 · 𝑋))
47 simp-4l 825 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ oMnd)
4817ad4antr 771 . . . . . . . . 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 827 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋𝐵)
51 simpr 479 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
5248, 51, 50, 35syl3anc 1477 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ∈ 𝐵)
53 simplr 809 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 𝑋)
54 eqid 2760 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
5518, 22, 54omndadd 30036 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5647, 49, 50, 52, 53, 55syl131anc 1490 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5718, 54, 19mndlid 17532 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑛 · 𝑋) ∈ 𝐵) → ( 0 (+g𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
5848, 52, 57syl2anc 696 . . . . . . 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 17792 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (1 ∈ ℕ0𝑛 ∈ ℕ0𝑋𝐵)) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
6148, 59, 51, 50, 60syl13anc 1479 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
62 1cnd 10268 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 1 ∈ ℂ)
63 simpr3 1238 . . . . . . . . . . . 12 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0)
6463nn0cnd 11565 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℂ)
6562, 64addcomd 10450 . . . . . . . . . 10 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → (1 + 𝑛) = (𝑛 + 1))
66653anassrs 1454 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 + 𝑛) = (𝑛 + 1))
6766oveq1d 6829 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
6818, 26mulg1 17769 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
6950, 68syl 17 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
7069oveq1d 6829 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)) = (𝑋(+g𝑀)(𝑛 · 𝑋)))
7161, 67, 703eqtr3rd 2803 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑋(+g𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
7256, 58, 713brtr3d 4835 . . . . . 6 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))
7372adantr 472 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))
7418, 22postr 17174 . . . . . 6 ((𝑀 ∈ Poset ∧ ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) → (( 0 (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋)) → 0 ((𝑛 + 1) · 𝑋)))
7574imp 444 . . . . 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 1478 . . . 4 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 ((𝑛 + 1) · 𝑋))
777, 9, 11, 13, 29, 76nn0indd 11686 . . 3 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑁 ∈ ℕ0) → 0 (𝑁 · 𝑋))
785, 77mpdan 705 . 2 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))
794, 78sylbi 207 1 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (𝑁 · 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072   = wceq 1632  wcel 2139   class class class wbr 4804  cfv 6049  (class class class)co 6814  0cc0 10148  1c1 10149   + caddc 10151  0cn0 11504  Basecbs 16079  +gcplusg 16163  lecple 16170  0gc0g 16322  Posetcpo 17161  Tosetctos 17254  Mndcmnd 17515  .gcmg 17761  oMndcomnd 30027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-seq 13016  df-0g 16324  df-preset 17149  df-poset 17167  df-toset 17255  df-mgm 17463  df-sgrp 17505  df-mnd 17516  df-mulg 17762  df-omnd 30029
This theorem is referenced by:  omndmul3  30043
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