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Theorem omndmul2 30052
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 1073 . . 3 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0) ∧ 0 𝑋) ↔ ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)) ∧ 0 𝑋))
2 anass 459 . . . 4 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ↔ (𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0)))
32anbi1i 610 . . 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 752 . . 3 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 𝑁 ∈ ℕ0)
6 oveq1 6800 . . . . 5 (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
76breq2d 4798 . . . 4 (𝑚 = 0 → ( 0 (𝑚 · 𝑋) ↔ 0 (0 · 𝑋)))
8 oveq1 6800 . . . . 5 (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
98breq2d 4798 . . . 4 (𝑚 = 𝑛 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑛 · 𝑋)))
10 oveq1 6800 . . . . 5 (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
1110breq2d 4798 . . . 4 (𝑚 = (𝑛 + 1) → ( 0 (𝑚 · 𝑋) ↔ 0 ((𝑛 + 1) · 𝑋)))
12 oveq1 6800 . . . . 5 (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
1312breq2d 4798 . . . 4 (𝑚 = 𝑁 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑁 · 𝑋)))
14 omndtos 30045 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15 tospos 29998 . . . . . . . 8 (𝑀 ∈ Toset → 𝑀 ∈ Poset)
1614, 15syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17 omndmnd 30044 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18 omndmul.0 . . . . . . . . 9 𝐵 = (Base‘𝑀)
19 omndmul2.3 . . . . . . . . 9 0 = (0g𝑀)
2018, 19mndidcl 17516 . . . . . . . 8 (𝑀 ∈ Mnd → 0𝐵)
2117, 20syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 0𝐵)
22 omndmul.1 . . . . . . . 8 = (le‘𝑀)
2318, 22posref 17159 . . . . . . 7 ((𝑀 ∈ Poset ∧ 0𝐵) → 0 0 )
2416, 21, 23syl2anc 573 . . . . . 6 (𝑀 ∈ oMnd → 0 0 )
2524ad3antrrr 709 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 0 )
26 omndmul2.2 . . . . . . 7 · = (.g𝑀)
2718, 19, 26mulg0 17754 . . . . . 6 (𝑋𝐵 → (0 · 𝑋) = 0 )
2827ad3antlr 710 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → (0 · 𝑋) = 0 )
2925, 28breqtrrd 4814 . . . 4 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → 0 (0 · 𝑋))
3016ad5antr 716 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑀 ∈ Poset)
3117ad5antr 716 . . . . . . 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 752 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑛 ∈ ℕ0)
34 simp-5r 774 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 𝑋𝐵)
3518, 26mulgnn0cl 17766 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑋𝐵) → (𝑛 · 𝑋) ∈ 𝐵)
3631, 33, 34, 35syl3anc 1476 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
37 simpr32 1346 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ0)
38 1nn0 11510 . . . . . . . . . . 11 1 ∈ ℕ0
3938a1i 11 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 1 ∈ ℕ0)
4037, 39nn0addcld 11557 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ0)
41403anassrs 1453 . . . . . . . 8 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ0)
42413anassrs 1453 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ0)
4318, 26mulgnn0cl 17766 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑛 + 1) ∈ ℕ0𝑋𝐵) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4431, 42, 34, 43syl3anc 1476 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4532, 36, 443jca 1122 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵))
46 simpr 471 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 (𝑛 · 𝑋))
47 simp-4l 768 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑀 ∈ oMnd)
4817ad4antr 712 . . . . . . . . 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 770 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑋𝐵)
51 simpr 471 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
5248, 51, 50, 35syl3anc 1476 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ∈ 𝐵)
53 simplr 752 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → 0 𝑋)
54 eqid 2771 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
5518, 22, 54omndadd 30046 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5647, 49, 50, 52, 53, 55syl131anc 1489 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5718, 54, 19mndlid 17519 . . . . . . . 8 ((𝑀 ∈ Mnd ∧ (𝑛 · 𝑋) ∈ 𝐵) → ( 0 (+g𝑀)(𝑛 · 𝑋)) = (𝑛 · 𝑋))
5848, 52, 57syl2anc 573 . . . . . . 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 17779 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (1 ∈ ℕ0𝑛 ∈ ℕ0𝑋𝐵)) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
6148, 59, 51, 50, 60syl13anc 1478 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
62 1cnd 10258 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 1 ∈ ℂ)
63 simpr3 1237 . . . . . . . . . . . 12 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0)
6463nn0cnd 11555 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℂ)
6562, 64addcomd 10440 . . . . . . . . . 10 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → (1 + 𝑛) = (𝑛 + 1))
66653anassrs 1453 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 + 𝑛) = (𝑛 + 1))
6766oveq1d 6808 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
6818, 26mulg1 17756 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
6950, 68syl 17 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
7069oveq1d 6808 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)) = (𝑋(+g𝑀)(𝑛 · 𝑋)))
7161, 67, 703eqtr3rd 2814 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑋(+g𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
7256, 58, 713brtr3d 4817 . . . . . 6 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))
7372adantr 466 . . . . 5 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋))
7418, 22postr 17161 . . . . . 6 ((𝑀 ∈ Poset ∧ ( 0𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵 ∧ ((𝑛 + 1) · 𝑋) ∈ 𝐵)) → (( 0 (𝑛 · 𝑋) ∧ (𝑛 · 𝑋) ((𝑛 + 1) · 𝑋)) → 0 ((𝑛 + 1) · 𝑋)))
7574imp 393 . . . . 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 1477 . . . 4 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 ((𝑛 + 1) · 𝑋))
777, 9, 11, 13, 29, 76nn0indd 11676 . . 3 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑁 ∈ ℕ0) → 0 (𝑁 · 𝑋))
785, 77mpdan 667 . 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 382  w3a 1071   = wceq 1631  wcel 2145   class class class wbr 4786  cfv 6031  (class class class)co 6793  0cc0 10138  1c1 10139   + caddc 10141  0cn0 11494  Basecbs 16064  +gcplusg 16149  lecple 16156  0gc0g 16308  Posetcpo 17148  Tosetctos 17241  Mndcmnd 17502  .gcmg 17748  oMndcomnd 30037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-seq 13009  df-0g 16310  df-preset 17136  df-poset 17154  df-toset 17242  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-mulg 17749  df-omnd 30039
This theorem is referenced by:  omndmul3  30053
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