Mathbox for Thierry Arnoux < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  omndmul2 Structured version   Visualization version   GIF version

Theorem omndmul2 30369
 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 1082 . . 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 7030 . . . . 5 (𝑚 = 0 → (𝑚 · 𝑋) = (0 · 𝑋))
76breq2d 4980 . . . 4 (𝑚 = 0 → ( 0 (𝑚 · 𝑋) ↔ 0 (0 · 𝑋)))
8 oveq1 7030 . . . . 5 (𝑚 = 𝑛 → (𝑚 · 𝑋) = (𝑛 · 𝑋))
98breq2d 4980 . . . 4 (𝑚 = 𝑛 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑛 · 𝑋)))
10 oveq1 7030 . . . . 5 (𝑚 = (𝑛 + 1) → (𝑚 · 𝑋) = ((𝑛 + 1) · 𝑋))
1110breq2d 4980 . . . 4 (𝑚 = (𝑛 + 1) → ( 0 (𝑚 · 𝑋) ↔ 0 ((𝑛 + 1) · 𝑋)))
12 oveq1 7030 . . . . 5 (𝑚 = 𝑁 → (𝑚 · 𝑋) = (𝑁 · 𝑋))
1312breq2d 4980 . . . 4 (𝑚 = 𝑁 → ( 0 (𝑚 · 𝑋) ↔ 0 (𝑁 · 𝑋)))
14 omndtos 30362 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Toset)
15 tospos 30315 . . . . . . . 8 (𝑀 ∈ Toset → 𝑀 ∈ Poset)
1614, 15syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 𝑀 ∈ Poset)
17 omndmnd 30361 . . . . . . . 8 (𝑀 ∈ oMnd → 𝑀 ∈ Mnd)
18 omndmul.0 . . . . . . . . 9 𝐵 = (Base‘𝑀)
19 omndmul2.3 . . . . . . . . 9 0 = (0g𝑀)
2018, 19mndidcl 17751 . . . . . . . 8 (𝑀 ∈ Mnd → 0𝐵)
2117, 20syl 17 . . . . . . 7 (𝑀 ∈ oMnd → 0𝐵)
22 omndmul.1 . . . . . . . 8 = (le‘𝑀)
2318, 22posref 17394 . . . . . . 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 17992 . . . . . 6 (𝑋𝐵 → (0 · 𝑋) = 0 )
2827ad3antlr 727 . . . . 5 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) → (0 · 𝑋) = 0 )
2925, 28breqtrrd 4996 . . . 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 18003 . . . . . . 7 ((𝑀 ∈ Mnd ∧ 𝑛 ∈ ℕ0𝑋𝐵) → (𝑛 · 𝑋) ∈ 𝐵)
3631, 33, 34, 35syl3anc 1364 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 · 𝑋) ∈ 𝐵)
37 simpr32 1257 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 𝑛 ∈ ℕ0)
38 1nn0 11767 . . . . . . . . . . 11 1 ∈ ℕ0
3938a1i 11 . . . . . . . . . 10 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → 1 ∈ ℕ0)
4037, 39nn0addcld 11813 . . . . . . . . 9 ((𝑀 ∈ oMnd ∧ (𝑋𝐵𝑁 ∈ ℕ0 ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋)))) → (𝑛 + 1) ∈ ℕ0)
41403anassrs 1353 . . . . . . . 8 ((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ ( 0 𝑋𝑛 ∈ ℕ00 (𝑛 · 𝑋))) → (𝑛 + 1) ∈ ℕ0)
42413anassrs 1353 . . . . . . 7 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → (𝑛 + 1) ∈ ℕ0)
4318, 26mulgnn0cl 18003 . . . . . . 7 ((𝑀 ∈ Mnd ∧ (𝑛 + 1) ∈ ℕ0𝑋𝐵) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4431, 42, 34, 43syl3anc 1364 . . . . . 6 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → ((𝑛 + 1) · 𝑋) ∈ 𝐵)
4532, 36, 443jca 1121 . . . . 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 1364 . . . . . . . 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 2797 . . . . . . . . 9 (+g𝑀) = (+g𝑀)
5518, 22, 54omndadd 30363 . . . . . . . 8 ((𝑀 ∈ oMnd ∧ ( 0𝐵𝑋𝐵 ∧ (𝑛 · 𝑋) ∈ 𝐵) ∧ 0 𝑋) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5647, 49, 50, 52, 53, 55syl131anc 1376 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ( 0 (+g𝑀)(𝑛 · 𝑋)) (𝑋(+g𝑀)(𝑛 · 𝑋)))
5718, 54, 19mndlid 17754 . . . . . . . 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 18015 . . . . . . . . 9 ((𝑀 ∈ Mnd ∧ (1 ∈ ℕ0𝑛 ∈ ℕ0𝑋𝐵)) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
6148, 59, 51, 50, 60syl13anc 1365 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)))
62 1cnd 10489 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 1 ∈ ℂ)
63 simpr3 1189 . . . . . . . . . . . 12 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℕ0)
6463nn0cnd 11811 . . . . . . . . . . 11 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → 𝑛 ∈ ℂ)
6562, 64addcomd 10695 . . . . . . . . . 10 (((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ (𝑁 ∈ ℕ00 𝑋𝑛 ∈ ℕ0)) → (1 + 𝑛) = (𝑛 + 1))
66653anassrs 1353 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 + 𝑛) = (𝑛 + 1))
6766oveq1d 7038 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 + 𝑛) · 𝑋) = ((𝑛 + 1) · 𝑋))
6818, 26mulg1 17994 . . . . . . . . . 10 (𝑋𝐵 → (1 · 𝑋) = 𝑋)
6950, 68syl 17 . . . . . . . . 9 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (1 · 𝑋) = 𝑋)
7069oveq1d 7038 . . . . . . . 8 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → ((1 · 𝑋)(+g𝑀)(𝑛 · 𝑋)) = (𝑋(+g𝑀)(𝑛 · 𝑋)))
7161, 67, 703eqtr3rd 2842 . . . . . . 7 (((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) → (𝑋(+g𝑀)(𝑛 · 𝑋)) = ((𝑛 + 1) · 𝑋))
7256, 58, 713brtr3d 4999 . . . . . 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 17396 . . . . . 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 835 . . . 4 ((((((𝑀 ∈ oMnd ∧ 𝑋𝐵) ∧ 𝑁 ∈ ℕ0) ∧ 0 𝑋) ∧ 𝑛 ∈ ℕ0) ∧ 0 (𝑛 · 𝑋)) → 0 ((𝑛 + 1) · 𝑋))
777, 9, 11, 13, 29, 76nn0indd 11933 . . 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 1080   = wceq 1525   ∈ wcel 2083   class class class wbr 4968  ‘cfv 6232  (class class class)co 7023  0cc0 10390  1c1 10391   + caddc 10393  ℕ0cn0 11751  Basecbs 16316  +gcplusg 16398  lecple 16405  0gc0g 16546  Posetcpo 17383  Tosetctos 17476  Mndcmnd 17737  .gcmg 17985  oMndcomnd 30354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-cnex 10446  ax-resscn 10447  ax-1cn 10448  ax-icn 10449  ax-addcl 10450  ax-addrcl 10451  ax-mulcl 10452  ax-mulrcl 10453  ax-mulcom 10454  ax-addass 10455  ax-mulass 10456  ax-distr 10457  ax-i2m1 10458  ax-1ne0 10459  ax-1rid 10460  ax-rnegex 10461  ax-rrecex 10462  ax-cnre 10463  ax-pre-lttri 10464  ax-pre-lttrn 10465  ax-pre-ltadd 10466  ax-pre-mulgt0 10467 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-pred 6030  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-riota 6984  df-ov 7026  df-oprab 7027  df-mpo 7028  df-om 7444  df-1st 7552  df-2nd 7553  df-wrecs 7805  df-recs 7867  df-rdg 7905  df-er 8146  df-en 8365  df-dom 8366  df-sdom 8367  df-pnf 10530  df-mnf 10531  df-xr 10532  df-ltxr 10533  df-le 10534  df-sub 10725  df-neg 10726  df-nn 11493  df-n0 11752  df-z 11836  df-uz 12098  df-fz 12747  df-seq 13224  df-0g 16548  df-proset 17371  df-poset 17389  df-toset 17477  df-mgm 17685  df-sgrp 17727  df-mnd 17738  df-mulg 17986  df-omnd 30356 This theorem is referenced by:  omndmul3  30370
 Copyright terms: Public domain W3C validator