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Theorem omndmul2 27862
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  |-  .x.  =  (.g
`  M )
omndmul2.3  |-  .0.  =  ( 0g `  M )
Assertion
Ref Expression
omndmul2  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( N  .x.  X ) )

Proof of Theorem omndmul2
Dummy variables  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-3an 975 . . 3  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 )  /\  .0.  .<_  X )  <->  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 ) )  /\  .0.  .<_  X ) )
2 anass 649 . . . 4  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 ) 
<->  ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 ) ) )
32anbi1i 695 . . 3  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  <->  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e. 
NN0 ) )  /\  .0.  .<_  X ) )
41, 3bitr4i 252 . 2  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 )  /\  .0.  .<_  X )  <->  ( (
( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X ) )
5 simplr 755 . . 3  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  N  e.  NN0 )
6 oveq1 6303 . . . . 5  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
76breq2d 4468 . . . 4  |-  ( m  =  0  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( 0 
.x.  X ) ) )
8 oveq1 6303 . . . . 5  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
98breq2d 4468 . . . 4  |-  ( m  =  n  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( n 
.x.  X ) ) )
10 oveq1 6303 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
1110breq2d 4468 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( ( n  +  1 ) 
.x.  X ) ) )
12 oveq1 6303 . . . . 5  |-  ( m  =  N  ->  (
m  .x.  X )  =  ( N  .x.  X ) )
1312breq2d 4468 . . . 4  |-  ( m  =  N  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( N 
.x.  X ) ) )
14 omndtos 27855 . . . . . . . 8  |-  ( M  e. oMnd  ->  M  e. Toset )
15 tospos 27806 . . . . . . . 8  |-  ( M  e. Toset  ->  M  e.  Poset )
1614, 15syl 16 . . . . . . 7  |-  ( M  e. oMnd  ->  M  e.  Poset )
17 omndmnd 27854 . . . . . . . 8  |-  ( M  e. oMnd  ->  M  e.  Mnd )
18 omndmul.0 . . . . . . . . 9  |-  B  =  ( Base `  M
)
19 omndmul2.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  M )
2018, 19mndidcl 16065 . . . . . . . 8  |-  ( M  e.  Mnd  ->  .0.  e.  B )
2117, 20syl 16 . . . . . . 7  |-  ( M  e. oMnd  ->  .0.  e.  B
)
22 omndmul.1 . . . . . . . 8  |-  .<_  =  ( le `  M )
2318, 22posref 15707 . . . . . . 7  |-  ( ( M  e.  Poset  /\  .0.  e.  B )  ->  .0.  .<_  .0.  )
2416, 21, 23syl2anc 661 . . . . . 6  |-  ( M  e. oMnd  ->  .0.  .<_  .0.  )
2524ad3antrrr 729 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  .0.  )
26 omndmul2.2 . . . . . . 7  |-  .x.  =  (.g
`  M )
2718, 19, 26mulg0 16274 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
2827ad3antlr 730 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  ( 0 
.x.  X )  =  .0.  )
2925, 28breqtrrd 4482 . . . 4  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( 0 
.x.  X ) )
3016ad5antr 733 . . . . 5  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  M  e.  Poset
)
3117ad5antr 733 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  M  e.  Mnd )
3231, 20syl 16 . . . . . 6  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  .0.  e.  B )
33 simplr 755 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  n  e.  NN0 )
34 simp-5r 770 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  X  e.  B )
3518, 26mulgnn0cl 16285 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
3631, 33, 34, 35syl3anc 1228 . . . . . 6  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  ( n  .x.  X )  e.  B
)
37 simpr32 1087 . . . . . . . . . 10  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  ->  n  e.  NN0 )
38 1nn0 10832 . . . . . . . . . . 11  |-  1  e.  NN0
3938a1i 11 . . . . . . . . . 10  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  -> 
1  e.  NN0 )
4037, 39nn0addcld 10877 . . . . . . . . 9  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  -> 
( n  +  1 )  e.  NN0 )
41403anassrs 1218 . . . . . . . 8  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  (  .0. 
.<_  X  /\  n  e. 
NN0  /\  .0.  .<_  ( n 
.x.  X ) ) )  ->  ( n  +  1 )  e. 
NN0 )
42413anassrs 1218 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  ( n  +  1 )  e. 
NN0 )
4318, 26mulgnn0cl 16285 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( n  +  1
)  e.  NN0  /\  X  e.  B )  ->  ( ( n  + 
1 )  .x.  X
)  e.  B )
4431, 42, 34, 43syl3anc 1228 . . . . . 6  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  ( (
n  +  1 ) 
.x.  X )  e.  B )
4532, 36, 443jca 1176 . . . . 5  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  (  .0.  e.  B  /\  (
n  .x.  X )  e.  B  /\  (
( n  +  1 )  .x.  X )  e.  B ) )
46 simpr 461 . . . . 5  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  .0.  .<_  ( n 
.x.  X ) )
47 simp-4l 767 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  M  e. oMnd )
4817ad4antr 731 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  M  e.  Mnd )
4948, 20syl 16 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  .0.  e.  B )
50 simp-4r 768 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  X  e.  B )
51 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
5248, 51, 50, 35syl3anc 1228 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
n  .x.  X )  e.  B )
53 simplr 755 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  .0.  .<_  X )
54 eqid 2457 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
5518, 22, 54omndadd 27856 . . . . . . . 8  |-  ( ( M  e. oMnd  /\  (  .0.  e.  B  /\  X  e.  B  /\  (
n  .x.  X )  e.  B )  /\  .0.  .<_  X )  ->  (  .0.  ( +g  `  M
) ( n  .x.  X ) )  .<_  ( X ( +g  `  M
) ( n  .x.  X ) ) )
5647, 49, 50, 52, 53, 55syl131anc 1241 . . . . . . 7  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (  .0.  ( +g  `  M
) ( n  .x.  X ) )  .<_  ( X ( +g  `  M
) ( n  .x.  X ) ) )
5718, 54, 19mndlid 16068 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  ( n  .x.  X )  e.  B )  -> 
(  .0.  ( +g  `  M ) ( n 
.x.  X ) )  =  ( n  .x.  X ) )
5848, 52, 57syl2anc 661 . . . . . . 7  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (  .0.  ( +g  `  M
) ( n  .x.  X ) )  =  ( n  .x.  X
) )
5938a1i 11 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  1  e.  NN0 )
6018, 26, 54mulgnn0dir 16292 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( 1  e.  NN0  /\  n  e.  NN0  /\  X  e.  B )
)  ->  ( (
1  +  n ) 
.x.  X )  =  ( ( 1  .x. 
X ) ( +g  `  M ) ( n 
.x.  X ) ) )
6148, 59, 51, 50, 60syl13anc 1230 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
( 1  +  n
)  .x.  X )  =  ( ( 1 
.x.  X ) ( +g  `  M ) ( n  .x.  X
) ) )
62 1cnd 9629 . . . . . . . . . . 11  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  -> 
1  e.  CC )
63 simpr3 1004 . . . . . . . . . . . 12  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  ->  n  e.  NN0 )
6463nn0cnd 10875 . . . . . . . . . . 11  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  ->  n  e.  CC )
6562, 64addcomd 9799 . . . . . . . . . 10  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  -> 
( 1  +  n
)  =  ( n  +  1 ) )
66653anassrs 1218 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
1  +  n )  =  ( n  + 
1 ) )
6766oveq1d 6311 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
( 1  +  n
)  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
6818, 26mulg1 16276 . . . . . . . . . 10  |-  ( X  e.  B  ->  (
1  .x.  X )  =  X )
6950, 68syl 16 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
1  .x.  X )  =  X )
7069oveq1d 6311 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
( 1  .x.  X
) ( +g  `  M
) ( n  .x.  X ) )  =  ( X ( +g  `  M ) ( n 
.x.  X ) ) )
7161, 67, 703eqtr3rd 2507 . . . . . . 7  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  ( X ( +g  `  M
) ( n  .x.  X ) )  =  ( ( n  + 
1 )  .x.  X
) )
7256, 58, 713brtr3d 4485 . . . . . 6  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
n  .x.  X )  .<_  ( ( n  + 
1 )  .x.  X
) )
7372adantr 465 . . . . 5  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  ( n  .x.  X )  .<_  ( ( n  +  1 ) 
.x.  X ) )
7418, 22postr 15710 . . . . . 6  |-  ( ( M  e.  Poset  /\  (  .0.  e.  B  /\  (
n  .x.  X )  e.  B  /\  (
( n  +  1 )  .x.  X )  e.  B ) )  ->  ( (  .0. 
.<_  ( n  .x.  X
)  /\  ( n  .x.  X )  .<_  ( ( n  +  1 ) 
.x.  X ) )  ->  .0.  .<_  ( ( n  +  1 ) 
.x.  X ) ) )
7574imp 429 . . . . 5  |-  ( ( ( M  e.  Poset  /\  (  .0.  e.  B  /\  ( n  .x.  X
)  e.  B  /\  ( ( n  + 
1 )  .x.  X
)  e.  B ) )  /\  (  .0. 
.<_  ( n  .x.  X
)  /\  ( n  .x.  X )  .<_  ( ( n  +  1 ) 
.x.  X ) ) )  ->  .0.  .<_  ( ( n  +  1 ) 
.x.  X ) )
7630, 45, 46, 73, 75syl22anc 1229 . . . 4  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  .0.  .<_  ( ( n  +  1 ) 
.x.  X ) )
777, 9, 11, 13, 29, 76nn0indd 10982 . . 3  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  N  e.  NN0 )  ->  .0.  .<_  ( N  .x.  X ) )
785, 77mpdan 668 . 2  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( N 
.x.  X ) )
794, 78sylbi 195 1  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( N  .x.  X ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   NN0cn0 10816   Basecbs 14644   +g cplusg 14712   lecple 14719   0gc0g 14857   Posetcpo 15696  Tosetctos 15790   Mndcmnd 16046  .gcmg 16183  oMndcomnd 27847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12111  df-0g 14859  df-preset 15684  df-poset 15702  df-toset 15791  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mulg 16187  df-omnd 27849
This theorem is referenced by:  omndmul3  27863
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