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Theorem omndmul2 28482
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 988 . . 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 659 . . . 4  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 ) 
<->  ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 ) ) )
32anbi1i 706 . . 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 260 . 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 767 . . 3  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  N  e.  NN0 )
6 oveq1 6283 . . . . 5  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
76breq2d 4386 . . . 4  |-  ( m  =  0  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( 0 
.x.  X ) ) )
8 oveq1 6283 . . . . 5  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
98breq2d 4386 . . . 4  |-  ( m  =  n  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( n 
.x.  X ) ) )
10 oveq1 6283 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
1110breq2d 4386 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( ( n  +  1 ) 
.x.  X ) ) )
12 oveq1 6283 . . . . 5  |-  ( m  =  N  ->  (
m  .x.  X )  =  ( N  .x.  X ) )
1312breq2d 4386 . . . 4  |-  ( m  =  N  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( N 
.x.  X ) ) )
14 omndtos 28475 . . . . . . . 8  |-  ( M  e. oMnd  ->  M  e. Toset )
15 tospos 28427 . . . . . . . 8  |-  ( M  e. Toset  ->  M  e.  Poset )
1614, 15syl 17 . . . . . . 7  |-  ( M  e. oMnd  ->  M  e.  Poset )
17 omndmnd 28474 . . . . . . . 8  |-  ( M  e. oMnd  ->  M  e.  Mnd )
18 omndmul.0 . . . . . . . . 9  |-  B  =  ( Base `  M
)
19 omndmul2.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  M )
2018, 19mndidcl 16565 . . . . . . . 8  |-  ( M  e.  Mnd  ->  .0.  e.  B )
2117, 20syl 17 . . . . . . 7  |-  ( M  e. oMnd  ->  .0.  e.  B
)
22 omndmul.1 . . . . . . . 8  |-  .<_  =  ( le `  M )
2318, 22posref 16207 . . . . . . 7  |-  ( ( M  e.  Poset  /\  .0.  e.  B )  ->  .0.  .<_  .0.  )
2416, 21, 23syl2anc 671 . . . . . 6  |-  ( M  e. oMnd  ->  .0.  .<_  .0.  )
2524ad3antrrr 741 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  .0.  )
26 omndmul2.2 . . . . . . 7  |-  .x.  =  (.g
`  M )
2718, 19, 26mulg0 16774 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
2827ad3antlr 742 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  ( 0 
.x.  X )  =  .0.  )
2925, 28breqtrrd 4401 . . . 4  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( 0 
.x.  X ) )
3016ad5antr 745 . . . . 5  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  M  e.  Poset
)
3117ad5antr 745 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  M  e.  Mnd )
3231, 20syl 17 . . . . . 6  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  .0.  e.  B )
33 simplr 767 . . . . . . 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 784 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  X  e.  B )
3518, 26mulgnn0cl 16785 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
3631, 33, 34, 35syl3anc 1271 . . . . . 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 1100 . . . . . . . . . 10  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  ->  n  e.  NN0 )
38 1nn0 10875 . . . . . . . . . . 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 10919 . . . . . . . . 9  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  -> 
( n  +  1 )  e.  NN0 )
41403anassrs 1235 . . . . . . . 8  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  (  .0. 
.<_  X  /\  n  e. 
NN0  /\  .0.  .<_  ( n 
.x.  X ) ) )  ->  ( n  +  1 )  e. 
NN0 )
42413anassrs 1235 . . . . . . 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 16785 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( n  +  1
)  e.  NN0  /\  X  e.  B )  ->  ( ( n  + 
1 )  .x.  X
)  e.  B )
4431, 42, 34, 43syl3anc 1271 . . . . . 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 1189 . . . . 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 467 . . . . 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 781 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  M  e. oMnd )
4817ad4antr 743 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  M  e.  Mnd )
4948, 20syl 17 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  .0.  e.  B )
50 simp-4r 782 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  X  e.  B )
51 simpr 467 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
5248, 51, 50, 35syl3anc 1271 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
n  .x.  X )  e.  B )
53 simplr 767 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  .0.  .<_  X )
54 eqid 2452 . . . . . . . . 9  |-  ( +g  `  M )  =  ( +g  `  M )
5518, 22, 54omndadd 28476 . . . . . . . 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 1284 . . . . . . 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 16568 . . . . . . . 8  |-  ( ( M  e.  Mnd  /\  ( n  .x.  X )  e.  B )  -> 
(  .0.  ( +g  `  M ) ( n 
.x.  X ) )  =  ( n  .x.  X ) )
5848, 52, 57syl2anc 671 . . . . . . 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 16792 . . . . . . . . 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 1273 . . . . . . . 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 9646 . . . . . . . . . . 11  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  -> 
1  e.  CC )
63 simpr3 1017 . . . . . . . . . . . 12  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  ->  n  e.  NN0 )
6463nn0cnd 10917 . . . . . . . . . . 11  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  ->  n  e.  CC )
6562, 64addcomd 9822 . . . . . . . . . 10  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  -> 
( 1  +  n
)  =  ( n  +  1 ) )
66653anassrs 1235 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
1  +  n )  =  ( n  + 
1 ) )
6766oveq1d 6291 . . . . . . . 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 16776 . . . . . . . . . 10  |-  ( X  e.  B  ->  (
1  .x.  X )  =  X )
6950, 68syl 17 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
1  .x.  X )  =  X )
7069oveq1d 6291 . . . . . . . 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 2495 . . . . . . 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 4404 . . . . . 6  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
n  .x.  X )  .<_  ( ( n  + 
1 )  .x.  X
) )
7372adantr 471 . . . . 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 16210 . . . . . 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 435 . . . . 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 1272 . . . 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 11022 . . 3  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  N  e.  NN0 )  ->  .0.  .<_  ( N  .x.  X ) )
785, 77mpdan 679 . 2  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( N 
.x.  X ) )
794, 78sylbi 200 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 375    /\ w3a 986    = wceq 1448    e. wcel 1891   class class class wbr 4374   ` cfv 5561  (class class class)co 6276   0cc0 9526   1c1 9527    + caddc 9529   NN0cn0 10859   Basecbs 15132   +g cplusg 15201   lecple 15208   0gc0g 15349   Posetcpo 16196  Tosetctos 16290   Mndcmnd 16546  .gcmg 16683  oMndcomnd 28467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-rep 4487  ax-sep 4497  ax-nul 4506  ax-pow 4554  ax-pr 4612  ax-un 6571  ax-inf2 8133  ax-cnex 9582  ax-resscn 9583  ax-1cn 9584  ax-icn 9585  ax-addcl 9586  ax-addrcl 9587  ax-mulcl 9588  ax-mulrcl 9589  ax-mulcom 9590  ax-addass 9591  ax-mulass 9592  ax-distr 9593  ax-i2m1 9594  ax-1ne0 9595  ax-1rid 9596  ax-rnegex 9597  ax-rrecex 9598  ax-cnre 9599  ax-pre-lttri 9600  ax-pre-lttrn 9601  ax-pre-ltadd 9602  ax-pre-mulgt0 9603
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3015  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3700  df-if 3850  df-pw 3921  df-sn 3937  df-pr 3939  df-tp 3941  df-op 3943  df-uni 4169  df-iun 4250  df-br 4375  df-opab 4434  df-mpt 4435  df-tr 4470  df-eprel 4723  df-id 4727  df-po 4733  df-so 4734  df-fr 4771  df-we 4773  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-pred 5359  df-ord 5405  df-on 5406  df-lim 5407  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-riota 6238  df-ov 6279  df-oprab 6280  df-mpt2 6281  df-om 6681  df-1st 6781  df-2nd 6782  df-wrecs 7015  df-recs 7077  df-rdg 7115  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9664  df-mnf 9665  df-xr 9666  df-ltxr 9667  df-le 9668  df-sub 9849  df-neg 9850  df-nn 10599  df-n0 10860  df-z 10928  df-uz 11150  df-fz 11776  df-seq 12208  df-0g 15351  df-preset 16184  df-poset 16202  df-toset 16291  df-mgm 16499  df-sgrp 16538  df-mnd 16548  df-mulg 16687  df-omnd 28469
This theorem is referenced by:  omndmul3  28483
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