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Theorem omndmul2 26108
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 962 . . 3  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 )  /\  .0.  .<_  X )  <->  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 ) )  /\  .0.  .<_  X ) )
2 df-3an 962 . . . . 5  |-  ( ( M  e. oMnd  /\  X  e.  B  /\  N  e. 
NN0 )  <->  ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 ) )
3 3anass 964 . . . . 5  |-  ( ( M  e. oMnd  /\  X  e.  B  /\  N  e. 
NN0 )  <->  ( M  e. oMnd  /\  ( X  e.  B  /\  N  e. 
NN0 ) ) )
42, 3bitr3i 251 . . . 4  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 ) 
<->  ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 ) ) )
54anbi1i 690 . . 3  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  <->  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e. 
NN0 ) )  /\  .0.  .<_  X ) )
61, 5bitr4i 252 . 2  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0 )  /\  .0.  .<_  X )  <->  ( (
( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X ) )
7 simplr 749 . . 3  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  N  e.  NN0 )
8 oveq1 6097 . . . . 5  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
98breq2d 4301 . . . 4  |-  ( m  =  0  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( 0 
.x.  X ) ) )
10 oveq1 6097 . . . . 5  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
1110breq2d 4301 . . . 4  |-  ( m  =  n  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( n 
.x.  X ) ) )
12 oveq1 6097 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
1312breq2d 4301 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( ( n  +  1 ) 
.x.  X ) ) )
14 oveq1 6097 . . . . 5  |-  ( m  =  N  ->  (
m  .x.  X )  =  ( N  .x.  X ) )
1514breq2d 4301 . . . 4  |-  ( m  =  N  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( N 
.x.  X ) ) )
16 omndtos 26101 . . . . . . . 8  |-  ( M  e. oMnd  ->  M  e. Toset )
17 tospos 26052 . . . . . . . 8  |-  ( M  e. Toset  ->  M  e.  Poset )
1816, 17syl 16 . . . . . . 7  |-  ( M  e. oMnd  ->  M  e.  Poset )
19 omndmnd 26100 . . . . . . . 8  |-  ( M  e. oMnd  ->  M  e.  Mnd )
20 omndmul.0 . . . . . . . . 9  |-  B  =  ( Base `  M
)
21 omndmul2.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  M )
2220, 21mndidcl 15435 . . . . . . . 8  |-  ( M  e.  Mnd  ->  .0.  e.  B )
2319, 22syl 16 . . . . . . 7  |-  ( M  e. oMnd  ->  .0.  e.  B
)
24 omndmul.1 . . . . . . . 8  |-  .<_  =  ( le `  M )
2520, 24posref 15117 . . . . . . 7  |-  ( ( M  e.  Poset  /\  .0.  e.  B )  ->  .0.  .<_  .0.  )
2618, 23, 25syl2anc 656 . . . . . 6  |-  ( M  e. oMnd  ->  .0.  .<_  .0.  )
2726ad3antrrr 724 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  .0.  )
28 omndmul2.2 . . . . . . 7  |-  .x.  =  (.g
`  M )
2920, 21, 28mulg0 15625 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
3029ad3antlr 725 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  ( 0 
.x.  X )  =  .0.  )
3127, 30breqtrrd 4315 . . . 4  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( 0 
.x.  X ) )
3218ad5antr 728 . . . . 5  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  M  e.  Poset
)
3319ad5antr 728 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  M  e.  Mnd )
3433, 22syl 16 . . . . . 6  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  .0.  e.  B )
35 simplr 749 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  n  e.  NN0 )
36 simp-5r 763 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  X  e.  B )
3720, 28mulgnn0cl 15636 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
3833, 35, 36, 37syl3anc 1213 . . . . . 6  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  ( n  .x.  X )  e.  B
)
39 simpr32 1074 . . . . . . . . . 10  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  ->  n  e.  NN0 )
40 1nn0 10591 . . . . . . . . . . 11  |-  1  e.  NN0
4140a1i 11 . . . . . . . . . 10  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  -> 
1  e.  NN0 )
4239, 41nn0addcld 10636 . . . . . . . . 9  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  -> 
( n  +  1 )  e.  NN0 )
43423anassrs 1204 . . . . . . . 8  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  (  .0. 
.<_  X  /\  n  e. 
NN0  /\  .0.  .<_  ( n 
.x.  X ) ) )  ->  ( n  +  1 )  e. 
NN0 )
44433anassrs 1204 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  ( n  +  1 )  e. 
NN0 )
4520, 28mulgnn0cl 15636 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( n  +  1
)  e.  NN0  /\  X  e.  B )  ->  ( ( n  + 
1 )  .x.  X
)  e.  B )
4633, 44, 36, 45syl3anc 1213 . . . . . 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 )
4734, 38, 463jca 1163 . . . . 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 ) )
48 simpr 458 . . . . . 6  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  .0.  .<_  ( n 
.x.  X ) )
49 simp-4l 760 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  M  e. oMnd )
5019ad4antr 726 . . . . . . . . . 10  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  M  e.  Mnd )
5150, 22syl 16 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  .0.  e.  B )
52 simp-4r 761 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  X  e.  B )
53 simpr 458 . . . . . . . . . 10  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
5450, 53, 52, 37syl3anc 1213 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
n  .x.  X )  e.  B )
55 simplr 749 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  .0.  .<_  X )
56 eqid 2441 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
5720, 24, 56omndadd 26102 . . . . . . . . 9  |-  ( ( 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 ) ) )
5849, 51, 52, 54, 55, 57syl131anc 1226 . . . . . . . 8  |-  ( ( ( ( ( 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 ) ) )
5920, 56, 21mndlid 15437 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( n  .x.  X )  e.  B )  -> 
(  .0.  ( +g  `  M ) ( n 
.x.  X ) )  =  ( n  .x.  X ) )
6050, 54, 59syl2anc 656 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (  .0.  ( +g  `  M
) ( n  .x.  X ) )  =  ( n  .x.  X
) )
6140a1i 11 . . . . . . . . . 10  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  1  e.  NN0 )
6220, 28, 56mulgnn0dir 15643 . . . . . . . . . 10  |-  ( ( M  e.  Mnd  /\  ( 1  e.  NN0  /\  n  e.  NN0  /\  X  e.  B )
)  ->  ( (
1  +  n ) 
.x.  X )  =  ( ( 1  .x. 
X ) ( +g  `  M ) ( n 
.x.  X ) ) )
6350, 61, 53, 52, 62syl13anc 1215 . . . . . . . . 9  |-  ( ( ( ( ( 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
) ) )
64 ax-1cn 9336 . . . . . . . . . . . . 13  |-  1  e.  CC
6564a1i 11 . . . . . . . . . . . 12  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  -> 
1  e.  CC )
66 simpr3 991 . . . . . . . . . . . . 13  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  ->  n  e.  NN0 )
6766nn0cnd 10634 . . . . . . . . . . . 12  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  ->  n  e.  CC )
6865, 67addcomd 9567 . . . . . . . . . . 11  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  -> 
( 1  +  n
)  =  ( n  +  1 ) )
69683anassrs 1204 . . . . . . . . . 10  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
1  +  n )  =  ( n  + 
1 ) )
7069oveq1d 6105 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
( 1  +  n
)  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
7120, 28mulg1 15627 . . . . . . . . . . 11  |-  ( X  e.  B  ->  (
1  .x.  X )  =  X )
7252, 71syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
1  .x.  X )  =  X )
7372oveq1d 6105 . . . . . . . . 9  |-  ( ( ( ( ( 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 ) ) )
7463, 70, 733eqtr3rd 2482 . . . . . . . 8  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  ( X ( +g  `  M
) ( n  .x.  X ) )  =  ( ( n  + 
1 )  .x.  X
) )
7558, 60, 743brtr3d 4318 . . . . . . 7  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
n  .x.  X )  .<_  ( ( n  + 
1 )  .x.  X
) )
7675adantr 462 . . . . . 6  |-  ( ( ( ( ( ( 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 ) )
7748, 76jca 529 . . . . 5  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  (  .0.  .<_  ( n  .x.  X )  /\  ( n  .x.  X )  .<_  ( ( n  +  1 ) 
.x.  X ) ) )
7820, 24postr 15119 . . . . . 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 ) ) )
7978imp 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 ) )
8032, 47, 77, 79syl21anc 1212 . . . 4  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  .0.  .<_  ( ( n  +  1 ) 
.x.  X ) )
819, 11, 13, 15, 31, 80nn0indd 26021 . . 3  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  N  e.  NN0 )  ->  .0.  .<_  ( N  .x.  X ) )
827, 81mpdan 663 . 2  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( N 
.x.  X ) )
836, 82sylbi 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 960    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   CCcc 9276   0cc0 9278   1c1 9279    + caddc 9281   NN0cn0 10575   Basecbs 14170   +g cplusg 14234   lecple 14241   0gc0g 14374   Posetcpo 15106  Tosetctos 15199   Mndcmnd 15405  .gcmg 15410  oMndcomnd 26093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-1st 6576  df-2nd 6577  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-n0 10576  df-z 10643  df-uz 10858  df-fz 11434  df-seq 11803  df-0g 14376  df-poset 15112  df-toset 15200  df-mnd 15411  df-mulg 15541  df-omnd 26095
This theorem is referenced by:  omndmul3  26109
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