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

Theorem omndmul2 26315
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 967 . . 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 967 . . . . 5  |-  ( ( M  e. oMnd  /\  X  e.  B  /\  N  e. 
NN0 )  <->  ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 ) )
3 3anass 969 . . . . 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 695 . . 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 754 . . 3  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  N  e.  NN0 )
8 oveq1 6202 . . . . 5  |-  ( m  =  0  ->  (
m  .x.  X )  =  ( 0  .x. 
X ) )
98breq2d 4407 . . . 4  |-  ( m  =  0  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( 0 
.x.  X ) ) )
10 oveq1 6202 . . . . 5  |-  ( m  =  n  ->  (
m  .x.  X )  =  ( n  .x.  X ) )
1110breq2d 4407 . . . 4  |-  ( m  =  n  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( n 
.x.  X ) ) )
12 oveq1 6202 . . . . 5  |-  ( m  =  ( n  + 
1 )  ->  (
m  .x.  X )  =  ( ( n  +  1 )  .x.  X ) )
1312breq2d 4407 . . . 4  |-  ( m  =  ( n  + 
1 )  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( ( n  +  1 ) 
.x.  X ) ) )
14 oveq1 6202 . . . . 5  |-  ( m  =  N  ->  (
m  .x.  X )  =  ( N  .x.  X ) )
1514breq2d 4407 . . . 4  |-  ( m  =  N  ->  (  .0.  .<_  ( m  .x.  X )  <->  .0.  .<_  ( N 
.x.  X ) ) )
16 omndtos 26308 . . . . . . . 8  |-  ( M  e. oMnd  ->  M  e. Toset )
17 tospos 26259 . . . . . . . 8  |-  ( M  e. Toset  ->  M  e.  Poset )
1816, 17syl 16 . . . . . . 7  |-  ( M  e. oMnd  ->  M  e.  Poset )
19 omndmnd 26307 . . . . . . . 8  |-  ( M  e. oMnd  ->  M  e.  Mnd )
20 omndmul.0 . . . . . . . . 9  |-  B  =  ( Base `  M
)
21 omndmul2.3 . . . . . . . . 9  |-  .0.  =  ( 0g `  M )
2220, 21mndidcl 15553 . . . . . . . 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 15235 . . . . . . 7  |-  ( ( M  e.  Poset  /\  .0.  e.  B )  ->  .0.  .<_  .0.  )
2618, 23, 25syl2anc 661 . . . . . 6  |-  ( M  e. oMnd  ->  .0.  .<_  .0.  )
2726ad3antrrr 729 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  .0.  )
28 omndmul2.2 . . . . . . 7  |-  .x.  =  (.g
`  M )
2920, 21, 28mulg0 15746 . . . . . 6  |-  ( X  e.  B  ->  (
0  .x.  X )  =  .0.  )
3029ad3antlr 730 . . . . 5  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  ( 0 
.x.  X )  =  .0.  )
3127, 30breqtrrd 4421 . . . 4  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  .0.  .<_  X )  ->  .0.  .<_  ( 0 
.x.  X ) )
3218ad5antr 733 . . . . 5  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  M  e.  Poset
)
3319ad5antr 733 . . . . . . 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 754 . . . . . . 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 768 . . . . . . 7  |-  ( ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  /\  .0.  .<_  ( n  .x.  X ) )  ->  X  e.  B )
3720, 28mulgnn0cl 15757 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  n  e.  NN0  /\  X  e.  B )  ->  (
n  .x.  X )  e.  B )
3833, 35, 36, 37syl3anc 1219 . . . . . 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 1079 . . . . . . . . . 10  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  ->  n  e.  NN0 )
40 1nn0 10701 . . . . . . . . . . 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 10746 . . . . . . . . 9  |-  ( ( M  e. oMnd  /\  ( X  e.  B  /\  N  e.  NN0  /\  (  .0.  .<_  X  /\  n  e.  NN0  /\  .0.  .<_  ( n  .x.  X ) ) ) )  -> 
( n  +  1 )  e.  NN0 )
43423anassrs 1210 . . . . . . . 8  |-  ( ( ( ( M  e. oMnd  /\  X  e.  B
)  /\  N  e.  NN0 )  /\  (  .0. 
.<_  X  /\  n  e. 
NN0  /\  .0.  .<_  ( n 
.x.  X ) ) )  ->  ( n  +  1 )  e. 
NN0 )
44433anassrs 1210 . . . . . . 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 15757 . . . . . . 7  |-  ( ( M  e.  Mnd  /\  ( n  +  1
)  e.  NN0  /\  X  e.  B )  ->  ( ( n  + 
1 )  .x.  X
)  e.  B )
4633, 44, 36, 45syl3anc 1219 . . . . . 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 1168 . . . . 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 461 . . . . . 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 765 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  M  e. oMnd )
5019ad4antr 731 . . . . . . . . . 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 766 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  X  e.  B )
53 simpr 461 . . . . . . . . . 10  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  n  e.  NN0 )
5450, 53, 52, 37syl3anc 1219 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
n  .x.  X )  e.  B )
55 simplr 754 . . . . . . . . 9  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  .0.  .<_  X )
56 eqid 2452 . . . . . . . . . 10  |-  ( +g  `  M )  =  ( +g  `  M )
5720, 24, 56omndadd 26309 . . . . . . . . 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 1232 . . . . . . . 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 15555 . . . . . . . . 9  |-  ( ( M  e.  Mnd  /\  ( n  .x.  X )  e.  B )  -> 
(  .0.  ( +g  `  M ) ( n 
.x.  X ) )  =  ( n  .x.  X ) )
6050, 54, 59syl2anc 661 . . . . . . . 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 15764 . . . . . . . . . 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 1221 . . . . . . . . 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 9446 . . . . . . . . . . . . 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 996 . . . . . . . . . . . . 13  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  ->  n  e.  NN0 )
6766nn0cnd 10744 . . . . . . . . . . . 12  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  ->  n  e.  CC )
6865, 67addcomd 9677 . . . . . . . . . . 11  |-  ( ( ( M  e. oMnd  /\  X  e.  B )  /\  ( N  e.  NN0  /\  .0.  .<_  X  /\  n  e.  NN0 ) )  -> 
( 1  +  n
)  =  ( n  +  1 ) )
69683anassrs 1210 . . . . . . . . . 10  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
1  +  n )  =  ( n  + 
1 ) )
7069oveq1d 6210 . . . . . . . . 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 15748 . . . . . . . . . . 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 6210 . . . . . . . . 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 2502 . . . . . . . 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 4424 . . . . . . 7  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  n  e.  NN0 )  ->  (
n  .x.  X )  .<_  ( ( n  + 
1 )  .x.  X
) )
7675adantr 465 . . . . . 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 532 . . . . 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 15237 . . . . . 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 1218 . . . 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 26228 . . 3  |-  ( ( ( ( ( M  e. oMnd  /\  X  e.  B )  /\  N  e.  NN0 )  /\  .0.  .<_  X )  /\  N  e.  NN0 )  ->  .0.  .<_  ( N  .x.  X ) )
827, 81mpdan 668 . 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4395   ` cfv 5521  (class class class)co 6195   CCcc 9386   0cc0 9388   1c1 9389    + caddc 9391   NN0cn0 10685   Basecbs 14287   +g cplusg 14352   lecple 14359   0gc0g 14492   Posetcpo 15224  Tosetctos 15317   Mndcmnd 15523  .gcmg 15528  oMndcomnd 26300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-inf2 7953  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-mulcom 9452  ax-addass 9453  ax-mulass 9454  ax-distr 9455  ax-i2m1 9456  ax-1ne0 9457  ax-1rid 9458  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463  ax-pre-ltadd 9464  ax-pre-mulgt0 9465
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-pss 3447  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4195  df-iun 4276  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4489  df-eprel 4735  df-id 4739  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-ord 4825  df-on 4826  df-lim 4827  df-suc 4828  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-riota 6156  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-om 6582  df-1st 6682  df-2nd 6683  df-recs 6937  df-rdg 6971  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529  df-le 9530  df-sub 9703  df-neg 9704  df-nn 10429  df-n0 10686  df-z 10753  df-uz 10968  df-fz 11550  df-seq 11919  df-0g 14494  df-poset 15230  df-toset 15318  df-mnd 15529  df-mulg 15662  df-omnd 26302
This theorem is referenced by:  omndmul3  26316
  Copyright terms: Public domain W3C validator