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Theorem xrsmulgzz 27532
Description: The "multiple" function in the extended real numbers structure. (Contributed by Thierry Arnoux, 14-Jun-2017.)
Assertion
Ref Expression
xrsmulgzz  |-  ( ( A  e.  ZZ  /\  B  e.  RR* )  -> 
( A (.g `  RR*s
) B )  =  ( A xe B ) )

Proof of Theorem xrsmulgzz
Dummy variables  n  m  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6284 . . . 4  |-  ( n  =  0  ->  (
n (.g `  RR*s ) B )  =  ( 0 (.g `  RR*s ) B ) )
2 oveq1 6284 . . . 4  |-  ( n  =  0  ->  (
n xe B )  =  ( 0 xe B ) )
31, 2eqeq12d 2463 . . 3  |-  ( n  =  0  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( 0 (.g `  RR*s ) B )  =  ( 0 xe B ) ) )
4 oveq1 6284 . . . 4  |-  ( n  =  m  ->  (
n (.g `  RR*s ) B )  =  ( m (.g `  RR*s ) B ) )
5 oveq1 6284 . . . 4  |-  ( n  =  m  ->  (
n xe B )  =  ( m xe B ) )
64, 5eqeq12d 2463 . . 3  |-  ( n  =  m  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( m
(.g `  RR*s ) B )  =  ( m xe B ) ) )
7 oveq1 6284 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  (
n (.g `  RR*s ) B )  =  ( ( m  +  1 ) (.g `  RR*s ) B ) )
8 oveq1 6284 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  (
n xe B )  =  ( ( m  +  1 ) xe B ) )
97, 8eqeq12d 2463 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( (
m  +  1 ) (.g `  RR*s ) B )  =  ( ( m  +  1 ) xe B ) ) )
10 oveq1 6284 . . . 4  |-  ( n  =  -u m  ->  (
n (.g `  RR*s ) B )  =  ( -u m (.g `  RR*s ) B ) )
11 oveq1 6284 . . . 4  |-  ( n  =  -u m  ->  (
n xe B )  =  ( -u m xe B ) )
1210, 11eqeq12d 2463 . . 3  |-  ( n  =  -u m  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( -u m
(.g `  RR*s ) B )  =  ( -u m xe B ) ) )
13 oveq1 6284 . . . 4  |-  ( n  =  A  ->  (
n (.g `  RR*s ) B )  =  ( A (.g `  RR*s ) B ) )
14 oveq1 6284 . . . 4  |-  ( n  =  A  ->  (
n xe B )  =  ( A xe B ) )
1513, 14eqeq12d 2463 . . 3  |-  ( n  =  A  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( A
(.g `  RR*s ) B )  =  ( A xe B ) ) )
16 xrsbas 18302 . . . . 5  |-  RR*  =  ( Base `  RR*s )
17 xrs0 27529 . . . . 5  |-  0  =  ( 0g `  RR*s )
18 eqid 2441 . . . . 5  |-  (.g `  RR*s
)  =  (.g `  RR*s
)
1916, 17, 18mulg0 16016 . . . 4  |-  ( B  e.  RR*  ->  ( 0 (.g `  RR*s ) B )  =  0 )
20 xmul02 11464 . . . 4  |-  ( B  e.  RR*  ->  ( 0 xe B )  =  0 )
2119, 20eqtr4d 2485 . . 3  |-  ( B  e.  RR*  ->  ( 0 (.g `  RR*s ) B )  =  ( 0 xe B ) )
22 simpr 461 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( m (.g `  RR*s
) B )  =  ( m xe B ) )
2322oveq1d 6292 . . . . 5  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( ( m (.g ` 
RR*s ) B ) +e B )  =  ( ( m xe B ) +e B ) )
24 simpr 461 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  e.  NN )  ->  m  e.  NN )
25 simpll 753 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  e.  NN )  ->  B  e.  RR* )
26 xrsadd 18303 . . . . . . . . 9  |-  +e 
=  ( +g  `  RR*s
)
2716, 18, 26mulgnnp1 16019 . . . . . . . 8  |-  ( ( m  e.  NN  /\  B  e.  RR* )  -> 
( ( m  + 
1 ) (.g `  RR*s
) B )  =  ( ( m (.g ` 
RR*s ) B ) +e B ) )
2824, 25, 27syl2anc 661 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  e.  NN )  ->  ( ( m  +  1 ) (.g ` 
RR*s ) B )  =  ( ( m (.g `  RR*s ) B ) +e B ) )
29 simpr 461 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  =  0
)  ->  m  = 
0 )
30 simpll 753 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  =  0
)  ->  B  e.  RR* )
31 xaddid2 11443 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( 0 +e B )  =  B )
3231adantl 466 . . . . . . . . 9  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( 0 +e
B )  =  B )
33 simpl 457 . . . . . . . . . . . 12  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  m  =  0 )
3433oveq1d 6292 . . . . . . . . . . 11  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( m (.g `  RR*s
) B )  =  ( 0 (.g `  RR*s
) B ) )
3519adantl 466 . . . . . . . . . . 11  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( 0 (.g `  RR*s
) B )  =  0 )
3634, 35eqtrd 2482 . . . . . . . . . 10  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( m (.g `  RR*s
) B )  =  0 )
3736oveq1d 6292 . . . . . . . . 9  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( ( m (.g ` 
RR*s ) B ) +e B )  =  ( 0 +e B ) )
3833oveq1d 6292 . . . . . . . . . . . 12  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( m  +  1 )  =  ( 0  +  1 ) )
39 0p1e1 10648 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
4038, 39syl6eq 2498 . . . . . . . . . . 11  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( m  +  1 )  =  1 )
4140oveq1d 6292 . . . . . . . . . 10  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( ( m  + 
1 ) (.g `  RR*s
) B )  =  ( 1 (.g `  RR*s
) B ) )
4216, 18mulg1 16018 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( 1 (.g `  RR*s ) B )  =  B )
4342adantl 466 . . . . . . . . . 10  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( 1 (.g `  RR*s
) B )  =  B )
4441, 43eqtrd 2482 . . . . . . . . 9  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( ( m  + 
1 ) (.g `  RR*s
) B )  =  B )
4532, 37, 443eqtr4rd 2493 . . . . . . . 8  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( ( m  + 
1 ) (.g `  RR*s
) B )  =  ( ( m (.g ` 
RR*s ) B ) +e B ) )
4629, 30, 45syl2anc 661 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  =  0
)  ->  ( (
m  +  1 ) (.g `  RR*s ) B )  =  ( ( m (.g `  RR*s ) B ) +e B ) )
47 simpr 461 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN0 )  ->  m  e.  NN0 )
48 elnn0 10798 . . . . . . . 8  |-  ( m  e.  NN0  <->  ( m  e.  NN  \/  m  =  0 ) )
4947, 48sylib 196 . . . . . . 7  |-  ( ( B  e.  RR*  /\  m  e.  NN0 )  ->  (
m  e.  NN  \/  m  =  0 ) )
5028, 46, 49mpjaodan 784 . . . . . 6  |-  ( ( B  e.  RR*  /\  m  e.  NN0 )  ->  (
( m  +  1 ) (.g `  RR*s ) B )  =  ( ( m (.g `  RR*s ) B ) +e B ) )
5150adantr 465 . . . . 5  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( ( m  + 
1 ) (.g `  RR*s
) B )  =  ( ( m (.g ` 
RR*s ) B ) +e B ) )
52 nn0ssre 10800 . . . . . . . . 9  |-  NN0  C_  RR
53 ressxr 9635 . . . . . . . . 9  |-  RR  C_  RR*
5452, 53sstri 3495 . . . . . . . 8  |-  NN0  C_  RR*
5547adantr 465 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  ->  m  e.  NN0 )
5654, 55sseldi 3484 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  ->  m  e.  RR* )
57 nn0ge0 10822 . . . . . . . 8  |-  ( m  e.  NN0  ->  0  <_  m )
5857ad2antlr 726 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
0  <_  m )
59 1re 9593 . . . . . . . . 9  |-  1  e.  RR
6059rexri 9644 . . . . . . . 8  |-  1  e.  RR*
6160a1i 11 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
1  e.  RR* )
62 0le1 10077 . . . . . . . 8  |-  0  <_  1
6362a1i 11 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
0  <_  1 )
64 simpll 753 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  ->  B  e.  RR* )
65 xadddi2r 11494 . . . . . . 7  |-  ( ( ( m  e.  RR*  /\  0  <_  m )  /\  ( 1  e.  RR*  /\  0  <_  1 )  /\  B  e.  RR* )  ->  ( ( m +e 1 ) xe B )  =  ( ( m xe B ) +e ( 1 xe B ) ) )
6656, 58, 61, 63, 64, 65syl221anc 1238 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( ( m +e 1 ) xe B )  =  ( ( m xe B ) +e ( 1 xe B ) ) )
6752, 55sseldi 3484 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  ->  m  e.  RR )
6859a1i 11 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
1  e.  RR )
69 rexadd 11435 . . . . . . . 8  |-  ( ( m  e.  RR  /\  1  e.  RR )  ->  ( m +e 1 )  =  ( m  +  1 ) )
7067, 68, 69syl2anc 661 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( m +e 1 )  =  ( m  +  1 ) )
7170oveq1d 6292 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( ( m +e 1 ) xe B )  =  ( ( m  + 
1 ) xe B ) )
72 xmulid2 11476 . . . . . . . 8  |-  ( B  e.  RR*  ->  ( 1 xe B )  =  B )
7364, 72syl 16 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( 1 xe B )  =  B )
7473oveq2d 6293 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( ( m xe B ) +e ( 1 xe B ) )  =  ( ( m xe B ) +e B ) )
7566, 71, 743eqtr3d 2490 . . . . 5  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( ( m  + 
1 ) xe B )  =  ( ( m xe B ) +e
B ) )
7623, 51, 753eqtr4d 2492 . . . 4  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( ( m  + 
1 ) (.g `  RR*s
) B )  =  ( ( m  + 
1 ) xe B ) )
7776exp31 604 . . 3  |-  ( B  e.  RR*  ->  ( m  e.  NN0  ->  ( ( m (.g `  RR*s ) B )  =  ( m xe B )  ->  ( ( m  +  1 ) (.g ` 
RR*s ) B )  =  ( ( m  +  1 ) xe B ) ) ) )
78 xnegeq 11410 . . . . . 6  |-  ( ( m (.g `  RR*s ) B )  =  ( m xe B )  ->  -e ( m (.g `  RR*s ) B )  =  -e
( m xe B ) )
7978adantl 466 . . . . 5  |-  ( ( ( B  e.  RR*  /\  m  e.  NN )  /\  ( m (.g ` 
RR*s ) B )  =  ( m xe B ) )  ->  -e ( m (.g `  RR*s ) B )  =  -e
( m xe B ) )
80 eqid 2441 . . . . . . . . 9  |-  ( invg `  RR*s
)  =  ( invg `  RR*s
)
8116, 18, 80mulgnegnn 16021 . . . . . . . 8  |-  ( ( m  e.  NN  /\  B  e.  RR* )  -> 
( -u m (.g `  RR*s
) B )  =  ( ( invg `  RR*s ) `  ( m (.g `  RR*s
) B ) ) )
8281ancoms 453 . . . . . . 7  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  ( -u m (.g `  RR*s ) B )  =  ( ( invg `  RR*s
) `  ( m
(.g `  RR*s ) B ) ) )
83 xrsex 18301 . . . . . . . . . . . 12  |-  RR*s 
e.  _V
8483a1i 11 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  RR*s 
e.  _V )
85 ssid 3505 . . . . . . . . . . . 12  |-  RR*  C_  RR*
8685a1i 11 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  RR*  C_  RR* )
87 simp2 996 . . . . . . . . . . . 12  |-  ( ( m  e.  NN  /\  x  e.  RR*  /\  y  e.  RR* )  ->  x  e.  RR* )
88 simp3 997 . . . . . . . . . . . 12  |-  ( ( m  e.  NN  /\  x  e.  RR*  /\  y  e.  RR* )  ->  y  e.  RR* )
8987, 88xaddcld 11497 . . . . . . . . . . 11  |-  ( ( m  e.  NN  /\  x  e.  RR*  /\  y  e.  RR* )  ->  (
x +e y )  e.  RR* )
9016, 18, 26, 84, 86, 89mulgnnsubcl 16023 . . . . . . . . . 10  |-  ( ( m  e.  NN  /\  m  e.  NN  /\  B  e.  RR* )  ->  (
m (.g `  RR*s ) B )  e.  RR* )
91903anidm12 1284 . . . . . . . . 9  |-  ( ( m  e.  NN  /\  B  e.  RR* )  -> 
( m (.g `  RR*s
) B )  e. 
RR* )
9291ancoms 453 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  (
m (.g `  RR*s ) B )  e.  RR* )
93 xrsinvgval 27531 . . . . . . . 8  |-  ( ( m (.g `  RR*s ) B )  e.  RR*  ->  ( ( invg `  RR*s ) `  (
m (.g `  RR*s ) B ) )  =  -e ( m (.g ` 
RR*s ) B ) )
9492, 93syl 16 . . . . . . 7  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  (
( invg `  RR*s ) `  (
m (.g `  RR*s ) B ) )  =  -e ( m (.g ` 
RR*s ) B ) )
9582, 94eqtrd 2482 . . . . . 6  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  ( -u m (.g `  RR*s ) B )  =  -e
( m (.g `  RR*s
) B ) )
9695adantr 465 . . . . 5  |-  ( ( ( B  e.  RR*  /\  m  e.  NN )  /\  ( m (.g ` 
RR*s ) B )  =  ( m xe B ) )  ->  ( -u m
(.g `  RR*s ) B )  =  -e
( m (.g `  RR*s
) B ) )
97 nnre 10544 . . . . . . . . . 10  |-  ( m  e.  NN  ->  m  e.  RR )
9897adantl 466 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  m  e.  RR )
99 rexneg 11414 . . . . . . . . 9  |-  ( m  e.  RR  ->  -e
m  =  -u m
)
10098, 99syl 16 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  -e
m  =  -u m
)
101100oveq1d 6292 . . . . . . 7  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  (  -e m xe B )  =  (
-u m xe B ) )
102 nnssre 10541 . . . . . . . . . 10  |-  NN  C_  RR
103102, 53sstri 3495 . . . . . . . . 9  |-  NN  C_  RR*
104 simpr 461 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  m  e.  NN )
105103, 104sseldi 3484 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  m  e.  RR* )
106 simpl 457 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  B  e.  RR* )
107 xmulneg1 11465 . . . . . . . 8  |-  ( ( m  e.  RR*  /\  B  e.  RR* )  ->  (  -e m xe B )  =  -e ( m xe B ) )
108105, 106, 107syl2anc 661 . . . . . . 7  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  (  -e m xe B )  =  -e ( m xe B ) )
109101, 108eqtr3d 2484 . . . . . 6  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  ( -u m xe B )  =  -e
( m xe B ) )
110109adantr 465 . . . . 5  |-  ( ( ( B  e.  RR*  /\  m  e.  NN )  /\  ( m (.g ` 
RR*s ) B )  =  ( m xe B ) )  ->  ( -u m xe B )  =  -e ( m xe B ) )
11179, 96, 1103eqtr4d 2492 . . . 4  |-  ( ( ( B  e.  RR*  /\  m  e.  NN )  /\  ( m (.g ` 
RR*s ) B )  =  ( m xe B ) )  ->  ( -u m
(.g `  RR*s ) B )  =  ( -u m xe B ) )
112111exp31 604 . . 3  |-  ( B  e.  RR*  ->  ( m  e.  NN  ->  (
( m (.g `  RR*s
) B )  =  ( m xe B )  ->  ( -u m (.g `  RR*s ) B )  =  ( -u m xe B ) ) ) )
1133, 6, 9, 12, 15, 21, 77, 112zindd 10965 . 2  |-  ( B  e.  RR*  ->  ( A  e.  ZZ  ->  ( A (.g `  RR*s ) B )  =  ( A xe B ) ) )
114113impcom 430 1  |-  ( ( A  e.  ZZ  /\  B  e.  RR* )  -> 
( A (.g `  RR*s
) B )  =  ( A xe B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   _Vcvv 3093    C_ wss 3458   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   RRcr 9489   0cc0 9490   1c1 9491    + caddc 9493   RR*cxr 9625    <_ cle 9627   -ucneg 9806   NNcn 10537   NN0cn0 10796   ZZcz 10865    -ecxne 11319   +ecxad 11320   xecxmu 11321   RR*scxrs 14769   invgcminusg 15923  .gcmg 15925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-inf2 8056  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-10 10603  df-n0 10797  df-z 10866  df-dec 10980  df-uz 11086  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-fz 11677  df-seq 12082  df-struct 14506  df-ndx 14507  df-slot 14508  df-base 14509  df-plusg 14582  df-mulr 14583  df-tset 14588  df-ple 14589  df-ds 14591  df-0g 14711  df-xrs 14771  df-minusg 15927  df-mulg 15929
This theorem is referenced by:  xrge0mulgnn0  27543  pnfinf  27593
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