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Theorem xrsmulgzz 27903
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 6277 . . . 4  |-  ( n  =  0  ->  (
n (.g `  RR*s ) B )  =  ( 0 (.g `  RR*s ) B ) )
2 oveq1 6277 . . . 4  |-  ( n  =  0  ->  (
n xe B )  =  ( 0 xe B ) )
31, 2eqeq12d 2476 . . 3  |-  ( n  =  0  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( 0 (.g `  RR*s ) B )  =  ( 0 xe B ) ) )
4 oveq1 6277 . . . 4  |-  ( n  =  m  ->  (
n (.g `  RR*s ) B )  =  ( m (.g `  RR*s ) B ) )
5 oveq1 6277 . . . 4  |-  ( n  =  m  ->  (
n xe B )  =  ( m xe B ) )
64, 5eqeq12d 2476 . . 3  |-  ( n  =  m  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( m
(.g `  RR*s ) B )  =  ( m xe B ) ) )
7 oveq1 6277 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  (
n (.g `  RR*s ) B )  =  ( ( m  +  1 ) (.g `  RR*s ) B ) )
8 oveq1 6277 . . . 4  |-  ( n  =  ( m  + 
1 )  ->  (
n xe B )  =  ( ( m  +  1 ) xe B ) )
97, 8eqeq12d 2476 . . 3  |-  ( n  =  ( m  + 
1 )  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( (
m  +  1 ) (.g `  RR*s ) B )  =  ( ( m  +  1 ) xe B ) ) )
10 oveq1 6277 . . . 4  |-  ( n  =  -u m  ->  (
n (.g `  RR*s ) B )  =  ( -u m (.g `  RR*s ) B ) )
11 oveq1 6277 . . . 4  |-  ( n  =  -u m  ->  (
n xe B )  =  ( -u m xe B ) )
1210, 11eqeq12d 2476 . . 3  |-  ( n  =  -u m  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( -u m
(.g `  RR*s ) B )  =  ( -u m xe B ) ) )
13 oveq1 6277 . . . 4  |-  ( n  =  A  ->  (
n (.g `  RR*s ) B )  =  ( A (.g `  RR*s ) B ) )
14 oveq1 6277 . . . 4  |-  ( n  =  A  ->  (
n xe B )  =  ( A xe B ) )
1513, 14eqeq12d 2476 . . 3  |-  ( n  =  A  ->  (
( n (.g `  RR*s
) B )  =  ( n xe B )  <->  ( A
(.g `  RR*s ) B )  =  ( A xe B ) ) )
16 xrsbas 18632 . . . . 5  |-  RR*  =  ( Base `  RR*s )
17 xrs0 27900 . . . . 5  |-  0  =  ( 0g `  RR*s )
18 eqid 2454 . . . . 5  |-  (.g `  RR*s
)  =  (.g `  RR*s
)
1916, 17, 18mulg0 16349 . . . 4  |-  ( B  e.  RR*  ->  ( 0 (.g `  RR*s ) B )  =  0 )
20 xmul02 11463 . . . 4  |-  ( B  e.  RR*  ->  ( 0 xe B )  =  0 )
2119, 20eqtr4d 2498 . . 3  |-  ( B  e.  RR*  ->  ( 0 (.g `  RR*s ) B )  =  ( 0 xe B ) )
22 simpr 459 . . . . . 6  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( m (.g `  RR*s
) B )  =  ( m xe B ) )
2322oveq1d 6285 . . . . 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 459 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  e.  NN )  ->  m  e.  NN )
25 simpll 751 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  e.  NN )  ->  B  e.  RR* )
26 xrsadd 18633 . . . . . . . . 9  |-  +e 
=  ( +g  `  RR*s
)
2716, 18, 26mulgnnp1 16352 . . . . . . . 8  |-  ( ( m  e.  NN  /\  B  e.  RR* )  -> 
( ( m  + 
1 ) (.g `  RR*s
) B )  =  ( ( m (.g ` 
RR*s ) B ) +e B ) )
2824, 25, 27syl2anc 659 . . . . . . 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 459 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  =  0
)  ->  m  = 
0 )
30 simpll 751 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  =  0
)  ->  B  e.  RR* )
31 xaddid2 11442 . . . . . . . . . 10  |-  ( B  e.  RR*  ->  ( 0 +e B )  =  B )
3231adantl 464 . . . . . . . . 9  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( 0 +e
B )  =  B )
33 simpl 455 . . . . . . . . . . . 12  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  m  =  0 )
3433oveq1d 6285 . . . . . . . . . . 11  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( m (.g `  RR*s
) B )  =  ( 0 (.g `  RR*s
) B ) )
3519adantl 464 . . . . . . . . . . 11  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( 0 (.g `  RR*s
) B )  =  0 )
3634, 35eqtrd 2495 . . . . . . . . . 10  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( m (.g `  RR*s
) B )  =  0 )
3736oveq1d 6285 . . . . . . . . 9  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( ( m (.g ` 
RR*s ) B ) +e B )  =  ( 0 +e B ) )
3833oveq1d 6285 . . . . . . . . . . . 12  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( m  +  1 )  =  ( 0  +  1 ) )
39 0p1e1 10643 . . . . . . . . . . . 12  |-  ( 0  +  1 )  =  1
4038, 39syl6eq 2511 . . . . . . . . . . 11  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( m  +  1 )  =  1 )
4140oveq1d 6285 . . . . . . . . . 10  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( ( m  + 
1 ) (.g `  RR*s
) B )  =  ( 1 (.g `  RR*s
) B ) )
4216, 18mulg1 16351 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( 1 (.g `  RR*s ) B )  =  B )
4342adantl 464 . . . . . . . . . 10  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( 1 (.g `  RR*s
) B )  =  B )
4441, 43eqtrd 2495 . . . . . . . . 9  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( ( m  + 
1 ) (.g `  RR*s
) B )  =  B )
4532, 37, 443eqtr4rd 2506 . . . . . . . 8  |-  ( ( m  =  0  /\  B  e.  RR* )  ->  ( ( m  + 
1 ) (.g `  RR*s
) B )  =  ( ( m (.g ` 
RR*s ) B ) +e B ) )
4629, 30, 45syl2anc 659 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  m  =  0
)  ->  ( (
m  +  1 ) (.g `  RR*s ) B )  =  ( ( m (.g `  RR*s ) B ) +e B ) )
47 simpr 459 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN0 )  ->  m  e.  NN0 )
48 elnn0 10793 . . . . . . . 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 463 . . . . 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 10795 . . . . . . . . 9  |-  NN0  C_  RR
53 ressxr 9626 . . . . . . . . 9  |-  RR  C_  RR*
5452, 53sstri 3498 . . . . . . . 8  |-  NN0  C_  RR*
5547adantr 463 . . . . . . . 8  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  ->  m  e.  NN0 )
5654, 55sseldi 3487 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  ->  m  e.  RR* )
57 nn0ge0 10817 . . . . . . . 8  |-  ( m  e.  NN0  ->  0  <_  m )
5857ad2antlr 724 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
0  <_  m )
59 1re 9584 . . . . . . . . 9  |-  1  e.  RR
6059rexri 9635 . . . . . . . 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 10072 . . . . . . . 8  |-  0  <_  1
6362a1i 11 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
0  <_  1 )
64 simpll 751 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  ->  B  e.  RR* )
65 xadddi2r 11493 . . . . . . 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 1237 . . . . . 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 3487 . . . . . . . 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 11434 . . . . . . . 8  |-  ( ( m  e.  RR  /\  1  e.  RR )  ->  ( m +e 1 )  =  ( m  +  1 ) )
7067, 68, 69syl2anc 659 . . . . . . 7  |-  ( ( ( B  e.  RR*  /\  m  e.  NN0 )  /\  ( m (.g `  RR*s
) B )  =  ( m xe B ) )  -> 
( m +e 1 )  =  ( m  +  1 ) )
7170oveq1d 6285 . . . . . 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 11475 . . . . . . . 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 6286 . . . . . 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 2503 . . . . 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 2505 . . . 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 602 . . 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 11409 . . . . . 6  |-  ( ( m (.g `  RR*s ) B )  =  ( m xe B )  ->  -e ( m (.g `  RR*s ) B )  =  -e
( m xe B ) )
7978adantl 464 . . . . 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 2454 . . . . . . . . 9  |-  ( invg `  RR*s
)  =  ( invg `  RR*s
)
8116, 18, 80mulgnegnn 16354 . . . . . . . 8  |-  ( ( m  e.  NN  /\  B  e.  RR* )  -> 
( -u m (.g `  RR*s
) B )  =  ( ( invg `  RR*s ) `  ( m (.g `  RR*s
) B ) ) )
8281ancoms 451 . . . . . . 7  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  ( -u m (.g `  RR*s ) B )  =  ( ( invg `  RR*s
) `  ( m
(.g `  RR*s ) B ) ) )
83 xrsex 18631 . . . . . . . . . . . 12  |-  RR*s 
e.  _V
8483a1i 11 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  RR*s 
e.  _V )
85 ssid 3508 . . . . . . . . . . . 12  |-  RR*  C_  RR*
8685a1i 11 . . . . . . . . . . 11  |-  ( m  e.  NN  ->  RR*  C_  RR* )
87 simp2 995 . . . . . . . . . . . 12  |-  ( ( m  e.  NN  /\  x  e.  RR*  /\  y  e.  RR* )  ->  x  e.  RR* )
88 simp3 996 . . . . . . . . . . . 12  |-  ( ( m  e.  NN  /\  x  e.  RR*  /\  y  e.  RR* )  ->  y  e.  RR* )
8987, 88xaddcld 11496 . . . . . . . . . . 11  |-  ( ( m  e.  NN  /\  x  e.  RR*  /\  y  e.  RR* )  ->  (
x +e y )  e.  RR* )
9016, 18, 26, 84, 86, 89mulgnnsubcl 16356 . . . . . . . . . 10  |-  ( ( m  e.  NN  /\  m  e.  NN  /\  B  e.  RR* )  ->  (
m (.g `  RR*s ) B )  e.  RR* )
91903anidm12 1283 . . . . . . . . 9  |-  ( ( m  e.  NN  /\  B  e.  RR* )  -> 
( m (.g `  RR*s
) B )  e. 
RR* )
9291ancoms 451 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  (
m (.g `  RR*s ) B )  e.  RR* )
93 xrsinvgval 27902 . . . . . . . 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 2495 . . . . . 6  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  ( -u m (.g `  RR*s ) B )  =  -e
( m (.g `  RR*s
) B ) )
9695adantr 463 . . . . 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 10538 . . . . . . . . . 10  |-  ( m  e.  NN  ->  m  e.  RR )
9897adantl 464 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  m  e.  RR )
99 rexneg 11413 . . . . . . . . 9  |-  ( m  e.  RR  ->  -e
m  =  -u m
)
10098, 99syl 16 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  -e
m  =  -u m
)
101100oveq1d 6285 . . . . . . 7  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  (  -e m xe B )  =  (
-u m xe B ) )
102 nnssre 10535 . . . . . . . . . 10  |-  NN  C_  RR
103102, 53sstri 3498 . . . . . . . . 9  |-  NN  C_  RR*
104 simpr 459 . . . . . . . . 9  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  m  e.  NN )
105103, 104sseldi 3487 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  m  e.  RR* )
106 simpl 455 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  B  e.  RR* )
107 xmulneg1 11464 . . . . . . . 8  |-  ( ( m  e.  RR*  /\  B  e.  RR* )  ->  (  -e m xe B )  =  -e ( m xe B ) )
108105, 106, 107syl2anc 659 . . . . . . 7  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  (  -e m xe B )  =  -e ( m xe B ) )
109101, 108eqtr3d 2497 . . . . . 6  |-  ( ( B  e.  RR*  /\  m  e.  NN )  ->  ( -u m xe B )  =  -e
( m xe B ) )
110109adantr 463 . . . . 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 2505 . . . 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 602 . . 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 10961 . 2  |-  ( B  e.  RR*  ->  ( A  e.  ZZ  ->  ( A (.g `  RR*s ) B )  =  ( A xe B ) ) )
114113impcom 428 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 366    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   _Vcvv 3106    C_ wss 3461   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484   RR*cxr 9616    <_ cle 9618   -ucneg 9797   NNcn 10531   NN0cn0 10791   ZZcz 10860    -ecxne 11318   +ecxad 11319   xecxmu 11320   RR*scxrs 14992   invgcminusg 16256  .gcmg 16258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-fz 11676  df-seq 12093  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-plusg 14800  df-mulr 14801  df-tset 14806  df-ple 14807  df-ds 14809  df-0g 14934  df-xrs 14994  df-minusg 16260  df-mulg 16262
This theorem is referenced by:  xrge0mulgnn0  27914  pnfinf  27964
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