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Theorem lgsdirnn0 14115
Description: Variation on lgsdir 14103 valid for all  A ,  B but only for positive  N. (The exact location of the failure of this law is for  A  =  0,  B  <  0,  N  =  -u 1 in which case  ( 0  /L -u 1
)  =  1 but  ( B  /L -u 1 )  = 
-u 1.) (Contributed by Mario Carneiro, 28-Apr-2016.)
Assertion
Ref Expression
lgsdirnn0  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )

Proof of Theorem lgsdirnn0
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 oveq1 5876 . . . . . . . . 9  |-  ( x  =  B  ->  (
x  /L N )  =  ( B  /L N ) )
21oveq1d 5884 . . . . . . . 8  |-  ( x  =  B  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( B  /L
N )  x.  (
0  /L N ) ) )
32eqeq2d 2189 . . . . . . 7  |-  ( x  =  B  ->  (
( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  <->  ( 0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) ) )
4 id 19 . . . . . . . . . . . . . . 15  |-  ( x  e.  ZZ  ->  x  e.  ZZ )
5 nn0z 9262 . . . . . . . . . . . . . . 15  |-  ( N  e.  NN0  ->  N  e.  ZZ )
6 lgscl 14082 . . . . . . . . . . . . . . 15  |-  ( ( x  e.  ZZ  /\  N  e.  ZZ )  ->  ( x  /L
N )  e.  ZZ )
74, 5, 6syl2anr 290 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L
N )  e.  ZZ )
87zcnd 9365 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L
N )  e.  CC )
98adantr 276 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
x  /L N )  e.  CC )
109mul01d 8340 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
( x  /L
N )  x.  0 )  =  0 )
11 simpr 110 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
0  /L N )  =  0 )
1211oveq2d 5885 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( x  /L
N )  x.  0 ) )
1310, 12, 113eqtr4rd 2221 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =  0 )  ->  (
0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L N ) ) )
14 0z 9253 . . . . . . . . . . . . . . 15  |-  0  e.  ZZ
155adantr 276 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  N  e.  ZZ )
16 lgsne0 14106 . . . . . . . . . . . . . . 15  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( ( 0  /L N )  =/=  0  <->  ( 0  gcd 
N )  =  1 ) )
1714, 15, 16sylancr 414 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  /L N )  =/=  0  <->  ( 0  gcd 
N )  =  1 ) )
18 gcdcom 11957 . . . . . . . . . . . . . . . . . 18  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
1914, 15, 18sylancr 414 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  ( N  gcd  0 ) )
20 nn0gcdid0 11965 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN0  ->  ( N  gcd  0 )  =  N )
2120adantr 276 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  gcd  0
)  =  N )
2219, 21eqtrd 2210 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  gcd  N
)  =  N )
2322eqeq1d 2186 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  <-> 
N  =  1 ) )
24 lgs1 14112 . . . . . . . . . . . . . . . . 17  |-  ( x  e.  ZZ  ->  (
x  /L 1 )  =  1 )
2524adantl 277 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( x  /L 1 )  =  1 )
26 oveq2 5877 . . . . . . . . . . . . . . . . 17  |-  ( N  =  1  ->  (
x  /L N )  =  ( x  /L 1 ) )
2726eqeq1d 2186 . . . . . . . . . . . . . . . 16  |-  ( N  =  1  ->  (
( x  /L
N )  =  1  <-> 
( x  /L 1 )  =  1 ) )
2825, 27syl5ibrcom 157 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( N  =  1  ->  ( x  /L N )  =  1 ) )
2923, 28sylbid 150 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  gcd 
N )  =  1  ->  ( x  /L N )  =  1 ) )
3017, 29sylbid 150 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  /L N )  =/=  0  ->  ( x  /L N )  =  1 ) )
3130imp 124 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
x  /L N )  =  1 )
3231oveq1d 5884 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( 1  x.  ( 0  /L N ) ) )
335ad2antrr 488 . . . . . . . . . . . . . 14  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  N  e.  ZZ )
34 lgscl 14082 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  ZZ  /\  N  e.  ZZ )  ->  ( 0  /L
N )  e.  ZZ )
3514, 33, 34sylancr 414 . . . . . . . . . . . . 13  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  e.  ZZ )
3635zcnd 9365 . . . . . . . . . . . 12  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  e.  CC )
3736mulid2d 7966 . . . . . . . . . . 11  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
1  x.  ( 0  /L N ) )  =  ( 0  /L N ) )
3832, 37eqtr2d 2211 . . . . . . . . . 10  |-  ( ( ( N  e.  NN0  /\  x  e.  ZZ )  /\  ( 0  /L N )  =/=  0 )  ->  (
0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L N ) ) )
3914, 15, 34sylancr 414 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  /L
N )  e.  ZZ )
40 zdceq 9317 . . . . . . . . . . . 12  |-  ( ( ( 0  /L
N )  e.  ZZ  /\  0  e.  ZZ )  -> DECID 
( 0  /L
N )  =  0 )
4139, 14, 40sylancl 413 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  -> DECID  ( 0  /L N )  =  0 )
42 dcne 2358 . . . . . . . . . . 11  |-  (DECID  ( 0  /L N )  =  0  <->  ( (
0  /L N )  =  0  \/  ( 0  /L
N )  =/=  0
) )
4341, 42sylib 122 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( ( 0  /L N )  =  0  \/  ( 0  /L N )  =/=  0 ) )
4413, 38, 43mpjaodan 798 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  x  e.  ZZ )  ->  ( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) ) )
4544ralrimiva 2550 . . . . . . . 8  |-  ( N  e.  NN0  ->  A. x  e.  ZZ  ( 0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L
N ) ) )
46453ad2ant3 1020 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A. x  e.  ZZ  ( 0  /L N )  =  ( ( x  /L N )  x.  ( 0  /L
N ) ) )
47 simp2 998 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  ZZ )
483, 46, 47rspcdva 2846 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) )
4948adantr 276 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  =  ( ( B  /L N )  x.  ( 0  /L
N ) ) )
5053ad2ant3 1020 . . . . . . . . 9  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  N  e.  ZZ )
5114, 50, 34sylancr 414 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  e.  ZZ )
5251zcnd 9365 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  e.  CC )
5352adantr 276 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  e.  CC )
54 lgscl 14082 . . . . . . . . 9  |-  ( ( B  e.  ZZ  /\  N  e.  ZZ )  ->  ( B  /L
N )  e.  ZZ )
5547, 50, 54syl2anc 411 . . . . . . . 8  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  /L N )  e.  ZZ )
5655zcnd 9365 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( B  /L N )  e.  CC )
5756adantr 276 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( B  /L N )  e.  CC )
5853, 57mulcomd 7969 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( 0  /L N )  x.  ( B  /L N ) )  =  ( ( B  /L N )  x.  ( 0  /L N ) ) )
5949, 58eqtr4d 2213 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( 0  /L N )  =  ( ( 0  /L N )  x.  ( B  /L
N ) ) )
60 oveq1 5876 . . . . . 6  |-  ( A  =  0  ->  ( A  x.  B )  =  ( 0  x.  B ) )
61 zcn 9247 . . . . . . . 8  |-  ( B  e.  ZZ  ->  B  e.  CC )
62613ad2ant2 1019 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  B  e.  CC )
6362mul02d 8339 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  x.  B )  =  0 )
6460, 63sylan9eqr 2232 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  x.  B )  =  0 )
6564oveq1d 5884 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( 0  /L
N ) )
66 simpr 110 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  A  =  0 )
6766oveq1d 5884 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( A  /L N )  =  ( 0  /L
N ) )
6867oveq1d 5884 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  /L N )  x.  ( B  /L N ) )  =  ( ( 0  /L N )  x.  ( B  /L N ) ) )
6959, 65, 683eqtr4d 2220 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  A  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( ( A  /L N )  x.  ( B  /L
N ) ) )
70 oveq1 5876 . . . . . . . 8  |-  ( x  =  A  ->  (
x  /L N )  =  ( A  /L N ) )
7170oveq1d 5884 . . . . . . 7  |-  ( x  =  A  ->  (
( x  /L
N )  x.  (
0  /L N ) )  =  ( ( A  /L
N )  x.  (
0  /L N ) ) )
7271eqeq2d 2189 . . . . . 6  |-  ( x  =  A  ->  (
( 0  /L
N )  =  ( ( x  /L
N )  x.  (
0  /L N ) )  <->  ( 0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) ) )
73 simp1 997 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  ZZ )
7472, 46, 73rspcdva 2846 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) )
7574adantr 276 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( 0  /L N )  =  ( ( A  /L N )  x.  ( 0  /L
N ) ) )
76 oveq2 5877 . . . . . 6  |-  ( B  =  0  ->  ( A  x.  B )  =  ( A  x.  0 ) )
7773zcnd 9365 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  A  e.  CC )
7877mul01d 8340 . . . . . 6  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  ( A  x.  0 )  =  0 )
7976, 78sylan9eqr 2232 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( A  x.  B )  =  0 )
8079oveq1d 5884 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( 0  /L
N ) )
81 simpr 110 . . . . . 6  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  B  =  0 )
8281oveq1d 5884 . . . . 5  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( B  /L N )  =  ( 0  /L
N ) )
8382oveq2d 5885 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  /L N )  x.  ( B  /L N ) )  =  ( ( A  /L N )  x.  ( 0  /L N ) ) )
8475, 80, 833eqtr4d 2220 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  B  =  0 )  ->  ( ( A  x.  B )  /L N )  =  ( ( A  /L N )  x.  ( B  /L
N ) ) )
8569, 84jaodan 797 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  ( A  =  0  \/  B  =  0
) )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
86 neanior 2434 . . 3  |-  ( ( A  =/=  0  /\  B  =/=  0 )  <->  -.  ( A  =  0  \/  B  =  0 ) )
87 lgsdir 14103 . . . 4  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  ZZ )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
885, 87syl3anl3 1288 . . 3  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  ( A  =/=  0  /\  B  =/=  0
) )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
8986, 88sylan2br 288 . 2  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  /\  -.  ( A  =  0  \/  B  =  0 ) )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
90 zdceq 9317 . . . . 5  |-  ( ( A  e.  ZZ  /\  0  e.  ZZ )  -> DECID  A  =  0 )
9173, 14, 90sylancl 413 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  -> DECID  A  =  0
)
92 zdceq 9317 . . . . 5  |-  ( ( B  e.  ZZ  /\  0  e.  ZZ )  -> DECID  B  =  0 )
9347, 14, 92sylancl 413 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  -> DECID  B  =  0
)
94 dcor 935 . . . 4  |-  (DECID  A  =  0  ->  (DECID  B  = 
0  -> DECID  ( A  =  0  \/  B  =  0 ) ) )
9591, 93, 94sylc 62 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  -> DECID  ( A  =  0  \/  B  =  0 ) )
96 exmiddc 836 . . 3  |-  (DECID  ( A  =  0  \/  B  =  0 )  -> 
( ( A  =  0  \/  B  =  0 )  \/  -.  ( A  =  0  \/  B  =  0
) ) )
9795, 96syl 14 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
( A  =  0  \/  B  =  0 )  \/  -.  ( A  =  0  \/  B  =  0 ) ) )
9885, 89, 97mpjaodan 798 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  N  e.  NN0 )  ->  (
( A  x.  B
)  /L N )  =  ( ( A  /L N )  x.  ( B  /L N ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   A.wral 2455  (class class class)co 5869   CCcc 7800   0cc0 7802   1c1 7803    x. cmul 7807   NN0cn0 9165   ZZcz 9242    gcd cgcd 11926    /Lclgs 14065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-frec 6386  df-1o 6411  df-2o 6412  df-oadd 6415  df-er 6529  df-en 6735  df-dom 6736  df-fin 6737  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-ihash 10740  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-clim 11271  df-proddc 11543  df-dvds 11779  df-gcd 11927  df-prm 12091  df-phi 12194  df-pc 12268  df-lgs 14066
This theorem is referenced by: (None)
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