MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pcgcd1 Structured version   Unicode version

Theorem pcgcd1 14789
Description: The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.)
Assertion
Ref Expression
pcgcd1  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )

Proof of Theorem pcgcd1
StepHypRef Expression
1 oveq2 6313 . . . 4  |-  ( B  =  0  ->  ( A  gcd  B )  =  ( A  gcd  0
) )
21oveq2d 6321 . . 3  |-  ( B  =  0  ->  ( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  ( A  gcd  0 ) ) )
32eqeq1d 2431 . 2  |-  ( B  =  0  ->  (
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A )  <->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P 
pCnt  A ) ) )
4 simpl1 1008 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  P  e.  Prime )
5 simp2 1006 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  A  e.  ZZ )
65adantr 466 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  e.  ZZ )
7 simpl3 1010 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  B  e.  ZZ )
8 simprr 764 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  B  =/=  0 )
9 simpr 462 . . . . . . . . 9  |-  ( ( A  =  0  /\  B  =  0 )  ->  B  =  0 )
109necon3ai 2659 . . . . . . . 8  |-  ( B  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
118, 10syl 17 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -.  ( A  =  0  /\  B  =  0 ) )
12 gcdn0cl 14450 . . . . . . 7  |-  ( ( ( A  e.  ZZ  /\  B  e.  ZZ )  /\  -.  ( A  =  0  /\  B  =  0 ) )  ->  ( A  gcd  B )  e.  NN )
136, 7, 11, 12syl21anc 1263 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  e.  NN )
1413nnzd 11039 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  e.  ZZ )
15 gcddvds 14451 . . . . . . 7  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
166, 7, 15syl2anc 665 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( A  gcd  B )  ||  A  /\  ( A  gcd  B ) 
||  B ) )
1716simpld 460 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  gcd  B
)  ||  A )
18 pcdvdstr 14788 . . . . 5  |-  ( ( P  e.  Prime  /\  (
( A  gcd  B
)  e.  ZZ  /\  A  e.  ZZ  /\  ( A  gcd  B )  ||  A ) )  -> 
( P  pCnt  ( A  gcd  B ) )  <_  ( P  pCnt  A ) )
194, 14, 6, 17, 18syl13anc 1266 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  gcd  B ) )  <_  ( P  pCnt  A ) )
20 zq 11270 . . . . . . . . . . 11  |-  ( A  e.  ZZ  ->  A  e.  QQ )
216, 20syl 17 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  e.  QQ )
22 pcxcl 14773 . . . . . . . . . 10  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  A )  e. 
RR* )
234, 21, 22syl2anc 665 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  RR* )
24 pczcl 14761 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  ( B  e.  ZZ  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  NN0 )
254, 7, 8, 24syl12anc 1262 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  NN0 )
2625nn0red 10926 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  B
)  e.  RR )
27 pcge0 14774 . . . . . . . . . . 11  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  0  <_  ( P  pCnt  A
) )
284, 6, 27syl2anc 665 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
0  <_  ( P  pCnt  A ) )
29 ge0gtmnf 11467 . . . . . . . . . 10  |-  ( ( ( P  pCnt  A
)  e.  RR*  /\  0  <_  ( P  pCnt  A
) )  -> -oo  <  ( P  pCnt  A )
)
3023, 28, 29syl2anc 665 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> -oo  <  ( P  pCnt  A ) )
31 simprl 762 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  <_  ( P  pCnt  B ) )
32 xrre 11464 . . . . . . . . 9  |-  ( ( ( ( P  pCnt  A )  e.  RR*  /\  ( P  pCnt  B )  e.  RR )  /\  ( -oo  <  ( P  pCnt  A )  /\  ( P 
pCnt  A )  <_  ( P  pCnt  B ) ) )  ->  ( P  pCnt  A )  e.  RR )
3323, 26, 30, 31, 32syl22anc 1265 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  RR )
34 pnfnre 9681 . . . . . . . . . . . 12  |- +oo  e/  RR
3534neli 2767 . . . . . . . . . . 11  |-  -. +oo  e.  RR
36 pc0 14767 . . . . . . . . . . . . 13  |-  ( P  e.  Prime  ->  ( P 
pCnt  0 )  = +oo )
374, 36syl 17 . . . . . . . . . . . 12  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  0
)  = +oo )
3837eleq1d 2498 . . . . . . . . . . 11  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  0 )  e.  RR  <-> +oo  e.  RR ) )
3935, 38mtbiri 304 . . . . . . . . . 10  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  -.  ( P  pCnt  0
)  e.  RR )
40 oveq2 6313 . . . . . . . . . . . 12  |-  ( A  =  0  ->  ( P  pCnt  A )  =  ( P  pCnt  0
) )
4140eleq1d 2498 . . . . . . . . . . 11  |-  ( A  =  0  ->  (
( P  pCnt  A
)  e.  RR  <->  ( P  pCnt  0 )  e.  RR ) )
4241notbid 295 . . . . . . . . . 10  |-  ( A  =  0  ->  ( -.  ( P  pCnt  A
)  e.  RR  <->  -.  ( P  pCnt  0 )  e.  RR ) )
4339, 42syl5ibrcom 225 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( A  =  0  ->  -.  ( P  pCnt  A )  e.  RR ) )
4443necon2ad 2644 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  A )  e.  RR  ->  A  =/=  0 ) )
4533, 44mpd 15 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  A  =/=  0 )
46 pczdvds 14775 . . . . . . 7  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
474, 6, 45, 46syl12anc 1262 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  A )
48 pczcl 14761 . . . . . . . . 9  |-  ( ( P  e.  Prime  /\  ( A  e.  ZZ  /\  A  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
494, 6, 45, 48syl12anc 1262 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  e.  NN0 )
50 pcdvdsb 14781 . . . . . . . 8  |-  ( ( P  e.  Prime  /\  B  e.  ZZ  /\  ( P 
pCnt  A )  e.  NN0 )  ->  ( ( P 
pCnt  A )  <_  ( P  pCnt  B )  <->  ( P ^ ( P  pCnt  A ) )  ||  B
) )
514, 7, 49, 50syl3anc 1264 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  A )  <_  ( P  pCnt  B )  <->  ( P ^ ( P  pCnt  A ) )  ||  B
) )
5231, 51mpbid 213 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  B )
53 prmnn 14596 . . . . . . . . . 10  |-  ( P  e.  Prime  ->  P  e.  NN )
544, 53syl 17 . . . . . . . . 9  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  ->  P  e.  NN )
5554, 49nnexpcld 12434 . . . . . . . 8  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) )  e.  NN )
5655nnzd 11039 . . . . . . 7  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) )  e.  ZZ )
57 dvdsgcd 14482 . . . . . . 7  |-  ( ( ( P ^ ( P  pCnt  A ) )  e.  ZZ  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  (
( ( P ^
( P  pCnt  A
) )  ||  A  /\  ( P ^ ( P  pCnt  A ) ) 
||  B )  -> 
( P ^ ( P  pCnt  A ) ) 
||  ( A  gcd  B ) ) )
5856, 6, 7, 57syl3anc 1264 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( ( P ^ ( P  pCnt  A ) )  ||  A  /\  ( P ^ ( P  pCnt  A ) ) 
||  B )  -> 
( P ^ ( P  pCnt  A ) ) 
||  ( A  gcd  B ) ) )
5947, 52, 58mp2and 683 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P ^ ( P  pCnt  A ) ) 
||  ( A  gcd  B ) )
60 pcdvdsb 14781 . . . . . 6  |-  ( ( P  e.  Prime  /\  ( A  gcd  B )  e.  ZZ  /\  ( P 
pCnt  A )  e.  NN0 )  ->  ( ( P 
pCnt  A )  <_  ( P  pCnt  ( A  gcd  B ) )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( A  gcd  B ) ) )
614, 14, 49, 60syl3anc 1264 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  A )  <_  ( P  pCnt  ( A  gcd  B
) )  <->  ( P ^ ( P  pCnt  A ) )  ||  ( A  gcd  B ) ) )
6259, 61mpbird 235 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  A
)  <_  ( P  pCnt  ( A  gcd  B
) ) )
634, 13pccld 14763 . . . . . 6  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  gcd  B ) )  e.  NN0 )
6463nn0red 10926 . . . . 5  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  gcd  B ) )  e.  RR )
6564, 33letri3d 9776 . . . 4  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( ( P  pCnt  ( A  gcd  B ) )  =  ( P 
pCnt  A )  <->  ( ( P  pCnt  ( A  gcd  B ) )  <_  ( P  pCnt  A )  /\  ( P  pCnt  A )  <_  ( P  pCnt  ( A  gcd  B ) ) ) ) )
6619, 62, 65mpbir2and 930 . . 3  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( ( P  pCnt  A )  <_  ( P  pCnt  B )  /\  B  =/=  0 ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )
6766anassrs 652 . 2  |-  ( ( ( ( P  e. 
Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A )  <_  ( P  pCnt  B ) )  /\  B  =/=  0 )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )
68 gcdid0 14462 . . . . . 6  |-  ( A  e.  ZZ  ->  ( A  gcd  0 )  =  ( abs `  A
) )
695, 68syl 17 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A  gcd  0 )  =  ( abs `  A
) )
7069oveq2d 6321 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  ( abs `  A ) ) )
71 pcabs 14787 . . . . . 6  |-  ( ( P  e.  Prime  /\  A  e.  QQ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
7220, 71sylan2 476 . . . . 5  |-  ( ( P  e.  Prime  /\  A  e.  ZZ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
73723adant3 1025 . . . 4  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( abs `  A
) )  =  ( P  pCnt  A )
)
7470, 73eqtrd 2470 . . 3  |-  ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  ->  ( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  A
) )
7574adantr 466 . 2  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  0 ) )  =  ( P  pCnt  A ) )
763, 67, 75pm2.61ne 2746 1  |-  ( ( ( P  e.  Prime  /\  A  e.  ZZ  /\  B  e.  ZZ )  /\  ( P  pCnt  A
)  <_  ( P  pCnt  B ) )  -> 
( P  pCnt  ( A  gcd  B ) )  =  ( P  pCnt  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426   ` cfv 5601  (class class class)co 6305   RRcr 9537   0cc0 9538   +oocpnf 9671   -oocmnf 9672   RR*cxr 9673    < clt 9674    <_ cle 9675   NNcn 10609   NN0cn0 10869   ZZcz 10937   QQcq 11264   ^cexp 12269   abscabs 13276    || cdvds 14283    gcd cgcd 14442   Primecprime 14593    pCnt cpc 14749
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265  df-rp 11303  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284  df-gcd 14443  df-prm 14594  df-pc 14750
This theorem is referenced by:  pcgcd  14790
  Copyright terms: Public domain W3C validator