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Theorem pcgcd 13108
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
pcgcd |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd B ) ) = if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) )
Proof of Theorem pcgcd
StepHypRef Expression
1 pcgcd1 13107 . . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt A ) )
2 iftrue 3645 . . . 4 |- ( ( P pCnt A ) <_ ( P pCnt B ) -> if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) = ( P pCnt A ) )
32adantl 277 . . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) = ( P pCnt A ) )
41, 3eqtr4d 2274 . 2 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) )
5 gcdcom 12750 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
653adant1 1046 . . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
76adantr 276 . . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( A gcd B ) = ( B gcd A ) )
87oveq2d 6101 . . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = ( P pCnt ( B gcd A ) ) )
9 iffalse 3648 . . . . 5 |- ( -. ( P pCnt A ) <_ ( P pCnt B ) -> if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) = ( P pCnt B ) )
109adantl 277 . . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) = ( P pCnt B ) )
11 pcxnn0cl 13089 . . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt A ) e. NN0* )
12113adant3 1048 . . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt A ) e. NN0* )
13 pcxnn0cl 13089 . . . . . . 7 |- ( ( P e. Prime /\ B e. ZZ ) -> ( P pCnt B ) e. NN0* )
14 xnn0letri 10205 . . . . . . 7 |- ( ( ( P pCnt A ) e. NN0* /\ ( P pCnt B ) e. NN0* ) -> ( ( P pCnt A ) <_ ( P pCnt B ) \/ ( P pCnt B ) <_ ( P pCnt A ) ) )
1512, 13, 143imp3i2an 1214 . . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( ( P pCnt A ) <_ ( P pCnt B ) \/ ( P pCnt B ) <_ ( P pCnt A ) ) )
1615orcanai 940 . . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt B ) <_ ( P pCnt A ) )
17 3ancomb 1017 . . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) <-> ( P e. Prime /\ B e. ZZ /\ A e. ZZ ) )
18 pcgcd1 13107 . . . . . 6 |- ( ( ( P e. Prime /\ B e. ZZ /\ A e. ZZ ) /\ ( P pCnt B ) <_ ( P pCnt A ) ) -> ( P pCnt ( B gcd A ) ) = ( P pCnt B ) )
1917, 18sylanb 284 . . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ ( P pCnt B ) <_ ( P pCnt A ) ) -> ( P pCnt ( B gcd A ) ) = ( P pCnt B ) )
2016, 19syldan 282 . . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( B gcd A ) ) = ( P pCnt B ) )
2110, 20eqtr4d 2274 . . 3 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) = ( P pCnt ( B gcd A ) ) )
228, 21eqtr4d 2274 . 2 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt ( A gcd B ) ) = if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) )
23 xnn0dcle 10204 . . . 4 |- ( ( ( P pCnt A ) e. NN0* /\ ( P pCnt B ) e. NN0* ) -> DECID ( P pCnt A ) <_ ( P pCnt B ) )
2412, 13, 233imp3i2an 1214 . . 3 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> DECID ( P pCnt A ) <_ ( P pCnt B ) )
25 exmiddc 848 . . 3 |- (DECID ( P pCnt A ) <_ ( P pCnt B ) -> ( ( P pCnt A ) <_ ( P pCnt B ) \/ -. ( P pCnt A ) <_ ( P pCnt B ) ) )
2624, 25syl 14 . 2 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( ( P pCnt A ) <_ ( P pCnt B ) \/ -. ( P pCnt A ) <_ ( P pCnt B ) ) )
274, 22, 26mpjaodan 810 1 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt ( A gcd B ) ) = if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720  DECID wdc 846   /\ w3a 1009   = wceq 1402   e. wcel 2209   ifcif 3638   class class class wbr 4130  (class class class)co 6085   <_ cle 8361  NN0*cxnn0 9630   ZZcz 9644   gcd cgcd 12730   Primecprime 12885   pCnt cpc 13063
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299
This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-2o 6688  df-er 6807  df-en 7023  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-xnn0 9631  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fz 10412  df-fzo 10550  df-fl 10705  df-mod 10760  df-seqfrec 10885  df-exp 10976  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-dvds 12555  df-gcd 12731  df-prm 12886  df-pc 13064
This theorem is used by:  pc2dvds  13109
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