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Theorem pcgcd 12470
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 12469 . . 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 3563 . . . 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 2229 . 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 12113 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
653adant1 1017 . . . . 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 5935 . . 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 3566 . . . . 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 12451 . . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt A ) e. NN0* )
12113adant3 1019 . . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt A ) e. NN0* )
13 pcxnn0cl 12451 . . . . . . 7 |- ( ( P e. Prime /\ B e. ZZ ) -> ( P pCnt B ) e. NN0* )
14 xnn0letri 9872 . . . . . . 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 1185 . . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( ( P pCnt A ) <_ ( P pCnt B ) \/ ( P pCnt B ) <_ ( P pCnt A ) ) )
1615orcanai 929 . . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt B ) <_ ( P pCnt A ) )
17 3ancomb 988 . . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) <-> ( P e. Prime /\ B e. ZZ /\ A e. ZZ ) )
18 pcgcd1 12469 . . . . . 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 2229 . . 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 2229 . 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 9871 . . . 4 |- ( ( ( P pCnt A ) e. NN0* /\ ( P pCnt B ) e. NN0* ) -> DECID ( P pCnt A ) <_ ( P pCnt B ) )
2412, 13, 233imp3i2an 1185 . . 3 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> DECID ( P pCnt A ) <_ ( P pCnt B ) )
25 exmiddc 837 . . 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 799 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
Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 709  DECID wdc 835   /\ w3a 980   = wceq 1364   e. wcel 2164   ifcif 3558   class class class wbr 4030  (class class class)co 5919   <_ cle 8057  NN0*cxnn0 9306   ZZcz 9320   gcd cgcd 12082   Primecprime 12248   pCnt cpc 12425
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-1o 6471  df-2o 6472  df-er 6589  df-en 6797  df-sup 7045  df-inf 7046  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-xnn0 9307  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-fz 10078  df-fzo 10212  df-fl 10342  df-mod 10397  df-seqfrec 10522  df-exp 10613  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-dvds 11934  df-gcd 12083  df-prm 12249  df-pc 12426
This theorem is referenced by:  pc2dvds  12471
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