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Theorem pcgcd 12282
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 12281 . . 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 3531 . . . 4 |- ( ( P pCnt A ) <_ ( P pCnt B ) -> if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) = ( P pCnt A ) )
32adantl 275 . . 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 2206 . 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 11928 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
653adant1 1010 . . . . 5 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
76adantr 274 . . . 4 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( A gcd B ) = ( B gcd A ) )
87oveq2d 5869 . . 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 3534 . . . . 5 |- ( -. ( P pCnt A ) <_ ( P pCnt B ) -> if ( ( P pCnt A ) <_ ( P pCnt B ) , ( P pCnt A ) , ( P pCnt B ) ) = ( P pCnt B ) )
109adantl 275 . . . 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 12264 . . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt A ) e. NN0* )
12113adant3 1012 . . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt A ) e. NN0* )
13 pcxnn0cl 12264 . . . . . . 7 |- ( ( P e. Prime /\ B e. ZZ ) -> ( P pCnt B ) e. NN0* )
14 xnn0letri 9760 . . . . . . 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 1178 . . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( ( P pCnt A ) <_ ( P pCnt B ) \/ ( P pCnt B ) <_ ( P pCnt A ) ) )
1615orcanai 923 . . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt B ) <_ ( P pCnt A ) )
17 3ancomb 981 . . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) <-> ( P e. Prime /\ B e. ZZ /\ A e. ZZ ) )
18 pcgcd1 12281 . . . . . 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 282 . . . . 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 280 . . . 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 2206 . . 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 2206 . 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 9759 . . . 4 |- ( ( ( P pCnt A ) e. NN0* /\ ( P pCnt B ) e. NN0* ) -> DECID ( P pCnt A ) <_ ( P pCnt B ) )
2412, 13, 233imp3i2an 1178 . . 3 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> DECID ( P pCnt A ) <_ ( P pCnt B ) )
25 exmiddc 831 . . 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 793 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 103   \/ wo 703  DECID wdc 829   /\ w3a 973   = wceq 1348   e. wcel 2141   ifcif 3526   class class class wbr 3989  (class class class)co 5853   <_ cle 7955  NN0*cxnn0 9198   ZZcz 9212   gcd cgcd 11897   Primecprime 12061   pCnt cpc 12238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-xnn0 9199  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898  df-prm 12062  df-pc 12239
This theorem is referenced by:  pc2dvds  12283
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