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Theorem pcgcd 12767
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 12766 . . 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 3584 . . . 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 2243 . 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 12409 . . . . . 6 |- ( ( A e. ZZ /\ B e. ZZ ) -> ( A gcd B ) = ( B gcd A ) )
653adant1 1018 . . . . 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 5983 . . 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 3587 . . . . 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 12748 . . . . . . . 8 |- ( ( P e. Prime /\ A e. ZZ ) -> ( P pCnt A ) e. NN0* )
12113adant3 1020 . . . . . . 7 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( P pCnt A ) e. NN0* )
13 pcxnn0cl 12748 . . . . . . 7 |- ( ( P e. Prime /\ B e. ZZ ) -> ( P pCnt B ) e. NN0* )
14 xnn0letri 9960 . . . . . . 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 1186 . . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> ( ( P pCnt A ) <_ ( P pCnt B ) \/ ( P pCnt B ) <_ ( P pCnt A ) ) )
1615orcanai 930 . . . . 5 |- ( ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) /\ -. ( P pCnt A ) <_ ( P pCnt B ) ) -> ( P pCnt B ) <_ ( P pCnt A ) )
17 3ancomb 989 . . . . . 6 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) <-> ( P e. Prime /\ B e. ZZ /\ A e. ZZ ) )
18 pcgcd1 12766 . . . . . 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 2243 . . 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 2243 . 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 9959 . . . 4 |- ( ( ( P pCnt A ) e. NN0* /\ ( P pCnt B ) e. NN0* ) -> DECID ( P pCnt A ) <_ ( P pCnt B ) )
2412, 13, 233imp3i2an 1186 . . 3 |- ( ( P e. Prime /\ A e. ZZ /\ B e. ZZ ) -> DECID ( P pCnt A ) <_ ( P pCnt B ) )
25 exmiddc 838 . . 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 800 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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2178   ifcif 3579   class class class wbr 4059  (class class class)co 5967   <_ cle 8143  NN0*cxnn0 9393   ZZcz 9407   gcd cgcd 12389   Primecprime 12544   pCnt cpc 12722
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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080
This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-isom 5299  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-2o 6526  df-er 6643  df-en 6851  df-sup 7112  df-inf 7113  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-xnn0 9394  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390  df-prm 12545  df-pc 12723
This theorem is referenced by:  pc2dvds  12768
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