pcgcd - Intuitionistic Logic Explorer

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Theorem pcgcd 12867

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 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt (𝐴 gcd 𝐵)) = if((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵), (𝑃 pCnt 𝐴), (𝑃 pCnt 𝐵)))

**Proof of Theorem pcgcd**
Step	Hyp	Ref	Expression
1		pcgcd1 12866	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴))
2		iftrue 3607	. . . 4 ⊢ ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) → if((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵), (𝑃 pCnt 𝐴), (𝑃 pCnt 𝐵)) = (𝑃 pCnt 𝐴))
3	2	adantl 277	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → if((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵), (𝑃 pCnt 𝐴), (𝑃 pCnt 𝐵)) = (𝑃 pCnt 𝐴))
4	1, 3	eqtr4d 2265	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (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 12509	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
6	5	3adant1 1039	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
7	6	adantr 276	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
8	7	oveq2d 6023	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt (𝐵 gcd 𝐴)))
9		iffalse 3610	. . . . 5 ⊢ (¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) → if((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵), (𝑃 pCnt 𝐴), (𝑃 pCnt 𝐵)) = (𝑃 pCnt 𝐵))
10	9	adantl 277	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → if((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵), (𝑃 pCnt 𝐴), (𝑃 pCnt 𝐵)) = (𝑃 pCnt 𝐵))
11		pcxnn0cl 12848	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt 𝐴) ∈ ℕ₀^*)
12	11	3adant3 1041	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt 𝐴) ∈ ℕ₀^*)
13		pcxnn0cl 12848	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt 𝐵) ∈ ℕ₀^*)
14		xnn0letri 10011	. . . . . . 7 ⊢ (((𝑃 pCnt 𝐴) ∈ ℕ₀^* ∧ (𝑃 pCnt 𝐵) ∈ ℕ₀^*) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∨ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
15	12, 13, 14	3imp3i2an 1207	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∨ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
16	15	orcanai 933	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))
17		3ancomb 1010	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
18		pcgcd1 12866	. . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)) → (𝑃 pCnt (𝐵 gcd 𝐴)) = (𝑃 pCnt 𝐵))
19	17, 18	sylanb 284	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)) → (𝑃 pCnt (𝐵 gcd 𝐴)) = (𝑃 pCnt 𝐵))
20	16, 19	syldan 282	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐵 gcd 𝐴)) = (𝑃 pCnt 𝐵))
21	10, 20	eqtr4d 2265	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → if((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵), (𝑃 pCnt 𝐴), (𝑃 pCnt 𝐵)) = (𝑃 pCnt (𝐵 gcd 𝐴)))
22	8, 21	eqtr4d 2265	. 2 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (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 10010	. . . 4 ⊢ (((𝑃 pCnt 𝐴) ∈ ℕ₀^* ∧ (𝑃 pCnt 𝐵) ∈ ℕ₀^*) → DECID (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
24	12, 13, 23	3imp3i2an 1207	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
25		exmiddc 841	. . 3 ⊢ (DECID (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∨ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)))
26	24, 25	syl 14	. 2 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∨ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)))
27	4, 22, 26	mpjaodan 803	1 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (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 713 DECID wdc 839 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ifcif 3602 class class class wbr 4083 (class class class)co 6007 ≤ cle 8193 ℕ₀^*cxnn0 9443 ℤcz 9457 gcd cgcd 12489 ℙcprime 12644 pCnt cpc 12822

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8101 ax-resscn 8102 ax-1cn 8103 ax-1re 8104 ax-icn 8105 ax-addcl 8106 ax-addrcl 8107 ax-mulcl 8108 ax-mulrcl 8109 ax-addcom 8110 ax-mulcom 8111 ax-addass 8112 ax-mulass 8113 ax-distr 8114 ax-i2m1 8115 ax-0lt1 8116 ax-1rid 8117 ax-0id 8118 ax-rnegex 8119 ax-precex 8120 ax-cnre 8121 ax-pre-ltirr 8122 ax-pre-ltwlin 8123 ax-pre-lttrn 8124 ax-pre-apti 8125 ax-pre-ltadd 8126 ax-pre-mulgt0 8127 ax-pre-mulext 8128 ax-arch 8129 ax-caucvg 8130

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-1o 6568 df-2o 6569 df-er 6688 df-en 6896 df-sup 7162 df-inf 7163 df-pnf 8194 df-mnf 8195 df-xr 8196 df-ltxr 8197 df-le 8198 df-sub 8330 df-neg 8331 df-reap 8733 df-ap 8740 df-div 8831 df-inn 9122 df-2 9180 df-3 9181 df-4 9182 df-n0 9381 df-xnn0 9444 df-z 9458 df-uz 9734 df-q 9827 df-rp 9862 df-fz 10217 df-fzo 10351 df-fl 10502 df-mod 10557 df-seqfrec 10682 df-exp 10773 df-cj 11368 df-re 11369 df-im 11370 df-rsqrt 11524 df-abs 11525 df-dvds 12314 df-gcd 12490 df-prm 12645 df-pc 12823

This theorem is referenced by: pc2dvds 12868

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