pcgcd - Intuitionistic Logic Explorer

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Theorem pcgcd 13086

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 13085	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴))
2		iftrue 3642	. . . 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 2274	. 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 12728	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
6	5	3adant1 1046	. . . . 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 6091	. . 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 3645	. . . . 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 13067	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt 𝐴) ∈ ℕ₀^*)
12	11	3adant3 1048	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt 𝐴) ∈ ℕ₀^*)
13		pcxnn0cl 13067	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt 𝐵) ∈ ℕ₀^*)
14		xnn0letri 10184	. . . . . . 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 1214	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∨ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
16	15	orcanai 940	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))
17		3ancomb 1017	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
18		pcgcd1 13085	. . . . . 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 2274	. . 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 2274	. 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 10183	. . . 4 ⊢ (((𝑃 pCnt 𝐴) ∈ ℕ₀^* ∧ (𝑃 pCnt 𝐵) ∈ ℕ₀^*) → DECID (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
24	12, 13, 23	3imp3i2an 1214	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → 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)))
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 810	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 720 DECID wdc 846 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ifcif 3635 class class class wbr 4125 (class class class)co 6075 ≤ cle 8351 ℕ₀^*cxnn0 9609 ℤcz 9623 gcd cgcd 12708 ℙcprime 12863 pCnt cpc 13041

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 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 4241 ax-sep 4244 ax-nul 4254 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-iinf 4730 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287 ax-arch 8288 ax-caucvg 8289

This theorem 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 3636 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-int 3966 df-iun 4009 df-br 4126 df-opab 4188 df-mpt 4189 df-tr 4225 df-id 4433 df-po 4436 df-iso 4437 df-iord 4506 df-on 4508 df-ilim 4509 df-suc 4511 df-iom 4733 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-f1 5377 df-fo 5378 df-f1o 5379 df-fv 5380 df-isom 5381 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-1st 6364 df-2nd 6365 df-recs 6566 df-frec 6652 df-1o 6677 df-2o 6678 df-er 6797 df-en 7013 df-sup 7314 df-inf 7315 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-inn 9284 df-2 9342 df-3 9343 df-4 9344 df-n0 9543 df-xnn0 9610 df-z 9624 df-uz 9901 df-q 9999 df-rp 10034 df-fz 10391 df-fzo 10528 df-fl 10683 df-mod 10738 df-seqfrec 10863 df-exp 10954 df-cj 11585 df-re 11586 df-im 11587 df-rsqrt 11742 df-abs 11743 df-dvds 12533 df-gcd 12709 df-prm 12864 df-pc 13042

This theorem is referenced by: pc2dvds 13087

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