pcgcd - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > pcgcd		GIF version

Theorem pcgcd 12237

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 12236	. . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴))
2		iftrue 3520	. . . 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 275	. . 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 2200	. 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 11891	. . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
6	5	3adant1 1004	. . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
7	6	adantr 274	. . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴))
8	7	oveq2d 5852	. . 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 3523	. . . . 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 275	. . . 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 12219	. . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt 𝐴) ∈ ℕ₀^*)
12	11	3adant3 1006	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt 𝐴) ∈ ℕ₀^*)
13		pcxnn0cl 12219	. . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt 𝐵) ∈ ℕ₀^*)
14		xnn0letri 9730	. . . . . . 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 1172	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∨ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)))
16	15	orcanai 918	. . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))
17		3ancomb 975	. . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ))
18		pcgcd1 12236	. . . . . 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 282	. . . . 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 280	. . . 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 2200	. . 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 2200	. 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 9729	. . . 4 ⊢ (((𝑃 pCnt 𝐴) ∈ ℕ₀^* ∧ (𝑃 pCnt 𝐵) ∈ ℕ₀^*) → DECID (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
24	12, 13, 23	3imp3i2an 1172	. . 3 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → DECID (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵))
25		exmiddc 826	. . 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 788	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 103 ∨ wo 698 DECID wdc 824 ∧ w3a 967 = wceq 1342 ∈ wcel 2135 ifcif 3515 class class class wbr 3976 (class class class)co 5836 ≤ cle 7925 ℕ₀^*cxnn0 9168 ℤcz 9182 gcd cgcd 11860 ℙcprime 12018 pCnt cpc 12193

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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864

This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-isom 5191 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-1o 6375 df-2o 6376 df-er 6492 df-en 6698 df-sup 6940 df-inf 6941 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-n0 9106 df-xnn0 9169 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-gcd 11861 df-prm 12019 df-pc 12194

This theorem is referenced by: pc2dvds 12238

W3C validator