![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pcgcd | Structured version Visualization version GIF version |
Description: The prime count of a GCD is the minimum of the prime counts of the arguments. (Contributed by Mario Carneiro, 3-Oct-2014.) |
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))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pcgcd1 16872 | . . 3 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐴 gcd 𝐵)) = (𝑃 pCnt 𝐴)) | |
2 | iftrue 4530 | . . . 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 480 | . . 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 2769 | . 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 16506 | . . . . . 6 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴)) | |
6 | 5 | 3adant1 1127 | . . . . 5 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴)) |
7 | 6 | adantr 479 | . . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝐴 gcd 𝐵) = (𝐵 gcd 𝐴)) |
8 | 7 | oveq2d 7430 | . . 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 4533 | . . . . 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 480 | . . . 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 | zq 12982 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℤ → 𝐴 ∈ ℚ) | |
12 | pcxcl 16856 | . . . . . . . . 9 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℚ) → (𝑃 pCnt 𝐴) ∈ ℝ*) | |
13 | 11, 12 | sylan2 591 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ) → (𝑃 pCnt 𝐴) ∈ ℝ*) |
14 | 13 | 3adant3 1129 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt 𝐴) ∈ ℝ*) |
15 | zq 12982 | . . . . . . . 8 ⊢ (𝐵 ∈ ℤ → 𝐵 ∈ ℚ) | |
16 | pcxcl 16856 | . . . . . . . 8 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℚ) → (𝑃 pCnt 𝐵) ∈ ℝ*) | |
17 | 15, 16 | sylan2 591 | . . . . . . 7 ⊢ ((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ) → (𝑃 pCnt 𝐵) ∈ ℝ*) |
18 | xrletri 13178 | . . . . . . 7 ⊢ (((𝑃 pCnt 𝐴) ∈ ℝ* ∧ (𝑃 pCnt 𝐵) ∈ ℝ*) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∨ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) | |
19 | 14, 17, 18 | 3imp3i2an 1342 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵) ∨ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴))) |
20 | 19 | orcanai 1000 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)) |
21 | 3ancomb 1096 | . . . . . 6 ⊢ ((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ↔ (𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ)) | |
22 | pcgcd1 16872 | . . . . . 6 ⊢ (((𝑃 ∈ ℙ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℤ) ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)) → (𝑃 pCnt (𝐵 gcd 𝐴)) = (𝑃 pCnt 𝐵)) | |
23 | 21, 22 | sylanb 579 | . . . . 5 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ (𝑃 pCnt 𝐵) ≤ (𝑃 pCnt 𝐴)) → (𝑃 pCnt (𝐵 gcd 𝐴)) = (𝑃 pCnt 𝐵)) |
24 | 20, 23 | syldan 589 | . . . 4 ⊢ (((𝑃 ∈ ℙ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ¬ (𝑃 pCnt 𝐴) ≤ (𝑃 pCnt 𝐵)) → (𝑃 pCnt (𝐵 gcd 𝐴)) = (𝑃 pCnt 𝐵)) |
25 | 10, 24 | eqtr4d 2769 | . . 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))) |
26 | 8, 25 | eqtr4d 2769 | . 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))) |
27 | 4, 26 | pm2.61dan 811 | 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 setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ifcif 4524 class class class wbr 5144 (class class class)co 7414 ℝ*cxr 11286 ≤ cle 11288 ℤcz 12602 ℚcq 12976 gcd cgcd 16487 ℙcprime 16665 pCnt cpc 16831 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7736 ax-cnex 11203 ax-resscn 11204 ax-1cn 11205 ax-icn 11206 ax-addcl 11207 ax-addrcl 11208 ax-mulcl 11209 ax-mulrcl 11210 ax-mulcom 11211 ax-addass 11212 ax-mulass 11213 ax-distr 11214 ax-i2m1 11215 ax-1ne0 11216 ax-1rid 11217 ax-rnegex 11218 ax-rrecex 11219 ax-cnre 11220 ax-pre-lttri 11221 ax-pre-lttrn 11222 ax-pre-ltadd 11223 ax-pre-mulgt0 11224 ax-pre-sup 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3365 df-reu 3366 df-rab 3421 df-v 3465 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4324 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4907 df-iun 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6303 df-ord 6369 df-on 6370 df-lim 6371 df-suc 6372 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7867 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9476 df-inf 9477 df-pnf 11289 df-mnf 11290 df-xr 11291 df-ltxr 11292 df-le 11293 df-sub 11485 df-neg 11486 df-div 11911 df-nn 12257 df-2 12319 df-3 12320 df-n0 12517 df-z 12603 df-uz 12867 df-q 12977 df-rp 13021 df-fl 13804 df-mod 13882 df-seq 14014 df-exp 14074 df-cj 15097 df-re 15098 df-im 15099 df-sqrt 15233 df-abs 15234 df-dvds 16250 df-gcd 16488 df-prm 16666 df-pc 16832 |
This theorem is referenced by: pc2dvds 16874 mumullem2 27203 |
Copyright terms: Public domain | W3C validator |