coprmdvds - Intuitionistic Logic Explorer

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Theorem coprmdvds 12663

Description: Euclid's Lemma (see ProofWiki "Euclid's Lemma", 10-Jul-2021, https://proofwiki.org/wiki/Euclid's_Lemma): If an integer divides the product of two integers and is coprime to one of them, then it divides the other. See also theorem 1.5 in [ApostolNT] p. 16. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by AV, 10-Jul-2021.)

**Assertion**
Ref	Expression
coprmdvds	⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 · 𝑁) ∧ (𝐾 gcd 𝑀) = 1) → 𝐾 ∥ 𝑁))

**Proof of Theorem coprmdvds**
Step	Hyp	Ref	Expression
1		zcn 9483	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ)
2		zcn 9483	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
3		mulcom 8160	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
4	1, 2, 3	syl2an 289	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) = (𝑁 · 𝑀))
5	4	breq2d 4100	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ (𝑁 · 𝑀)))
6		dvdsmulgcd 12595	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 ∥ (𝑁 · 𝑀) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾))))
7	6	ancoms 268	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑁 · 𝑀) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾))))
8	5, 7	bitrd 188	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾))))
9	8	3adant1 1041	. . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾))))
10	9	adantr 276	. . . . . 6 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾))))
11		gcdcom 12543	. . . . . . . . . . . 12 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝐾 gcd 𝑀) = (𝑀 gcd 𝐾))
12	11	3adant3 1043	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 gcd 𝑀) = (𝑀 gcd 𝐾))
13	12	eqeq1d 2240	. . . . . . . . . 10 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 gcd 𝑀) = 1 ↔ (𝑀 gcd 𝐾) = 1))
14		oveq2 6025	. . . . . . . . . 10 ⊢ ((𝑀 gcd 𝐾) = 1 → (𝑁 · (𝑀 gcd 𝐾)) = (𝑁 · 1))
15	13, 14	biimtrdi 163	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 gcd 𝑀) = 1 → (𝑁 · (𝑀 gcd 𝐾)) = (𝑁 · 1)))
16	15	imp 124	. . . . . . . 8 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝑁 · (𝑀 gcd 𝐾)) = (𝑁 · 1))
17	2	mulridd 8195	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → (𝑁 · 1) = 𝑁)
18	17	3ad2ant3 1046	. . . . . . . . 9 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 · 1) = 𝑁)
19	18	adantr 276	. . . . . . . 8 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝑁 · 1) = 𝑁)
20	16, 19	eqtrd 2264	. . . . . . 7 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝑁 · (𝑀 gcd 𝐾)) = 𝑁)
21	20	breq2d 4100	. . . . . 6 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 ∥ (𝑁 · (𝑀 gcd 𝐾)) ↔ 𝐾 ∥ 𝑁))
22	10, 21	bitrd 188	. . . . 5 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 ∥ (𝑀 · 𝑁) ↔ 𝐾 ∥ 𝑁))
23	22	biimpd 144	. . . 4 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 ∥ (𝑀 · 𝑁) → 𝐾 ∥ 𝑁))
24	23	ex 115	. . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 gcd 𝑀) = 1 → (𝐾 ∥ (𝑀 · 𝑁) → 𝐾 ∥ 𝑁)))
25	24	com23 78	. 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ (𝑀 · 𝑁) → ((𝐾 gcd 𝑀) = 1 → 𝐾 ∥ 𝑁)))
26	25	impd 254	1 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝐾 ∥ (𝑀 · 𝑁) ∧ (𝐾 gcd 𝑀) = 1) → 𝐾 ∥ 𝑁))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 (class class class)co 6017 ℂcc 8029 1c1 8032 · cmul 8036 ℤcz 9478 ∥ cdvds 12347 gcd cgcd 12523

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-mulrcl 8130 ax-addcom 8131 ax-mulcom 8132 ax-addass 8133 ax-mulass 8134 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-1rid 8138 ax-0id 8139 ax-rnegex 8140 ax-precex 8141 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-apti 8146 ax-pre-ltadd 8147 ax-pre-mulgt0 8148 ax-pre-mulext 8149 ax-arch 8150 ax-caucvg 8151

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-sup 7182 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-reap 8754 df-ap 8761 df-div 8852 df-inn 9143 df-2 9201 df-3 9202 df-4 9203 df-n0 9402 df-z 9479 df-uz 9755 df-q 9853 df-rp 9888 df-fz 10243 df-fzo 10377 df-fl 10529 df-mod 10584 df-seqfrec 10709 df-exp 10800 df-cj 11402 df-re 11403 df-im 11404 df-rsqrt 11558 df-abs 11559 df-dvds 12348 df-gcd 12524

This theorem is referenced by: coprmdvds2 12664 qredeq 12667 cncongr1 12674 euclemma 12717 eulerthlemh 12802 eulerthlemth 12803 prmdiveq 12807 prmpwdvds 12927 mpodvdsmulf1o 15713 perfectlem1 15722 lgseisenlem1 15798 lgseisenlem2 15799 lgsquadlem2 15806 lgsquadlem3 15807 2sqlem8 15851

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