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Mirrors > Home > MPE Home > Th. List > rpmulgcd | Structured version Visualization version GIF version |
Description: If 𝐾 and 𝑀 are relatively prime, then the GCD of 𝐾 and 𝑀 · 𝑁 is the GCD of 𝐾 and 𝑁. (Contributed by Scott Fenton, 12-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) |
Ref | Expression |
---|---|
rpmulgcd | ⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 gcd (𝑀 · 𝑁)) = (𝐾 gcd 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | gcdmultiple 16532 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐾 gcd (𝐾 · 𝑁)) = 𝐾) | |
2 | 1 | 3adant2 1128 | . . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐾 gcd (𝐾 · 𝑁)) = 𝐾) |
3 | 2 | oveq1d 7431 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐾 gcd (𝐾 · 𝑁)) gcd (𝑀 · 𝑁)) = (𝐾 gcd (𝑀 · 𝑁))) |
4 | nnz 12625 | . . . . . 6 ⊢ (𝐾 ∈ ℕ → 𝐾 ∈ ℤ) | |
5 | 4 | 3ad2ant1 1130 | . . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝐾 ∈ ℤ) |
6 | nnz 12625 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℤ) | |
7 | zmulcl 12657 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 · 𝑁) ∈ ℤ) | |
8 | 4, 6, 7 | syl2an 594 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐾 · 𝑁) ∈ ℤ) |
9 | 8 | 3adant2 1128 | . . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐾 · 𝑁) ∈ ℤ) |
10 | nnz 12625 | . . . . . . 7 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℤ) | |
11 | zmulcl 12657 | . . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 · 𝑁) ∈ ℤ) | |
12 | 10, 6, 11 | syl2an 594 | . . . . . 6 ⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) ∈ ℤ) |
13 | 12 | 3adant1 1127 | . . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝑀 · 𝑁) ∈ ℤ) |
14 | gcdass 16543 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ (𝐾 · 𝑁) ∈ ℤ ∧ (𝑀 · 𝑁) ∈ ℤ) → ((𝐾 gcd (𝐾 · 𝑁)) gcd (𝑀 · 𝑁)) = (𝐾 gcd ((𝐾 · 𝑁) gcd (𝑀 · 𝑁)))) | |
15 | 5, 9, 13, 14 | syl3anc 1368 | . . . 4 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐾 gcd (𝐾 · 𝑁)) gcd (𝑀 · 𝑁)) = (𝐾 gcd ((𝐾 · 𝑁) gcd (𝑀 · 𝑁)))) |
16 | 3, 15 | eqtr3d 2768 | . . 3 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → (𝐾 gcd (𝑀 · 𝑁)) = (𝐾 gcd ((𝐾 · 𝑁) gcd (𝑀 · 𝑁)))) |
17 | 16 | adantr 479 | . 2 ⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 gcd (𝑀 · 𝑁)) = (𝐾 gcd ((𝐾 · 𝑁) gcd (𝑀 · 𝑁)))) |
18 | nnnn0 12525 | . . . . . 6 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℕ0) | |
19 | mulgcdr 16546 | . . . . . 6 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ0) → ((𝐾 · 𝑁) gcd (𝑀 · 𝑁)) = ((𝐾 gcd 𝑀) · 𝑁)) | |
20 | 4, 10, 18, 19 | syl3an 1157 | . . . . 5 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐾 · 𝑁) gcd (𝑀 · 𝑁)) = ((𝐾 gcd 𝑀) · 𝑁)) |
21 | oveq1 7423 | . . . . 5 ⊢ ((𝐾 gcd 𝑀) = 1 → ((𝐾 gcd 𝑀) · 𝑁) = (1 · 𝑁)) | |
22 | 20, 21 | sylan9eq 2786 | . . . 4 ⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 gcd 𝑀) = 1) → ((𝐾 · 𝑁) gcd (𝑀 · 𝑁)) = (1 · 𝑁)) |
23 | nncn 12266 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ → 𝑁 ∈ ℂ) | |
24 | 23 | 3ad2ant3 1132 | . . . . . 6 ⊢ ((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℂ) |
25 | 24 | adantr 479 | . . . . 5 ⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 gcd 𝑀) = 1) → 𝑁 ∈ ℂ) |
26 | 25 | mullidd 11273 | . . . 4 ⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 gcd 𝑀) = 1) → (1 · 𝑁) = 𝑁) |
27 | 22, 26 | eqtrd 2766 | . . 3 ⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 gcd 𝑀) = 1) → ((𝐾 · 𝑁) gcd (𝑀 · 𝑁)) = 𝑁) |
28 | 27 | oveq2d 7432 | . 2 ⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 gcd ((𝐾 · 𝑁) gcd (𝑀 · 𝑁))) = (𝐾 gcd 𝑁)) |
29 | 17, 28 | eqtrd 2766 | 1 ⊢ (((𝐾 ∈ ℕ ∧ 𝑀 ∈ ℕ ∧ 𝑁 ∈ ℕ) ∧ (𝐾 gcd 𝑀) = 1) → (𝐾 gcd (𝑀 · 𝑁)) = (𝐾 gcd 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 (class class class)co 7416 ℂcc 11147 1c1 11150 · cmul 11154 ℕcn 12258 ℕ0cn0 12518 ℤcz 12604 gcd cgcd 16489 |
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 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 |
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 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-sup 9478 df-inf 9479 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-div 11913 df-nn 12259 df-2 12321 df-3 12322 df-n0 12519 df-z 12605 df-uz 12869 df-rp 13023 df-fl 13806 df-mod 13884 df-seq 14016 df-exp 14076 df-cj 15099 df-re 15100 df-im 15101 df-sqrt 15235 df-abs 15236 df-dvds 16252 df-gcd 16490 |
This theorem is referenced by: rplpwr 16554 coprmprod 16657 lgsquad2lem2 27411 |
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