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Mirrors > Home > MPE Home > Th. List > Mathboxes > coprmdvds2d | Structured version Visualization version GIF version |
Description: If an integer is divisible by two coprime integers, then it is divisible by their product, a deduction version. (Contributed by metakunt, 12-May-2024.) |
Ref | Expression |
---|---|
coprmdvds2d.1 | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
coprmdvds2d.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
coprmdvds2d.3 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
coprmdvds2d.4 | ⊢ (𝜑 → (𝐾 gcd 𝑀) = 1) |
coprmdvds2d.5 | ⊢ (𝜑 → 𝐾 ∥ 𝑁) |
coprmdvds2d.6 | ⊢ (𝜑 → 𝑀 ∥ 𝑁) |
Ref | Expression |
---|---|
coprmdvds2d | ⊢ (𝜑 → (𝐾 · 𝑀) ∥ 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coprmdvds2d.5 | . 2 ⊢ (𝜑 → 𝐾 ∥ 𝑁) | |
2 | coprmdvds2d.6 | . 2 ⊢ (𝜑 → 𝑀 ∥ 𝑁) | |
3 | coprmdvds2d.1 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
4 | coprmdvds2d.2 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | coprmdvds2d.3 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
6 | 3, 4, 5 | 3jca 1126 | . . . 4 ⊢ (𝜑 → (𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) |
7 | coprmdvds2d.4 | . . . 4 ⊢ (𝜑 → (𝐾 gcd 𝑀) = 1) | |
8 | 6, 7 | jca 511 | . . 3 ⊢ (𝜑 → ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1)) |
9 | coprmdvds2 16677 | . . 3 ⊢ (((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝐾 gcd 𝑀) = 1) → ((𝐾 ∥ 𝑁 ∧ 𝑀 ∥ 𝑁) → (𝐾 · 𝑀) ∥ 𝑁)) | |
10 | 8, 9 | syl 17 | . 2 ⊢ (𝜑 → ((𝐾 ∥ 𝑁 ∧ 𝑀 ∥ 𝑁) → (𝐾 · 𝑀) ∥ 𝑁)) |
11 | 1, 2, 10 | mp2and 698 | 1 ⊢ (𝜑 → (𝐾 · 𝑀) ∥ 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1535 ∈ wcel 2104 class class class wbr 5149 (class class class)co 7425 1c1 11147 · cmul 11151 ℤcz 12604 ∥ cdvds 16276 gcd cgcd 16517 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-om 7881 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-er 8738 df-en 8979 df-dom 8980 df-sdom 8981 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-n0 12518 df-z 12605 df-uz 12870 df-rp 13026 df-fl 13818 df-mod 13896 df-seq 14029 df-exp 14089 df-cj 15124 df-re 15125 df-im 15126 df-sqrt 15260 df-abs 15261 df-dvds 16277 df-gcd 16518 |
This theorem is referenced by: lcmineqlem14 41985 aks4d1p8d1 42027 aks4d1p8d2 42028 aks4d1p8 42030 |
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