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Theorem lcmdvdsb 16542

Description: Biconditional form of lcmdvds 16537. (Contributed by Steve Rodriguez, 20-Jan-2020.)

**Assertion**
Ref	Expression
lcmdvdsb	⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾) ↔ (𝑀 lcm 𝑁) ∥ 𝐾))

**Proof of Theorem lcmdvdsb**
Step	Hyp	Ref	Expression
1		lcmdvds 16537	. 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾) → (𝑀 lcm 𝑁) ∥ 𝐾))
2		dvdslcm 16527	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
3	2	simpld 494	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 lcm 𝑁))
4	3	3adant1 1130	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∥ (𝑀 lcm 𝑁))
5		simp2 1137	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ)
6		lcmcl 16530	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ₀)
7	6	nn0zd 12515	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℤ)
8	7	3adant1 1130	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℤ)
9		simp1 1136	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝐾 ∈ ℤ)
10		dvdstr 16223	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝐾) → 𝑀 ∥ 𝐾))
11	5, 8, 9, 10	syl3anc 1373	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝐾) → 𝑀 ∥ 𝐾))
12	4, 11	mpand 695	. . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) ∥ 𝐾 → 𝑀 ∥ 𝐾))
13	2	simprd 495	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 lcm 𝑁))
14	13	3adant1 1130	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∥ (𝑀 lcm 𝑁))
15		dvdstr 16223	. . . . . 6 ⊢ ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝐾) → 𝑁 ∥ 𝐾))
16	15	3com13 1124	. . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ (𝑀 lcm 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝐾) → 𝑁 ∥ 𝐾))
17	8, 16	syld3an2 1413	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 ∥ (𝑀 lcm 𝑁) ∧ (𝑀 lcm 𝑁) ∥ 𝐾) → 𝑁 ∥ 𝐾))
18	14, 17	mpand 695	. . 3 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) ∥ 𝐾 → 𝑁 ∥ 𝐾))
19	12, 18	jcad 512	. 2 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 lcm 𝑁) ∥ 𝐾 → (𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾)))
20	1, 19	impbid 212	1 ⊢ ((𝐾 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ 𝐾 ∧ 𝑁 ∥ 𝐾) ↔ (𝑀 lcm 𝑁) ∥ 𝐾))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2109 class class class wbr 5095 (class class class)co 7353 ℤcz 12489 ∥ cdvds 16181 lcm clcm 16517

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-rp 12912 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-dvds 16182 df-gcd 16424 df-lcm 16519

This theorem is referenced by: lcmass 16543

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