Mathbox for Steve Rodriguez < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nzin Structured version   Visualization version   GIF version

Theorem nzin 37539
 Description: The intersection of the set of multiples of m, mℤ, and those of n, nℤ, is the set of multiples of their least common multiple. Roughly Lemma 2.1(c) of https://www.mscs.dal.ca/~selinger/3343/handouts/ideals.pdf p. 5 and Problem 1(b) of https://people.math.binghamton.edu/mazur/teach/40107/40107h16sol.pdf p. 1, with mℤ and nℤ as images of the divides relation under m and n. (Contributed by Steve Rodriguez, 20-Jan-2020.)
Hypotheses
Ref Expression
nzin.m (𝜑𝑀 ∈ ℤ)
nzin.n (𝜑𝑁 ∈ ℤ)
Assertion
Ref Expression
nzin (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))

Proof of Theorem nzin
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 dvdszrcl 14826 . . . . . . . . 9 (𝑀𝑛 → (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ))
2 dvdszrcl 14826 . . . . . . . . 9 (𝑁𝑛 → (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ))
31, 2anim12i 588 . . . . . . . 8 ((𝑀𝑛𝑁𝑛) → ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ)))
4 anandir 868 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ) ↔ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ)))
53, 4sylibr 223 . . . . . . 7 ((𝑀𝑛𝑁𝑛) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ))
65ancomd 466 . . . . . 6 ((𝑀𝑛𝑁𝑛) → (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
7 lcmdvds 15159 . . . . . . 7 ((𝑛 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
873expb 1258 . . . . . 6 ((𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
96, 8mpcom 37 . . . . 5 ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛)
10 elin 3758 . . . . . 6 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})))
11 reldvds 37536 . . . . . . . 8 Rel ∥
12 elrelimasn 5408 . . . . . . . 8 (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀𝑛))
1311, 12ax-mp 5 . . . . . . 7 (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀𝑛)
14 elrelimasn 5408 . . . . . . . 8 (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁𝑛))
1511, 14ax-mp 5 . . . . . . 7 (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁𝑛)
1613, 15anbi12i 729 . . . . . 6 ((𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})) ↔ (𝑀𝑛𝑁𝑛))
1710, 16bitri 263 . . . . 5 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑀𝑛𝑁𝑛))
18 elrelimasn 5408 . . . . . 6 (Rel ∥ → (𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}) ↔ (𝑀 lcm 𝑁) ∥ 𝑛))
1911, 18ax-mp 5 . . . . 5 (𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}) ↔ (𝑀 lcm 𝑁) ∥ 𝑛)
209, 17, 193imtr4i 280 . . . 4 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) → 𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}))
2120ssriv 3572 . . 3 (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ⊆ ( ∥ “ {(𝑀 lcm 𝑁)})
2221a1i 11 . 2 (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ⊆ ( ∥ “ {(𝑀 lcm 𝑁)}))
23 nzin.m . . . . . 6 (𝜑𝑀 ∈ ℤ)
24 nzin.n . . . . . 6 (𝜑𝑁 ∈ ℤ)
25 dvdslcm 15149 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
2623, 24, 25syl2anc 691 . . . . 5 (𝜑 → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
2726simpld 474 . . . 4 (𝜑𝑀 ∥ (𝑀 lcm 𝑁))
28 lcmcl 15152 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0)
2923, 24, 28syl2anc 691 . . . . . 6 (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0)
3029nn0zd 11356 . . . . 5 (𝜑 → (𝑀 lcm 𝑁) ∈ ℤ)
3130, 23nzss 37538 . . . 4 (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}) ↔ 𝑀 ∥ (𝑀 lcm 𝑁)))
3227, 31mpbird 246 . . 3 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}))
3326simprd 478 . . . 4 (𝜑𝑁 ∥ (𝑀 lcm 𝑁))
3430, 24nzss 37538 . . . 4 (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ (𝑀 lcm 𝑁)))
3533, 34mpbird 246 . . 3 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}))
3632, 35ssind 3799 . 2 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})))
3722, 36eqssd 3585 1 (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  {csn 4125   class class class wbr 4583   “ cima 5041  Rel wrel 5043  (class class class)co 6549  ℕ0cn0 11169  ℤcz 11254   ∥ cdvds 14821   lcm clcm 15139 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-rp 11709  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-dvds 14822  df-gcd 15055  df-lcm 15141 This theorem is referenced by:  nzprmdif  37540
 Copyright terms: Public domain W3C validator