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Theorem nzin 40943
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 15615 . . . . . . . . 9 (𝑀𝑛 → (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ))
2 dvdszrcl 15615 . . . . . . . . 9 (𝑁𝑛 → (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ))
31, 2anim12i 615 . . . . . . . 8 ((𝑀𝑛𝑁𝑛) → ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ)))
4 anandir 676 . . . . . . . 8 (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ) ↔ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ)))
53, 4sylibr 237 . . . . . . 7 ((𝑀𝑛𝑁𝑛) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ))
65ancomd 465 . . . . . 6 ((𝑀𝑛𝑁𝑛) → (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
7 lcmdvds 15953 . . . . . . 7 ((𝑛 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
873expb 1117 . . . . . 6 ((𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
96, 8mpcom 38 . . . . 5 ((𝑀𝑛𝑁𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛)
10 elin 3936 . . . . . 6 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})))
11 reldvds 40940 . . . . . . . 8 Rel ∥
12 elrelimasn 5941 . . . . . . . 8 (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀𝑛))
1311, 12ax-mp 5 . . . . . . 7 (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀𝑛)
14 elrelimasn 5941 . . . . . . . 8 (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁𝑛))
1511, 14ax-mp 5 . . . . . . 7 (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁𝑛)
1613, 15anbi12i 629 . . . . . 6 ((𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})) ↔ (𝑀𝑛𝑁𝑛))
1710, 16bitri 278 . . . . 5 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑀𝑛𝑁𝑛))
18 elrelimasn 5941 . . . . . 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 295 . . . 4 (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) → 𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}))
2120ssriv 3958 . . 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 15943 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
2623, 24, 25syl2anc 587 . . . . 5 (𝜑 → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
2726simpld 498 . . . 4 (𝜑𝑀 ∥ (𝑀 lcm 𝑁))
28 lcmcl 15946 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0)
2923, 24, 28syl2anc 587 . . . . . 6 (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0)
3029nn0zd 12085 . . . . 5 (𝜑 → (𝑀 lcm 𝑁) ∈ ℤ)
3130, 23nzss 40942 . . . 4 (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}) ↔ 𝑀 ∥ (𝑀 lcm 𝑁)))
3227, 31mpbird 260 . . 3 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}))
3326simprd 499 . . . 4 (𝜑𝑁 ∥ (𝑀 lcm 𝑁))
3430, 24nzss 40942 . . . 4 (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ (𝑀 lcm 𝑁)))
3533, 34mpbird 260 . . 3 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}))
3632, 35ssind 4195 . 2 (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})))
3722, 36eqssd 3971 1 (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cin 3919  wss 3920  {csn 4551   class class class wbr 5053  cima 5546  Rel wrel 5548  (class class class)co 7150  0cn0 11897  cz 11981  cdvds 15610   lcm clcm 15933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7456  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613  ax-pre-sup 10614
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3483  df-sbc 3760  df-csb 3868  df-dif 3923  df-un 3925  df-in 3927  df-ss 3937  df-pss 3939  df-nul 4278  df-if 4452  df-pw 4525  df-sn 4552  df-pr 4554  df-tp 4556  df-op 4558  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7576  df-2nd 7686  df-wrecs 7944  df-recs 8005  df-rdg 8043  df-er 8286  df-en 8507  df-dom 8508  df-sdom 8509  df-sup 8904  df-inf 8905  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-nn 11638  df-2 11700  df-3 11701  df-n0 11898  df-z 11982  df-uz 12244  df-rp 12390  df-fl 13169  df-mod 13245  df-seq 13377  df-exp 13438  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-dvds 15611  df-gcd 15845  df-lcm 15935
This theorem is referenced by:  nzprmdif  40944
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