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	⊢ (𝜑 → 𝑀 ∈ ℤ)
nzin.n	⊢ (𝜑 → 𝑁 ∈ ℤ)

Assertion

Ref	Expression
nzin	⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))

Proof of Theorem nzin

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvdszrcl 14826	. . . . . . . . 9 ⊢ (𝑀 ∥ 𝑛 → (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ))
2		dvdszrcl 14826	. . . . . . . . 9 ⊢ (𝑁 ∥ 𝑛 → (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ))
3	1, 2	anim12i 588	. . . . . . . 8 ⊢ ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → ((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)))
5	3, 4	sylibr 223	. . . . . . 7 ⊢ ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ))
6	5	ancomd 466	. . . . . 6 ⊢ ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
7		lcmdvds 15159	. . . . . . 7 ⊢ ((𝑛 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
8	7	3expb 1258	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
9	6, 8	mpcom 37	. . . . 5 ⊢ ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (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}) ↔ 𝑀 ∥ 𝑛))
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀 ∥ 𝑛)
14		elrelimasn 5408	. . . . . . . 8 ⊢ (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑛))
15	11, 14	ax-mp 5	. . . . . . 7 ⊢ (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑛)
16	13, 15	anbi12i 729	. . . . . 6 ⊢ ((𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))
17	10, 16	bitri 263	. . . . 5 ⊢ (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))
18		elrelimasn 5408	. . . . . 6 ⊢ (Rel ∥ → (𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}) ↔ (𝑀 lcm 𝑁) ∥ 𝑛))
19	11, 18	ax-mp 5	. . . . 5 ⊢ (𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}) ↔ (𝑀 lcm 𝑁) ∥ 𝑛)
20	9, 17, 19	3imtr4i 280	. . . 4 ⊢ (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) → 𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}))
21	20	ssriv 3572	. . 3 ⊢ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ⊆ ( ∥ “ {(𝑀 lcm 𝑁)})
22	21	a1i 11	. 2 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ⊆ ( ∥ “ {(𝑀 lcm 𝑁)}))
23		nzin.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
24		nzin.n	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
25		dvdslcm 15149	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
26	23, 24, 25	syl2anc 691	. . . . 5 ⊢ (𝜑 → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
27	26	simpld 474	. . . 4 ⊢ (𝜑 → 𝑀 ∥ (𝑀 lcm 𝑁))
28		lcmcl 15152	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0)
29	23, 24, 28	syl2anc 691	. . . . . 6 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0)
30	29	nn0zd 11356	. . . . 5 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℤ)
31	30, 23	nzss 37538	. . . 4 ⊢ (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}) ↔ 𝑀 ∥ (𝑀 lcm 𝑁)))
32	27, 31	mpbird 246	. . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}))
33	26	simprd 478	. . . 4 ⊢ (𝜑 → 𝑁 ∥ (𝑀 lcm 𝑁))
34	30, 24	nzss 37538	. . . 4 ⊢ (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ (𝑀 lcm 𝑁)))
35	33, 34	mpbird 246	. . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}))
36	32, 35	ssind 3799	. 2 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})))
37	22, 36	eqssd 3585	1 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))

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