Theorem nzin 43591

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 16201	. . . . . . . . 9 ⊢ (𝑀 ∥ 𝑛 → (𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ))
2		dvdszrcl 16201	. . . . . . . . 9 ⊢ (𝑁 ∥ 𝑛 → (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ))
3	1, 2	anim12i 612	. . . . . . . 8 ⊢ ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ)))
4		anandir 674	. . . . . . . 8 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ) ↔ ((𝑀 ∈ ℤ ∧ 𝑛 ∈ ℤ) ∧ (𝑁 ∈ ℤ ∧ 𝑛 ∈ ℤ)))
5	3, 4	sylibr 233	. . . . . . 7 ⊢ ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝑛 ∈ ℤ))
6	5	ancomd 461	. . . . . 6 ⊢ ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)))
7		lcmdvds 16544	. . . . . . 7 ⊢ ((𝑛 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
8	7	3expb 1117	. . . . . 6 ⊢ ((𝑛 ∈ ℤ ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛))
9	6, 8	mpcom 38	. . . . 5 ⊢ ((𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛) → (𝑀 lcm 𝑁) ∥ 𝑛)
10		elin 3957	. . . . . 6 ⊢ (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})))
11		reldvds 43588	. . . . . . . 8 ⊢ Rel ∥
12		elrelimasn 6075	. . . . . . . 8 ⊢ (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀 ∥ 𝑛))
13	11, 12	ax-mp 5	. . . . . . 7 ⊢ (𝑛 ∈ ( ∥ “ {𝑀}) ↔ 𝑀 ∥ 𝑛)
14		elrelimasn 6075	. . . . . . . 8 ⊢ (Rel ∥ → (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑛))
15	11, 14	ax-mp 5	. . . . . . 7 ⊢ (𝑛 ∈ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ 𝑛)
16	13, 15	anbi12i 626	. . . . . 6 ⊢ ((𝑛 ∈ ( ∥ “ {𝑀}) ∧ 𝑛 ∈ ( ∥ “ {𝑁})) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))
17	10, 16	bitri 275	. . . . 5 ⊢ (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) ↔ (𝑀 ∥ 𝑛 ∧ 𝑁 ∥ 𝑛))
18		elrelimasn 6075	. . . . . 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 292	. . . 4 ⊢ (𝑛 ∈ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) → 𝑛 ∈ ( ∥ “ {(𝑀 lcm 𝑁)}))
21	20	ssriv 3979	. . 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 16534	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
26	23, 24, 25	syl2anc 583	. . . . 5 ⊢ (𝜑 → (𝑀 ∥ (𝑀 lcm 𝑁) ∧ 𝑁 ∥ (𝑀 lcm 𝑁)))
27	26	simpld 494	. . . 4 ⊢ (𝜑 → 𝑀 ∥ (𝑀 lcm 𝑁))
28		lcmcl 16537	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 lcm 𝑁) ∈ ℕ0)
29	23, 24, 28	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℕ0)
30	29	nn0zd 12582	. . . . 5 ⊢ (𝜑 → (𝑀 lcm 𝑁) ∈ ℤ)
31	30, 23	nzss 43590	. . . 4 ⊢ (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}) ↔ 𝑀 ∥ (𝑀 lcm 𝑁)))
32	27, 31	mpbird 257	. . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑀}))
33	26	simprd 495	. . . 4 ⊢ (𝜑 → 𝑁 ∥ (𝑀 lcm 𝑁))
34	30, 24	nzss 43590	. . . 4 ⊢ (𝜑 → (( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}) ↔ 𝑁 ∥ (𝑀 lcm 𝑁)))
35	33, 34	mpbird 257	. . 3 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ ( ∥ “ {𝑁}))
36	32, 35	ssind 4225	. 2 ⊢ (𝜑 → ( ∥ “ {(𝑀 lcm 𝑁)}) ⊆ (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})))
37	22, 36	eqssd 3992	1 ⊢ (𝜑 → (( ∥ “ {𝑀}) ∩ ( ∥ “ {𝑁})) = ( ∥ “ {(𝑀 lcm 𝑁)}))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098   ∩ cin 3940   ⊆ wss 3941  {csn 4621   class class class wbr 5139   “ cima 5670  Rel wrel 5672  (class class class)co 7402  ℕ₀cn0 12470  ℤcz 12556   ∥ cdvds 16196   lcm clcm 16524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-sup 9434  df-inf 9435  df-pnf 11248  df-mnf 11249  df-xr 11250  df-ltxr 11251  df-le 11252  df-sub 11444  df-neg 11445  df-div 11870  df-nn 12211  df-2 12273  df-3 12274  df-n0 12471  df-z 12557  df-uz 12821  df-rp 12973  df-fl 13755  df-mod 13833  df-seq 13965  df-exp 14026  df-cj 15044  df-re 15045  df-im 15046  df-sqrt 15180  df-abs 15181  df-dvds 16197  df-gcd 16435  df-lcm 16526

This theorem is referenced by:  nzprmdif  43592