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Theorem lcmass 12068
Description: Associative law for lcm operator. (Contributed by Steve Rodriguez, 20-Jan-2020.) (Proof shortened by AV, 16-Sep-2020.)
Assertion
Ref Expression
lcmass ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = (𝑁 lcm (𝑀 lcm 𝑃)))

Proof of Theorem lcmass
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 orass 767 . . 3 (((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0) ↔ (𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)))
2 anass 401 . . . . . 6 (((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥) ↔ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥)))
32a1i 9 . . . . 5 (𝑥 ∈ ℕ → (((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥) ↔ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))))
43rabbiia 2722 . . . 4 {𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)} = {𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}
54infeq1i 7006 . . 3 inf({𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}, ℝ, < )
61, 5ifbieq2i 3557 . 2 if(((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)}, ℝ, < )) = if((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)), 0, inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}, ℝ, < ))
7 lcmcl 12055 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
873adant3 1017 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
98nn0zd 9362 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℤ)
10 simp3 999 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℤ)
11 lcmval 12046 . . . 4 (((𝑁 lcm 𝑀) ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = if(((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)}, ℝ, < )))
129, 10, 11syl2anc 411 . . 3 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = if(((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)}, ℝ, < )))
13 lcmeq0 12054 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
14133adant3 1017 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
1514orbi1d 791 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0) ↔ ((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0)))
1615bicomd 141 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0) ↔ ((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0)))
17 nnz 9261 . . . . . . . . 9 (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
1817adantl 277 . . . . . . . 8 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℤ)
19 simp1 997 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑁 ∈ ℤ)
2019adantr 276 . . . . . . . 8 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
21 simpl2 1001 . . . . . . . 8 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑀 ∈ ℤ)
22 lcmdvdsb 12067 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁𝑥𝑀𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
2318, 20, 21, 22syl3anc 1238 . . . . . . 7 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑁𝑥𝑀𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
2423anbi1d 465 . . . . . 6 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → (((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥) ↔ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)))
2524rabbidva 2725 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → {𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)} = {𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)})
2625infeq1d 7005 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → inf({𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)}, ℝ, < ))
2716, 26ifbieq2d 3558 . . 3 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → if(((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)}, ℝ, < )) = if(((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)}, ℝ, < )))
2812, 27eqtr4d 2213 . 2 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = if(((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0), 0, inf({𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)}, ℝ, < )))
29 lcmcl 12055 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
30293adant1 1015 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
3130nn0zd 9362 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℤ)
32 lcmval 12046 . . . 4 ((𝑁 ∈ ℤ ∧ (𝑀 lcm 𝑃) ∈ ℤ) → (𝑁 lcm (𝑀 lcm 𝑃)) = if((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0), 0, inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < )))
3319, 31, 32syl2anc 411 . . 3 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm (𝑀 lcm 𝑃)) = if((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0), 0, inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < )))
34 lcmeq0 12054 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
35343adant1 1015 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
3635orbi2d 790 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0) ↔ (𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0))))
3736bicomd 141 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)) ↔ (𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0)))
3810adantr 276 . . . . . . . 8 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑃 ∈ ℤ)
39 lcmdvdsb 12067 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀𝑥𝑃𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
4018, 21, 38, 39syl3anc 1238 . . . . . . 7 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑀𝑥𝑃𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
4140anbi2d 464 . . . . . 6 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥)) ↔ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)))
4241rabbidva 2725 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → {𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))} = {𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)})
4342infeq1d 7005 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < ))
4437, 43ifbieq2d 3558 . . 3 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → if((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)), 0, inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}, ℝ, < )) = if((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0), 0, inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < )))
4533, 44eqtr4d 2213 . 2 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm (𝑀 lcm 𝑃)) = if((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)), 0, inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}, ℝ, < )))
466, 28, 453eqtr4a 2236 1 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = (𝑁 lcm (𝑀 lcm 𝑃)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  w3a 978   = wceq 1353  wcel 2148  {crab 2459  ifcif 3534   class class class wbr 4000  (class class class)co 5869  infcinf 6976  cr 7801  0cc0 7802   < clt 7982  cn 8908  0cn0 9165  cz 9242  cdvds 11778   lcm clcm 12043
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920  ax-arch 7921  ax-caucvg 7922
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-isom 5221  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-sup 6977  df-inf 6978  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-n0 9166  df-z 9243  df-uz 9518  df-q 9609  df-rp 9641  df-fz 9996  df-fzo 10129  df-fl 10256  df-mod 10309  df-seqfrec 10432  df-exp 10506  df-cj 10835  df-re 10836  df-im 10837  df-rsqrt 10991  df-abs 10992  df-dvds 11779  df-gcd 11927  df-lcm 12044
This theorem is referenced by: (None)
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