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Theorem lcmass 16490
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 920 . . 3 (((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0) ↔ (𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)))
2 anass 469 . . . . 5 (((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥) ↔ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥)))
32rabbii 3413 . . . 4 {𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)} = {𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}
43infeq1i 9414 . . 3 inf({𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}, ℝ, < )
51, 4ifbieq2i 4511 . 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, < ))
6 lcmcl 16477 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
763adant3 1132 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℕ0)
87nn0zd 12525 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑁 lcm 𝑀) ∈ ℤ)
9 simp3 1138 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑃 ∈ ℤ)
10 lcmval 16468 . . . 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, < )))
118, 9, 10syl2anc 584 . . 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, < )))
12 lcmeq0 16476 . . . . . . 7 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
13123adant3 1132 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) = 0 ↔ (𝑁 = 0 ∨ 𝑀 = 0)))
1413orbi1d 915 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0) ↔ ((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0)))
1514bicomd 222 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (((𝑁 = 0 ∨ 𝑀 = 0) ∨ 𝑃 = 0) ↔ ((𝑁 lcm 𝑀) = 0 ∨ 𝑃 = 0)))
16 nnz 12520 . . . . . . . . 9 (𝑥 ∈ ℕ → 𝑥 ∈ ℤ)
1716adantl 482 . . . . . . . 8 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑥 ∈ ℤ)
18 simp1 1136 . . . . . . . . 9 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → 𝑁 ∈ ℤ)
1918adantr 481 . . . . . . . 8 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑁 ∈ ℤ)
20 simpl2 1192 . . . . . . . 8 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑀 ∈ ℤ)
21 lcmdvdsb 16489 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → ((𝑁𝑥𝑀𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
2217, 19, 20, 21syl3anc 1371 . . . . . . 7 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑁𝑥𝑀𝑥) ↔ (𝑁 lcm 𝑀) ∥ 𝑥))
2322anbi1d 630 . . . . . 6 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → (((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥) ↔ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)))
2423rabbidva 3414 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → {𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)} = {𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)})
2524infeq1d 9413 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → inf({𝑥 ∈ ℕ ∣ ((𝑁𝑥𝑀𝑥) ∧ 𝑃𝑥)}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ ((𝑁 lcm 𝑀) ∥ 𝑥𝑃𝑥)}, ℝ, < ))
2615, 25ifbieq2d 4512 . . 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, < )))
2711, 26eqtr4d 2779 . 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, < )))
28 lcmcl 16477 . . . . . 6 ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
29283adant1 1130 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℕ0)
3029nn0zd 12525 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑀 lcm 𝑃) ∈ ℤ)
31 lcmval 16468 . . . 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, < )))
3218, 30, 31syl2anc 584 . . 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, < )))
33 lcmeq0 16476 . . . . . . 7 ((𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
34333adant1 1130 . . . . . 6 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀 lcm 𝑃) = 0 ↔ (𝑀 = 0 ∨ 𝑃 = 0)))
3534orbi2d 914 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0) ↔ (𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0))))
3635bicomd 222 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 = 0 ∨ (𝑀 = 0 ∨ 𝑃 = 0)) ↔ (𝑁 = 0 ∨ (𝑀 lcm 𝑃) = 0)))
379adantr 481 . . . . . . . 8 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → 𝑃 ∈ ℤ)
38 lcmdvdsb 16489 . . . . . . . 8 ((𝑥 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑀𝑥𝑃𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
3917, 20, 37, 38syl3anc 1371 . . . . . . 7 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑀𝑥𝑃𝑥) ↔ (𝑀 lcm 𝑃) ∥ 𝑥))
4039anbi2d 629 . . . . . 6 (((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) ∧ 𝑥 ∈ ℕ) → ((𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥)) ↔ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)))
4140rabbidva 3414 . . . . 5 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → {𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))} = {𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)})
4241infeq1d 9413 . . . 4 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀𝑥𝑃𝑥))}, ℝ, < ) = inf({𝑥 ∈ ℕ ∣ (𝑁𝑥 ∧ (𝑀 lcm 𝑃) ∥ 𝑥)}, ℝ, < ))
4336, 42ifbieq2d 4512 . . 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, < )))
4432, 43eqtr4d 2779 . 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, < )))
455, 27, 443eqtr4a 2802 1 ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑃 ∈ ℤ) → ((𝑁 lcm 𝑀) lcm 𝑃) = (𝑁 lcm (𝑀 lcm 𝑃)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 845  w3a 1087   = wceq 1541  wcel 2106  {crab 3407  ifcif 4486   class class class wbr 5105  (class class class)co 7357  infcinf 9377  cr 11050  0cc0 11051   < clt 11189  cn 12153  0cn0 12413  cz 12499  cdvds 16136   lcm clcm 16464
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-rp 12916  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-gcd 16375  df-lcm 16466
This theorem is referenced by:  lcmfunsnlem2lem2  16515  lcmfun  16521
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