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Mirrors > Home > MPE Home > Th. List > lcmfass | Structured version Visualization version GIF version |
Description: Associative law for the lcm function. (Contributed by AV, 27-Aug-2020.) |
Ref | Expression |
---|---|
lcmfass | ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘({(lcm‘𝑌)} ∪ 𝑍)) = (lcm‘(𝑌 ∪ {(lcm‘𝑍)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcmfcl 15542 | . . . . . 6 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℕ0) | |
2 | 1 | nn0zd 11680 | . . . . 5 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℤ) |
3 | lcmfsn 15549 | . . . . 5 ⊢ ((lcm‘𝑌) ∈ ℤ → (lcm‘{(lcm‘𝑌)}) = (abs‘(lcm‘𝑌))) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘{(lcm‘𝑌)}) = (abs‘(lcm‘𝑌))) |
5 | nn0re 11501 | . . . . . 6 ⊢ ((lcm‘𝑌) ∈ ℕ0 → (lcm‘𝑌) ∈ ℝ) | |
6 | nn0ge0 11518 | . . . . . 6 ⊢ ((lcm‘𝑌) ∈ ℕ0 → 0 ≤ (lcm‘𝑌)) | |
7 | 5, 6 | jca 501 | . . . . 5 ⊢ ((lcm‘𝑌) ∈ ℕ0 → ((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌))) |
8 | absid 14237 | . . . . 5 ⊢ (((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌)) → (abs‘(lcm‘𝑌)) = (lcm‘𝑌)) | |
9 | 1, 7, 8 | 3syl 18 | . . . 4 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (abs‘(lcm‘𝑌)) = (lcm‘𝑌)) |
10 | 4, 9 | eqtrd 2805 | . . 3 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘{(lcm‘𝑌)}) = (lcm‘𝑌)) |
11 | lcmfcl 15542 | . . . . . 6 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) ∈ ℕ0) | |
12 | 11 | nn0zd 11680 | . . . . 5 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) ∈ ℤ) |
13 | lcmfsn 15549 | . . . . 5 ⊢ ((lcm‘𝑍) ∈ ℤ → (lcm‘{(lcm‘𝑍)}) = (abs‘(lcm‘𝑍))) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘{(lcm‘𝑍)}) = (abs‘(lcm‘𝑍))) |
15 | nn0re 11501 | . . . . . 6 ⊢ ((lcm‘𝑍) ∈ ℕ0 → (lcm‘𝑍) ∈ ℝ) | |
16 | nn0ge0 11518 | . . . . . 6 ⊢ ((lcm‘𝑍) ∈ ℕ0 → 0 ≤ (lcm‘𝑍)) | |
17 | 15, 16 | jca 501 | . . . . 5 ⊢ ((lcm‘𝑍) ∈ ℕ0 → ((lcm‘𝑍) ∈ ℝ ∧ 0 ≤ (lcm‘𝑍))) |
18 | absid 14237 | . . . . 5 ⊢ (((lcm‘𝑍) ∈ ℝ ∧ 0 ≤ (lcm‘𝑍)) → (abs‘(lcm‘𝑍)) = (lcm‘𝑍)) | |
19 | 11, 17, 18 | 3syl 18 | . . . 4 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (abs‘(lcm‘𝑍)) = (lcm‘𝑍)) |
20 | 14, 19 | eqtr2d 2806 | . . 3 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) = (lcm‘{(lcm‘𝑍)})) |
21 | 10, 20 | oveqan12d 6810 | . 2 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → ((lcm‘{(lcm‘𝑌)}) lcm (lcm‘𝑍)) = ((lcm‘𝑌) lcm (lcm‘{(lcm‘𝑍)}))) |
22 | 2 | snssd 4475 | . . . 4 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → {(lcm‘𝑌)} ⊆ ℤ) |
23 | snfi 8192 | . . . 4 ⊢ {(lcm‘𝑌)} ∈ Fin | |
24 | 22, 23 | jctir 510 | . . 3 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ({(lcm‘𝑌)} ⊆ ℤ ∧ {(lcm‘𝑌)} ∈ Fin)) |
25 | lcmfun 15559 | . . 3 ⊢ ((({(lcm‘𝑌)} ⊆ ℤ ∧ {(lcm‘𝑌)} ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘({(lcm‘𝑌)} ∪ 𝑍)) = ((lcm‘{(lcm‘𝑌)}) lcm (lcm‘𝑍))) | |
26 | 24, 25 | sylan 569 | . 2 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘({(lcm‘𝑌)} ∪ 𝑍)) = ((lcm‘{(lcm‘𝑌)}) lcm (lcm‘𝑍))) |
27 | 12 | snssd 4475 | . . . 4 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → {(lcm‘𝑍)} ⊆ ℤ) |
28 | snfi 8192 | . . . 4 ⊢ {(lcm‘𝑍)} ∈ Fin | |
29 | 27, 28 | jctir 510 | . . 3 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ({(lcm‘𝑍)} ⊆ ℤ ∧ {(lcm‘𝑍)} ∈ Fin)) |
30 | lcmfun 15559 | . . 3 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ ({(lcm‘𝑍)} ⊆ ℤ ∧ {(lcm‘𝑍)} ∈ Fin)) → (lcm‘(𝑌 ∪ {(lcm‘𝑍)})) = ((lcm‘𝑌) lcm (lcm‘{(lcm‘𝑍)}))) | |
31 | 29, 30 | sylan2 580 | . 2 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘(𝑌 ∪ {(lcm‘𝑍)})) = ((lcm‘𝑌) lcm (lcm‘{(lcm‘𝑍)}))) |
32 | 21, 26, 31 | 3eqtr4d 2815 | 1 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘({(lcm‘𝑌)} ∪ 𝑍)) = (lcm‘(𝑌 ∪ {(lcm‘𝑍)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ∪ cun 3721 ⊆ wss 3723 {csn 4316 class class class wbr 4786 ‘cfv 6029 (class class class)co 6791 Fincfn 8107 ℝcr 10135 0cc0 10136 ≤ cle 10275 ℕ0cn0 11492 ℤcz 11577 abscabs 14175 lcm clcm 15502 lcmclcmf 15503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7094 ax-inf2 8700 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 ax-pre-sup 10214 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-isom 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-sup 8502 df-inf 8503 df-oi 8569 df-card 8963 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-div 10885 df-nn 11221 df-2 11279 df-3 11280 df-n0 11493 df-z 11578 df-uz 11887 df-rp 12029 df-fz 12527 df-fzo 12667 df-fl 12794 df-mod 12870 df-seq 13002 df-exp 13061 df-hash 13315 df-cj 14040 df-re 14041 df-im 14042 df-sqrt 14176 df-abs 14177 df-clim 14420 df-prod 14836 df-dvds 15183 df-gcd 15418 df-lcm 15504 df-lcmf 15505 |
This theorem is referenced by: (None) |
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