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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 16662 | . . . . . 6 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℕ0) | |
2 | 1 | nn0zd 12637 | . . . . 5 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘𝑌) ∈ ℤ) |
3 | lcmfsn 16669 | . . . . 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 12533 | . . . . . 6 ⊢ ((lcm‘𝑌) ∈ ℕ0 → (lcm‘𝑌) ∈ ℝ) | |
6 | nn0ge0 12549 | . . . . . 6 ⊢ ((lcm‘𝑌) ∈ ℕ0 → 0 ≤ (lcm‘𝑌)) | |
7 | 5, 6 | jca 511 | . . . . 5 ⊢ ((lcm‘𝑌) ∈ ℕ0 → ((lcm‘𝑌) ∈ ℝ ∧ 0 ≤ (lcm‘𝑌))) |
8 | absid 15332 | . . . . 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 2775 | . . 3 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → (lcm‘{(lcm‘𝑌)}) = (lcm‘𝑌)) |
11 | lcmfcl 16662 | . . . . . 6 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) ∈ ℕ0) | |
12 | 11 | nn0zd 12637 | . . . . 5 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) ∈ ℤ) |
13 | lcmfsn 16669 | . . . . 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 12533 | . . . . . 6 ⊢ ((lcm‘𝑍) ∈ ℕ0 → (lcm‘𝑍) ∈ ℝ) | |
16 | nn0ge0 12549 | . . . . . 6 ⊢ ((lcm‘𝑍) ∈ ℕ0 → 0 ≤ (lcm‘𝑍)) | |
17 | 15, 16 | jca 511 | . . . . 5 ⊢ ((lcm‘𝑍) ∈ ℕ0 → ((lcm‘𝑍) ∈ ℝ ∧ 0 ≤ (lcm‘𝑍))) |
18 | absid 15332 | . . . . 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 2776 | . . 3 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → (lcm‘𝑍) = (lcm‘{(lcm‘𝑍)})) |
21 | 10, 20 | oveqan12d 7450 | . 2 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → ((lcm‘{(lcm‘𝑌)}) lcm (lcm‘𝑍)) = ((lcm‘𝑌) lcm (lcm‘{(lcm‘𝑍)}))) |
22 | 2 | snssd 4814 | . . . 4 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → {(lcm‘𝑌)} ⊆ ℤ) |
23 | snfi 9082 | . . . 4 ⊢ {(lcm‘𝑌)} ∈ Fin | |
24 | 22, 23 | jctir 520 | . . 3 ⊢ ((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) → ({(lcm‘𝑌)} ⊆ ℤ ∧ {(lcm‘𝑌)} ∈ Fin)) |
25 | lcmfun 16679 | . . 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 580 | . 2 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘({(lcm‘𝑌)} ∪ 𝑍)) = ((lcm‘{(lcm‘𝑌)}) lcm (lcm‘𝑍))) |
27 | 12 | snssd 4814 | . . . 4 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → {(lcm‘𝑍)} ⊆ ℤ) |
28 | snfi 9082 | . . . 4 ⊢ {(lcm‘𝑍)} ∈ Fin | |
29 | 27, 28 | jctir 520 | . . 3 ⊢ ((𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin) → ({(lcm‘𝑍)} ⊆ ℤ ∧ {(lcm‘𝑍)} ∈ Fin)) |
30 | lcmfun 16679 | . . 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 593 | . 2 ⊢ (((𝑌 ⊆ ℤ ∧ 𝑌 ∈ Fin) ∧ (𝑍 ⊆ ℤ ∧ 𝑍 ∈ Fin)) → (lcm‘(𝑌 ∪ {(lcm‘𝑍)})) = ((lcm‘𝑌) lcm (lcm‘{(lcm‘𝑍)}))) |
32 | 21, 26, 31 | 3eqtr4d 2785 | 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 395 = wceq 1537 ∈ wcel 2106 ∪ cun 3961 ⊆ wss 3963 {csn 4631 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Fincfn 8984 ℝcr 11152 0cc0 11153 ≤ cle 11294 ℕ0cn0 12524 ℤcz 12611 abscabs 15270 lcm clcm 16622 lcmclcmf 16623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-prod 15937 df-dvds 16288 df-gcd 16529 df-lcm 16624 df-lcmf 16625 |
This theorem is referenced by: (None) |
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