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Mirrors > Home > MPE Home > Th. List > ex-lcm | Structured version Visualization version GIF version |
Description: Example for df-lcm 16034. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-lcm | ⊢ (6 lcm 9) = ;18 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 11808 | . . . . 5 ⊢ 6 ∈ ℕ | |
2 | 9nn 11817 | . . . . 5 ⊢ 9 ∈ ℕ | |
3 | 1, 2 | nnmulcli 11744 | . . . 4 ⊢ (6 · 9) ∈ ℕ |
4 | 3 | nncni 11729 | . . 3 ⊢ (6 · 9) ∈ ℂ |
5 | 1 | nnzi 12090 | . . . . 5 ⊢ 6 ∈ ℤ |
6 | 2 | nnzi 12090 | . . . . 5 ⊢ 9 ∈ ℤ |
7 | 5, 6 | pm3.2i 474 | . . . 4 ⊢ (6 ∈ ℤ ∧ 9 ∈ ℤ) |
8 | lcmcl 16045 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℕ0) | |
9 | 8 | nn0cnd 12041 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℂ) |
10 | 7, 9 | ax-mp 5 | . . 3 ⊢ (6 lcm 9) ∈ ℂ |
11 | neggcd 15969 | . . . . . . . 8 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (-6 gcd 9) = (6 gcd 9)) | |
12 | 7, 11 | ax-mp 5 | . . . . . . 7 ⊢ (-6 gcd 9) = (6 gcd 9) |
13 | 12 | eqcomi 2748 | . . . . . 6 ⊢ (6 gcd 9) = (-6 gcd 9) |
14 | ex-gcd 28397 | . . . . . 6 ⊢ (-6 gcd 9) = 3 | |
15 | 13, 14 | eqtri 2762 | . . . . 5 ⊢ (6 gcd 9) = 3 |
16 | 3cn 11800 | . . . . 5 ⊢ 3 ∈ ℂ | |
17 | 15, 16 | eqeltri 2830 | . . . 4 ⊢ (6 gcd 9) ∈ ℂ |
18 | 3ne0 11825 | . . . . 5 ⊢ 3 ≠ 0 | |
19 | 15, 18 | eqnetri 3005 | . . . 4 ⊢ (6 gcd 9) ≠ 0 |
20 | 17, 19 | pm3.2i 474 | . . 3 ⊢ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0) |
21 | 1, 2 | pm3.2i 474 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 9 ∈ ℕ) |
22 | lcmgcdnn 16055 | . . . . . . 7 ⊢ ((6 ∈ ℕ ∧ 9 ∈ ℕ) → ((6 lcm 9) · (6 gcd 9)) = (6 · 9)) | |
23 | 21, 22 | mp1i 13 | . . . . . 6 ⊢ (((6 · 9) ∈ ℂ ∧ (6 lcm 9) ∈ ℂ ∧ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0)) → ((6 lcm 9) · (6 gcd 9)) = (6 · 9)) |
24 | 23 | eqcomd 2745 | . . . . 5 ⊢ (((6 · 9) ∈ ℂ ∧ (6 lcm 9) ∈ ℂ ∧ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0)) → (6 · 9) = ((6 lcm 9) · (6 gcd 9))) |
25 | divmul3 11384 | . . . . 5 ⊢ (((6 · 9) ∈ ℂ ∧ (6 lcm 9) ∈ ℂ ∧ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0)) → (((6 · 9) / (6 gcd 9)) = (6 lcm 9) ↔ (6 · 9) = ((6 lcm 9) · (6 gcd 9)))) | |
26 | 24, 25 | mpbird 260 | . . . 4 ⊢ (((6 · 9) ∈ ℂ ∧ (6 lcm 9) ∈ ℂ ∧ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0)) → ((6 · 9) / (6 gcd 9)) = (6 lcm 9)) |
27 | 26 | eqcomd 2745 | . . 3 ⊢ (((6 · 9) ∈ ℂ ∧ (6 lcm 9) ∈ ℂ ∧ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0)) → (6 lcm 9) = ((6 · 9) / (6 gcd 9))) |
28 | 4, 10, 20, 27 | mp3an 1462 | . 2 ⊢ (6 lcm 9) = ((6 · 9) / (6 gcd 9)) |
29 | 15 | oveq2i 7184 | . 2 ⊢ ((6 · 9) / (6 gcd 9)) = ((6 · 9) / 3) |
30 | 6cn 11810 | . . . 4 ⊢ 6 ∈ ℂ | |
31 | 9cn 11819 | . . . 4 ⊢ 9 ∈ ℂ | |
32 | 30, 31, 16, 18 | divassi 11477 | . . 3 ⊢ ((6 · 9) / 3) = (6 · (9 / 3)) |
33 | 3t3e9 11886 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
34 | 33 | eqcomi 2748 | . . . . . 6 ⊢ 9 = (3 · 3) |
35 | 34 | oveq1i 7183 | . . . . 5 ⊢ (9 / 3) = ((3 · 3) / 3) |
36 | 16, 16, 18 | divcan3i 11467 | . . . . 5 ⊢ ((3 · 3) / 3) = 3 |
37 | 35, 36 | eqtri 2762 | . . . 4 ⊢ (9 / 3) = 3 |
38 | 37 | oveq2i 7184 | . . 3 ⊢ (6 · (9 / 3)) = (6 · 3) |
39 | 6t3e18 12287 | . . 3 ⊢ (6 · 3) = ;18 | |
40 | 32, 38, 39 | 3eqtri 2766 | . 2 ⊢ ((6 · 9) / 3) = ;18 |
41 | 28, 29, 40 | 3eqtri 2766 | 1 ⊢ (6 lcm 9) = ;18 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ≠ wne 2935 (class class class)co 7173 ℂcc 10616 0cc0 10618 1c1 10619 · cmul 10623 -cneg 10952 / cdiv 11378 ℕcn 11719 3c3 11775 6c6 11778 8c8 11780 9c9 11781 ℤcz 12065 ;cdc 12182 gcd cgcd 15940 lcm clcm 16032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 ax-pre-sup 10696 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-2nd 7718 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-sup 8982 df-inf 8983 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-div 11379 df-nn 11720 df-2 11782 df-3 11783 df-4 11784 df-5 11785 df-6 11786 df-7 11787 df-8 11788 df-9 11789 df-n0 11980 df-z 12066 df-dec 12183 df-uz 12328 df-rp 12476 df-fl 13256 df-mod 13332 df-seq 13464 df-exp 13525 df-cj 14551 df-re 14552 df-im 14553 df-sqrt 14687 df-abs 14688 df-dvds 15703 df-gcd 15941 df-lcm 16034 |
This theorem is referenced by: (None) |
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