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Mirrors > Home > MPE Home > Th. List > ex-lcm | Structured version Visualization version GIF version |
Description: Example for df-lcm 15922. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-lcm | ⊢ (6 lcm 9) = ;18 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 11714 | . . . . 5 ⊢ 6 ∈ ℕ | |
2 | 9nn 11723 | . . . . 5 ⊢ 9 ∈ ℕ | |
3 | 1, 2 | nnmulcli 11650 | . . . 4 ⊢ (6 · 9) ∈ ℕ |
4 | 3 | nncni 11636 | . . 3 ⊢ (6 · 9) ∈ ℂ |
5 | 1 | nnzi 11994 | . . . . 5 ⊢ 6 ∈ ℤ |
6 | 2 | nnzi 11994 | . . . . 5 ⊢ 9 ∈ ℤ |
7 | 5, 6 | pm3.2i 471 | . . . 4 ⊢ (6 ∈ ℤ ∧ 9 ∈ ℤ) |
8 | lcmcl 15933 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℕ0) | |
9 | 8 | nn0cnd 11945 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℂ) |
10 | 7, 9 | ax-mp 5 | . . 3 ⊢ (6 lcm 9) ∈ ℂ |
11 | neggcd 15859 | . . . . . . . 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 2827 | . . . . . 6 ⊢ (6 gcd 9) = (-6 gcd 9) |
14 | ex-gcd 28163 | . . . . . 6 ⊢ (-6 gcd 9) = 3 | |
15 | 13, 14 | eqtri 2841 | . . . . 5 ⊢ (6 gcd 9) = 3 |
16 | 3cn 11706 | . . . . 5 ⊢ 3 ∈ ℂ | |
17 | 15, 16 | eqeltri 2906 | . . . 4 ⊢ (6 gcd 9) ∈ ℂ |
18 | 3ne0 11731 | . . . . 5 ⊢ 3 ≠ 0 | |
19 | 15, 18 | eqnetri 3083 | . . . 4 ⊢ (6 gcd 9) ≠ 0 |
20 | 17, 19 | pm3.2i 471 | . . 3 ⊢ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0) |
21 | 1, 2 | pm3.2i 471 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 9 ∈ ℕ) |
22 | lcmgcdnn 15943 | . . . . . . 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 2824 | . . . . 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 11291 | . . . . 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 258 | . . . 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 2824 | . . 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 1452 | . 2 ⊢ (6 lcm 9) = ((6 · 9) / (6 gcd 9)) |
29 | 15 | oveq2i 7156 | . 2 ⊢ ((6 · 9) / (6 gcd 9)) = ((6 · 9) / 3) |
30 | 6cn 11716 | . . . 4 ⊢ 6 ∈ ℂ | |
31 | 9cn 11725 | . . . 4 ⊢ 9 ∈ ℂ | |
32 | 30, 31, 16, 18 | divassi 11384 | . . 3 ⊢ ((6 · 9) / 3) = (6 · (9 / 3)) |
33 | 3t3e9 11792 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
34 | 33 | eqcomi 2827 | . . . . . 6 ⊢ 9 = (3 · 3) |
35 | 34 | oveq1i 7155 | . . . . 5 ⊢ (9 / 3) = ((3 · 3) / 3) |
36 | 16, 16, 18 | divcan3i 11374 | . . . . 5 ⊢ ((3 · 3) / 3) = 3 |
37 | 35, 36 | eqtri 2841 | . . . 4 ⊢ (9 / 3) = 3 |
38 | 37 | oveq2i 7156 | . . 3 ⊢ (6 · (9 / 3)) = (6 · 3) |
39 | 6t3e18 12191 | . . 3 ⊢ (6 · 3) = ;18 | |
40 | 32, 38, 39 | 3eqtri 2845 | . 2 ⊢ ((6 · 9) / 3) = ;18 |
41 | 28, 29, 40 | 3eqtri 2845 | 1 ⊢ (6 lcm 9) = ;18 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 (class class class)co 7145 ℂcc 10523 0cc0 10525 1c1 10526 · cmul 10530 -cneg 10859 / cdiv 11285 ℕcn 11626 3c3 11681 6c6 11684 8c8 11686 9c9 11687 ℤcz 11969 ;cdc 12086 gcd cgcd 15831 lcm clcm 15920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-sup 8894 df-inf 8895 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-dvds 15596 df-gcd 15832 df-lcm 15922 |
This theorem is referenced by: (None) |
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