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Mirrors > Home > MPE Home > Th. List > ex-lcm | Structured version Visualization version GIF version |
Description: Example for df-lcm 16564. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-lcm | ⊢ (6 lcm 9) = ;18 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 12334 | . . . . 5 ⊢ 6 ∈ ℕ | |
2 | 9nn 12343 | . . . . 5 ⊢ 9 ∈ ℕ | |
3 | 1, 2 | nnmulcli 12270 | . . . 4 ⊢ (6 · 9) ∈ ℕ |
4 | 3 | nncni 12255 | . . 3 ⊢ (6 · 9) ∈ ℂ |
5 | 1 | nnzi 12619 | . . . . 5 ⊢ 6 ∈ ℤ |
6 | 2 | nnzi 12619 | . . . . 5 ⊢ 9 ∈ ℤ |
7 | 5, 6 | pm3.2i 469 | . . . 4 ⊢ (6 ∈ ℤ ∧ 9 ∈ ℤ) |
8 | lcmcl 16575 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℕ0) | |
9 | 8 | nn0cnd 12567 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℂ) |
10 | 7, 9 | ax-mp 5 | . . 3 ⊢ (6 lcm 9) ∈ ℂ |
11 | neggcd 16501 | . . . . . . . 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 2734 | . . . . . 6 ⊢ (6 gcd 9) = (-6 gcd 9) |
14 | ex-gcd 30339 | . . . . . 6 ⊢ (-6 gcd 9) = 3 | |
15 | 13, 14 | eqtri 2753 | . . . . 5 ⊢ (6 gcd 9) = 3 |
16 | 3cn 12326 | . . . . 5 ⊢ 3 ∈ ℂ | |
17 | 15, 16 | eqeltri 2821 | . . . 4 ⊢ (6 gcd 9) ∈ ℂ |
18 | 3ne0 12351 | . . . . 5 ⊢ 3 ≠ 0 | |
19 | 15, 18 | eqnetri 3000 | . . . 4 ⊢ (6 gcd 9) ≠ 0 |
20 | 17, 19 | pm3.2i 469 | . . 3 ⊢ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0) |
21 | 1, 2 | pm3.2i 469 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 9 ∈ ℕ) |
22 | lcmgcdnn 16585 | . . . . . . 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 2731 | . . . . 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 11910 | . . . . 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 256 | . . . 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 2731 | . . 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 1457 | . 2 ⊢ (6 lcm 9) = ((6 · 9) / (6 gcd 9)) |
29 | 15 | oveq2i 7430 | . 2 ⊢ ((6 · 9) / (6 gcd 9)) = ((6 · 9) / 3) |
30 | 6cn 12336 | . . . 4 ⊢ 6 ∈ ℂ | |
31 | 9cn 12345 | . . . 4 ⊢ 9 ∈ ℂ | |
32 | 30, 31, 16, 18 | divassi 12003 | . . 3 ⊢ ((6 · 9) / 3) = (6 · (9 / 3)) |
33 | 3t3e9 12412 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
34 | 33 | eqcomi 2734 | . . . . . 6 ⊢ 9 = (3 · 3) |
35 | 34 | oveq1i 7429 | . . . . 5 ⊢ (9 / 3) = ((3 · 3) / 3) |
36 | 16, 16, 18 | divcan3i 11993 | . . . . 5 ⊢ ((3 · 3) / 3) = 3 |
37 | 35, 36 | eqtri 2753 | . . . 4 ⊢ (9 / 3) = 3 |
38 | 37 | oveq2i 7430 | . . 3 ⊢ (6 · (9 / 3)) = (6 · 3) |
39 | 6t3e18 12815 | . . 3 ⊢ (6 · 3) = ;18 | |
40 | 32, 38, 39 | 3eqtri 2757 | . 2 ⊢ ((6 · 9) / 3) = ;18 |
41 | 28, 29, 40 | 3eqtri 2757 | 1 ⊢ (6 lcm 9) = ;18 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 (class class class)co 7419 ℂcc 11138 0cc0 11140 1c1 11141 · cmul 11145 -cneg 11477 / cdiv 11903 ℕcn 12245 3c3 12301 6c6 12304 8c8 12306 9c9 12307 ℤcz 12591 ;cdc 12710 gcd cgcd 16472 lcm clcm 16562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-dvds 16235 df-gcd 16473 df-lcm 16564 |
This theorem is referenced by: (None) |
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