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Mirrors > Home > MPE Home > Th. List > ex-lcm | Structured version Visualization version GIF version |
Description: Example for df-lcm 15676. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-lcm | ⊢ (6 lcm 9) = ;18 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 11443 | . . . . 5 ⊢ 6 ∈ ℕ | |
2 | 9nn 11455 | . . . . 5 ⊢ 9 ∈ ℕ | |
3 | 1, 2 | nnmulcli 11377 | . . . 4 ⊢ (6 · 9) ∈ ℕ |
4 | 3 | nncni 11361 | . . 3 ⊢ (6 · 9) ∈ ℂ |
5 | 1 | nnzi 11729 | . . . . 5 ⊢ 6 ∈ ℤ |
6 | 2 | nnzi 11729 | . . . . 5 ⊢ 9 ∈ ℤ |
7 | 5, 6 | pm3.2i 464 | . . . 4 ⊢ (6 ∈ ℤ ∧ 9 ∈ ℤ) |
8 | lcmcl 15687 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℕ0) | |
9 | 8 | nn0cnd 11680 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℂ) |
10 | 7, 9 | ax-mp 5 | . . 3 ⊢ (6 lcm 9) ∈ ℂ |
11 | neggcd 15617 | . . . . . . . 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 2834 | . . . . . 6 ⊢ (6 gcd 9) = (-6 gcd 9) |
14 | ex-gcd 27861 | . . . . . 6 ⊢ (-6 gcd 9) = 3 | |
15 | 13, 14 | eqtri 2849 | . . . . 5 ⊢ (6 gcd 9) = 3 |
16 | 3cn 11432 | . . . . 5 ⊢ 3 ∈ ℂ | |
17 | 15, 16 | eqeltri 2902 | . . . 4 ⊢ (6 gcd 9) ∈ ℂ |
18 | 3ne0 11464 | . . . . 5 ⊢ 3 ≠ 0 | |
19 | 15, 18 | eqnetri 3069 | . . . 4 ⊢ (6 gcd 9) ≠ 0 |
20 | 17, 19 | pm3.2i 464 | . . 3 ⊢ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0) |
21 | 1, 2 | pm3.2i 464 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 9 ∈ ℕ) |
22 | lcmgcdnn 15697 | . . . . . . 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 2831 | . . . . 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 11015 | . . . . 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 249 | . . . 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 2831 | . . 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 1589 | . 2 ⊢ (6 lcm 9) = ((6 · 9) / (6 gcd 9)) |
29 | 15 | oveq2i 6916 | . 2 ⊢ ((6 · 9) / (6 gcd 9)) = ((6 · 9) / 3) |
30 | 6cn 11445 | . . . 4 ⊢ 6 ∈ ℂ | |
31 | 9cn 11457 | . . . 4 ⊢ 9 ∈ ℂ | |
32 | 30, 31, 16, 18 | divassi 11107 | . . 3 ⊢ ((6 · 9) / 3) = (6 · (9 / 3)) |
33 | 3t3e9 11525 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
34 | 33 | eqcomi 2834 | . . . . . 6 ⊢ 9 = (3 · 3) |
35 | 34 | oveq1i 6915 | . . . . 5 ⊢ (9 / 3) = ((3 · 3) / 3) |
36 | 16, 16, 18 | divcan3i 11097 | . . . . 5 ⊢ ((3 · 3) / 3) = 3 |
37 | 35, 36 | eqtri 2849 | . . . 4 ⊢ (9 / 3) = 3 |
38 | 37 | oveq2i 6916 | . . 3 ⊢ (6 · (9 / 3)) = (6 · 3) |
39 | 6t3e18 11928 | . . 3 ⊢ (6 · 3) = ;18 | |
40 | 32, 38, 39 | 3eqtri 2853 | . 2 ⊢ ((6 · 9) / 3) = ;18 |
41 | 28, 29, 40 | 3eqtri 2853 | 1 ⊢ (6 lcm 9) = ;18 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 (class class class)co 6905 ℂcc 10250 0cc0 10252 1c1 10253 · cmul 10257 -cneg 10586 / cdiv 11009 ℕcn 11350 3c3 11407 6c6 11410 8c8 11412 9c9 11413 ℤcz 11704 ;cdc 11821 gcd cgcd 15589 lcm clcm 15674 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-pre-sup 10330 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-sup 8617 df-inf 8618 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-div 11010 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-7 11419 df-8 11420 df-9 11421 df-n0 11619 df-z 11705 df-dec 11822 df-uz 11969 df-rp 12113 df-fl 12888 df-mod 12964 df-seq 13096 df-exp 13155 df-cj 14216 df-re 14217 df-im 14218 df-sqrt 14352 df-abs 14353 df-dvds 15358 df-gcd 15590 df-lcm 15676 |
This theorem is referenced by: (None) |
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