| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > ex-lcm | Structured version Visualization version GIF version | ||
| Description: Example for df-lcm 16627. (Contributed by AV, 5-Sep-2021.) |
| Ref | Expression |
|---|---|
| ex-lcm | ⊢ (6 lcm 9) = ;18 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 6nn 12355 | . . . . 5 ⊢ 6 ∈ ℕ | |
| 2 | 9nn 12364 | . . . . 5 ⊢ 9 ∈ ℕ | |
| 3 | 1, 2 | nnmulcli 12291 | . . . 4 ⊢ (6 · 9) ∈ ℕ |
| 4 | 3 | nncni 12276 | . . 3 ⊢ (6 · 9) ∈ ℂ |
| 5 | 1 | nnzi 12641 | . . . . 5 ⊢ 6 ∈ ℤ |
| 6 | 2 | nnzi 12641 | . . . . 5 ⊢ 9 ∈ ℤ |
| 7 | 5, 6 | pm3.2i 470 | . . . 4 ⊢ (6 ∈ ℤ ∧ 9 ∈ ℤ) |
| 8 | lcmcl 16638 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℕ0) | |
| 9 | 8 | nn0cnd 12589 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℂ) |
| 10 | 7, 9 | ax-mp 5 | . . 3 ⊢ (6 lcm 9) ∈ ℂ |
| 11 | neggcd 16560 | . . . . . . . 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 2746 | . . . . . 6 ⊢ (6 gcd 9) = (-6 gcd 9) |
| 14 | ex-gcd 30476 | . . . . . 6 ⊢ (-6 gcd 9) = 3 | |
| 15 | 13, 14 | eqtri 2765 | . . . . 5 ⊢ (6 gcd 9) = 3 |
| 16 | 3cn 12347 | . . . . 5 ⊢ 3 ∈ ℂ | |
| 17 | 15, 16 | eqeltri 2837 | . . . 4 ⊢ (6 gcd 9) ∈ ℂ |
| 18 | 3ne0 12372 | . . . . 5 ⊢ 3 ≠ 0 | |
| 19 | 15, 18 | eqnetri 3011 | . . . 4 ⊢ (6 gcd 9) ≠ 0 |
| 20 | 17, 19 | pm3.2i 470 | . . 3 ⊢ ((6 gcd 9) ∈ ℂ ∧ (6 gcd 9) ≠ 0) |
| 21 | 1, 2 | pm3.2i 470 | . . . . . . 7 ⊢ (6 ∈ ℕ ∧ 9 ∈ ℕ) |
| 22 | lcmgcdnn 16648 | . . . . . . 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 2743 | . . . . 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 11927 | . . . . 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 257 | . . . 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 2743 | . . 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 1463 | . 2 ⊢ (6 lcm 9) = ((6 · 9) / (6 gcd 9)) |
| 29 | 15 | oveq2i 7442 | . 2 ⊢ ((6 · 9) / (6 gcd 9)) = ((6 · 9) / 3) |
| 30 | 6cn 12357 | . . . 4 ⊢ 6 ∈ ℂ | |
| 31 | 9cn 12366 | . . . 4 ⊢ 9 ∈ ℂ | |
| 32 | 30, 31, 16, 18 | divassi 12023 | . . 3 ⊢ ((6 · 9) / 3) = (6 · (9 / 3)) |
| 33 | 3t3e9 12433 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
| 34 | 33 | eqcomi 2746 | . . . . . 6 ⊢ 9 = (3 · 3) |
| 35 | 34 | oveq1i 7441 | . . . . 5 ⊢ (9 / 3) = ((3 · 3) / 3) |
| 36 | 16, 16, 18 | divcan3i 12013 | . . . . 5 ⊢ ((3 · 3) / 3) = 3 |
| 37 | 35, 36 | eqtri 2765 | . . . 4 ⊢ (9 / 3) = 3 |
| 38 | 37 | oveq2i 7442 | . . 3 ⊢ (6 · (9 / 3)) = (6 · 3) |
| 39 | 6t3e18 12838 | . . 3 ⊢ (6 · 3) = ;18 | |
| 40 | 32, 38, 39 | 3eqtri 2769 | . 2 ⊢ ((6 · 9) / 3) = ;18 |
| 41 | 28, 29, 40 | 3eqtri 2769 | 1 ⊢ (6 lcm 9) = ;18 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 (class class class)co 7431 ℂcc 11153 0cc0 11155 1c1 11156 · cmul 11160 -cneg 11493 / cdiv 11920 ℕcn 12266 3c3 12322 6c6 12325 8c8 12327 9c9 12328 ℤcz 12613 ;cdc 12733 gcd cgcd 16531 lcm clcm 16625 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-dvds 16291 df-gcd 16532 df-lcm 16627 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |