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Mirrors > Home > MPE Home > Th. List > ex-lcm | Structured version Visualization version GIF version |
Description: Example for df-lcm 15141. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-lcm | ⊢ (6 lcm 9) = ;18 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 11066 | . . . . 5 ⊢ 6 ∈ ℕ | |
2 | 9nn 11069 | . . . . 5 ⊢ 9 ∈ ℕ | |
3 | 1, 2 | nnmulcli 10921 | . . . 4 ⊢ (6 · 9) ∈ ℕ |
4 | 3 | nncni 10907 | . . 3 ⊢ (6 · 9) ∈ ℂ |
5 | 1 | nnzi 11278 | . . . . 5 ⊢ 6 ∈ ℤ |
6 | 2 | nnzi 11278 | . . . . 5 ⊢ 9 ∈ ℤ |
7 | 5, 6 | pm3.2i 470 | . . . 4 ⊢ (6 ∈ ℤ ∧ 9 ∈ ℤ) |
8 | lcmcl 15152 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℕ0) | |
9 | 8 | nn0cnd 11230 | . . . 4 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (6 lcm 9) ∈ ℂ) |
10 | 7, 9 | ax-mp 5 | . . 3 ⊢ (6 lcm 9) ∈ ℂ |
11 | neggcd 15082 | . . . . . . . 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 2619 | . . . . . 6 ⊢ (6 gcd 9) = (-6 gcd 9) |
14 | ex-gcd 26706 | . . . . . 6 ⊢ (-6 gcd 9) = 3 | |
15 | 13, 14 | eqtri 2632 | . . . . 5 ⊢ (6 gcd 9) = 3 |
16 | 3cn 10972 | . . . . 5 ⊢ 3 ∈ ℂ | |
17 | 15, 16 | eqeltri 2684 | . . . 4 ⊢ (6 gcd 9) ∈ ℂ |
18 | 3ne0 10992 | . . . . 5 ⊢ 3 ≠ 0 | |
19 | 15, 18 | eqnetri 2852 | . . . 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 15162 | . . . . . . 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 2616 | . . . . 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 10569 | . . . . 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 246 | . . . 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 2616 | . . 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 1416 | . 2 ⊢ (6 lcm 9) = ((6 · 9) / (6 gcd 9)) |
29 | 15 | oveq2i 6560 | . 2 ⊢ ((6 · 9) / (6 gcd 9)) = ((6 · 9) / 3) |
30 | 6cn 10979 | . . . 4 ⊢ 6 ∈ ℂ | |
31 | 9cn 10985 | . . . 4 ⊢ 9 ∈ ℂ | |
32 | 30, 31, 16, 18 | divassi 10660 | . . 3 ⊢ ((6 · 9) / 3) = (6 · (9 / 3)) |
33 | 3t3e9 11057 | . . . . . . 7 ⊢ (3 · 3) = 9 | |
34 | 33 | eqcomi 2619 | . . . . . 6 ⊢ 9 = (3 · 3) |
35 | 34 | oveq1i 6559 | . . . . 5 ⊢ (9 / 3) = ((3 · 3) / 3) |
36 | 16, 16, 18 | divcan3i 10650 | . . . . 5 ⊢ ((3 · 3) / 3) = 3 |
37 | 35, 36 | eqtri 2632 | . . . 4 ⊢ (9 / 3) = 3 |
38 | 37 | oveq2i 6560 | . . 3 ⊢ (6 · (9 / 3)) = (6 · 3) |
39 | 6t3e18 11518 | . . 3 ⊢ (6 · 3) = ;18 | |
40 | 32, 38, 39 | 3eqtri 2636 | . 2 ⊢ ((6 · 9) / 3) = ;18 |
41 | 28, 29, 40 | 3eqtri 2636 | 1 ⊢ (6 lcm 9) = ;18 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 · cmul 9820 -cneg 10146 / cdiv 10563 ℕcn 10897 3c3 10948 6c6 10951 8c8 10953 9c9 10954 ℤcz 11254 ;cdc 11369 gcd cgcd 15054 lcm clcm 15139 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 df-lcm 15141 |
This theorem is referenced by: (None) |
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