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Mirrors > Home > MPE Home > Th. List > Mathboxes > 420lcm8e840 | Structured version Visualization version GIF version |
Description: The lcm of 420 and 8 is 840. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
420lcm8e840 | ⊢ (;;420 lcm 8) = ;;840 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 4nn0 12439 | . . . 4 ⊢ 4 ∈ ℕ0 | |
2 | 2nn 12233 | . . . 4 ⊢ 2 ∈ ℕ | |
3 | 1, 2 | decnncl 12645 | . . 3 ⊢ ;42 ∈ ℕ |
4 | 3 | decnncl2 12649 | . 2 ⊢ ;;420 ∈ ℕ |
5 | 8nn 12255 | . 2 ⊢ 8 ∈ ℕ | |
6 | 4nn 12243 | . 2 ⊢ 4 ∈ ℕ | |
7 | 8nn0 12443 | . . . 4 ⊢ 8 ∈ ℕ0 | |
8 | 7, 6 | decnncl 12645 | . . 3 ⊢ ;84 ∈ ℕ |
9 | 8 | decnncl2 12649 | . 2 ⊢ ;;840 ∈ ℕ |
10 | 420gcd8e4 40492 | . 2 ⊢ (;;420 gcd 8) = 4 | |
11 | eqid 2737 | . 2 ⊢ (4 · ;;840) = (4 · ;;840) | |
12 | 4, 5 | mulcomnni 40474 | . . . . 5 ⊢ (;;420 · 8) = (8 · ;;420) |
13 | 4t2e8 12328 | . . . . . 6 ⊢ (4 · 2) = 8 | |
14 | 13 | oveq1i 7372 | . . . . 5 ⊢ ((4 · 2) · ;;420) = (8 · ;;420) |
15 | 12, 14 | eqtr4i 2768 | . . . 4 ⊢ (;;420 · 8) = ((4 · 2) · ;;420) |
16 | 6, 2, 4 | mulassnni 40473 | . . . 4 ⊢ ((4 · 2) · ;;420) = (4 · (2 · ;;420)) |
17 | 15, 16 | eqtri 2765 | . . 3 ⊢ (;;420 · 8) = (4 · (2 · ;;420)) |
18 | 2 | nnnn0i 12428 | . . . . 5 ⊢ 2 ∈ ℕ0 |
19 | 3 | nnnn0i 12428 | . . . . 5 ⊢ ;42 ∈ ℕ0 |
20 | 0nn0 12435 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
21 | eqid 2737 | . . . . 5 ⊢ ;;420 = ;;420 | |
22 | eqid 2737 | . . . . . . 7 ⊢ ;42 = ;42 | |
23 | 4cn 12245 | . . . . . . . . . 10 ⊢ 4 ∈ ℂ | |
24 | 2cn 12235 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
25 | 23, 24, 13 | mulcomli 11171 | . . . . . . . . 9 ⊢ (2 · 4) = 8 |
26 | 25 | oveq1i 7372 | . . . . . . . 8 ⊢ ((2 · 4) + 0) = (8 + 0) |
27 | 8cn 12257 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
28 | 27 | addid1i 11349 | . . . . . . . 8 ⊢ (8 + 0) = 8 |
29 | 26, 28 | eqtri 2765 | . . . . . . 7 ⊢ ((2 · 4) + 0) = 8 |
30 | 2t2e4 12324 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
31 | 1 | dec0h 12647 | . . . . . . . . 9 ⊢ 4 = ;04 |
32 | 31 | eqcomi 2746 | . . . . . . . 8 ⊢ ;04 = 4 |
33 | 30, 32 | eqtr4i 2768 | . . . . . . 7 ⊢ (2 · 2) = ;04 |
34 | 18, 1, 18, 22, 1, 20, 29, 33 | decmul2c 12691 | . . . . . 6 ⊢ (2 · ;42) = ;84 |
35 | 23 | addid1i 11349 | . . . . . 6 ⊢ (4 + 0) = 4 |
36 | 7, 1, 20, 34, 35 | decaddi 12685 | . . . . 5 ⊢ ((2 · ;42) + 0) = ;84 |
37 | 2t0e0 12329 | . . . . . 6 ⊢ (2 · 0) = 0 | |
38 | 20 | dec0h 12647 | . . . . . . 7 ⊢ 0 = ;00 |
39 | 38 | eqcomi 2746 | . . . . . 6 ⊢ ;00 = 0 |
40 | 37, 39 | eqtr4i 2768 | . . . . 5 ⊢ (2 · 0) = ;00 |
41 | 18, 19, 20, 21, 20, 20, 36, 40 | decmul2c 12691 | . . . 4 ⊢ (2 · ;;420) = ;;840 |
42 | 41 | oveq2i 7373 | . . 3 ⊢ (4 · (2 · ;;420)) = (4 · ;;840) |
43 | 17, 42 | eqtri 2765 | . 2 ⊢ (;;420 · 8) = (4 · ;;840) |
44 | 4, 5, 6, 9, 10, 11, 43 | lcmeprodgcdi 40493 | 1 ⊢ (;;420 lcm 8) = ;;840 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7362 0cc0 11058 + caddc 11061 · cmul 11063 2c2 12215 4c4 12217 8c8 12221 ;cdc 12625 lcm clcm 16471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-fl 13704 df-mod 13782 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-gcd 16382 df-lcm 16473 |
This theorem is referenced by: lcm8un 40506 |
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