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Mirrors > Home > MPE Home > Th. List > Mathboxes > 60lcm7e420 | Structured version Visualization version GIF version |
Description: The lcm of 60 and 7 is 420. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
60lcm7e420 | ⊢ (;60 lcm 7) = ;;420 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 12288 | . . 3 ⊢ 6 ∈ ℕ | |
2 | 1 | decnncl2 12688 | . 2 ⊢ ;60 ∈ ℕ |
3 | 7nn 12291 | . 2 ⊢ 7 ∈ ℕ | |
4 | 1nn 12210 | . 2 ⊢ 1 ∈ ℕ | |
5 | 4nn0 12478 | . . . 4 ⊢ 4 ∈ ℕ0 | |
6 | 2nn 12272 | . . . 4 ⊢ 2 ∈ ℕ | |
7 | 5, 6 | decnncl 12684 | . . 3 ⊢ ;42 ∈ ℕ |
8 | 7 | decnncl2 12688 | . 2 ⊢ ;;420 ∈ ℕ |
9 | 60gcd7e1 40776 | . 2 ⊢ (;60 gcd 7) = 1 | |
10 | 2nn0 12476 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
11 | 5, 10 | deccl 12679 | . . . . 5 ⊢ ;42 ∈ ℕ0 |
12 | 0nn0 12474 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
13 | 11, 12 | deccl 12679 | . . . 4 ⊢ ;;420 ∈ ℕ0 |
14 | 13 | nn0cni 12471 | . . 3 ⊢ ;;420 ∈ ℂ |
15 | 14 | mullidi 11206 | . 2 ⊢ (1 · ;;420) = ;;420 |
16 | 7nn0 12481 | . . 3 ⊢ 7 ∈ ℕ0 | |
17 | 6nn0 12480 | . . 3 ⊢ 6 ∈ ℕ0 | |
18 | eqid 2733 | . . 3 ⊢ ;60 = ;60 | |
19 | 7cn 12293 | . . . . 5 ⊢ 7 ∈ ℂ | |
20 | 6cn 12290 | . . . . 5 ⊢ 6 ∈ ℂ | |
21 | 7t6e42 12777 | . . . . 5 ⊢ (7 · 6) = ;42 | |
22 | 19, 20, 21 | mulcomli 11210 | . . . 4 ⊢ (6 · 7) = ;42 |
23 | 2cn 12274 | . . . . 5 ⊢ 2 ∈ ℂ | |
24 | 23 | addridi 11388 | . . . 4 ⊢ (2 + 0) = 2 |
25 | 5, 10, 12, 22, 24 | decaddi 12724 | . . 3 ⊢ ((6 · 7) + 0) = ;42 |
26 | 0cn 11193 | . . . 4 ⊢ 0 ∈ ℂ | |
27 | 19 | mul01i 11391 | . . . . 5 ⊢ (7 · 0) = 0 |
28 | 12 | dec0h 12686 | . . . . . 6 ⊢ 0 = ;00 |
29 | 28 | eqcomi 2742 | . . . . 5 ⊢ ;00 = 0 |
30 | 27, 29 | eqtr4i 2764 | . . . 4 ⊢ (7 · 0) = ;00 |
31 | 19, 26, 30 | mulcomli 11210 | . . 3 ⊢ (0 · 7) = ;00 |
32 | 16, 17, 12, 18, 12, 12, 25, 31 | decmul1c 12729 | . 2 ⊢ (;60 · 7) = ;;420 |
33 | 2, 3, 4, 8, 9, 15, 32 | lcmeprodgcdi 40778 | 1 ⊢ (;60 lcm 7) = ;;420 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7396 0cc0 11097 1c1 11098 · cmul 11102 2c2 12254 4c4 12256 6c6 12258 7c7 12259 ;cdc 12664 lcm clcm 16512 |
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 2704 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-pre-sup 11175 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-1st 7962 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-er 8691 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-sup 9424 df-inf 9425 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-div 11859 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-7 12267 df-8 12268 df-9 12269 df-n0 12460 df-z 12546 df-dec 12665 df-uz 12810 df-rp 12962 df-fz 13472 df-fl 13744 df-mod 13822 df-seq 13954 df-exp 14015 df-cj 15033 df-re 15034 df-im 15035 df-sqrt 15169 df-abs 15170 df-dvds 16185 df-gcd 16423 df-lcm 16514 df-prm 16596 |
This theorem is referenced by: lcm7un 40790 |
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