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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 11713 | . . 3 ⊢ 6 ∈ ℕ | |
2 | 1 | decnncl2 12109 | . 2 ⊢ ;60 ∈ ℕ |
3 | 7nn 11716 | . 2 ⊢ 7 ∈ ℕ | |
4 | 1nn 11635 | . 2 ⊢ 1 ∈ ℕ | |
5 | 4nn0 11903 | . . . 4 ⊢ 4 ∈ ℕ0 | |
6 | 2nn 11697 | . . . 4 ⊢ 2 ∈ ℕ | |
7 | 5, 6 | decnncl 12105 | . . 3 ⊢ ;42 ∈ ℕ |
8 | 7 | decnncl2 12109 | . 2 ⊢ ;;420 ∈ ℕ |
9 | 60gcd7e1 39143 | . 2 ⊢ (;60 gcd 7) = 1 | |
10 | 2nn0 11901 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
11 | 5, 10 | deccl 12100 | . . . . 5 ⊢ ;42 ∈ ℕ0 |
12 | 0nn0 11899 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
13 | 11, 12 | deccl 12100 | . . . 4 ⊢ ;;420 ∈ ℕ0 |
14 | 13 | nn0cni 11896 | . . 3 ⊢ ;;420 ∈ ℂ |
15 | 14 | mulid2i 10632 | . 2 ⊢ (1 · ;;420) = ;;420 |
16 | 7nn0 11906 | . . 3 ⊢ 7 ∈ ℕ0 | |
17 | 6nn0 11905 | . . 3 ⊢ 6 ∈ ℕ0 | |
18 | eqid 2821 | . . 3 ⊢ ;60 = ;60 | |
19 | 7cn 11718 | . . . . 5 ⊢ 7 ∈ ℂ | |
20 | 6cn 11715 | . . . . 5 ⊢ 6 ∈ ℂ | |
21 | 7t6e42 12198 | . . . . 5 ⊢ (7 · 6) = ;42 | |
22 | 19, 20, 21 | mulcomli 10636 | . . . 4 ⊢ (6 · 7) = ;42 |
23 | 2cn 11699 | . . . . 5 ⊢ 2 ∈ ℂ | |
24 | 23 | addid1i 10813 | . . . 4 ⊢ (2 + 0) = 2 |
25 | 5, 10, 12, 22, 24 | decaddi 12145 | . . 3 ⊢ ((6 · 7) + 0) = ;42 |
26 | 0cn 10619 | . . . 4 ⊢ 0 ∈ ℂ | |
27 | 19 | mul01i 10816 | . . . . 5 ⊢ (7 · 0) = 0 |
28 | 12 | dec0h 12107 | . . . . . 6 ⊢ 0 = ;00 |
29 | 28 | eqcomi 2830 | . . . . 5 ⊢ ;00 = 0 |
30 | 27, 29 | eqtr4i 2847 | . . . 4 ⊢ (7 · 0) = ;00 |
31 | 19, 26, 30 | mulcomli 10636 | . . 3 ⊢ (0 · 7) = ;00 |
32 | 16, 17, 12, 18, 12, 12, 25, 31 | decmul1c 12150 | . 2 ⊢ (;60 · 7) = ;;420 |
33 | 2, 3, 4, 8, 9, 15, 32 | lcmeprodgcdi 39145 | 1 ⊢ (;60 lcm 7) = ;;420 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 (class class class)co 7142 0cc0 10523 1c1 10524 · cmul 10528 2c2 11679 4c4 11681 6c6 11683 7c7 11684 ;cdc 12085 lcm clcm 15915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 ax-pre-sup 10601 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-er 8275 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-div 11284 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-rp 12377 df-fz 12883 df-fl 13152 df-mod 13228 df-seq 13360 df-exp 13420 df-cj 14443 df-re 14444 df-im 14445 df-sqrt 14579 df-abs 14580 df-dvds 15593 df-gcd 15827 df-lcm 15917 df-prm 15999 |
This theorem is referenced by: lcm7un 39166 |
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