| Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
||
| 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 12355 | . . 3 ⊢ 6 ∈ ℕ | |
| 2 | 1 | decnncl2 12757 | . 2 ⊢ ;60 ∈ ℕ |
| 3 | 7nn 12358 | . 2 ⊢ 7 ∈ ℕ | |
| 4 | 1nn 12277 | . 2 ⊢ 1 ∈ ℕ | |
| 5 | 4nn0 12545 | . . . 4 ⊢ 4 ∈ ℕ0 | |
| 6 | 2nn 12339 | . . . 4 ⊢ 2 ∈ ℕ | |
| 7 | 5, 6 | decnncl 12753 | . . 3 ⊢ ;42 ∈ ℕ |
| 8 | 7 | decnncl2 12757 | . 2 ⊢ ;;420 ∈ ℕ |
| 9 | 60gcd7e1 42006 | . 2 ⊢ (;60 gcd 7) = 1 | |
| 10 | 2nn0 12543 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
| 11 | 5, 10 | deccl 12748 | . . . . 5 ⊢ ;42 ∈ ℕ0 |
| 12 | 0nn0 12541 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 13 | 11, 12 | deccl 12748 | . . . 4 ⊢ ;;420 ∈ ℕ0 |
| 14 | 13 | nn0cni 12538 | . . 3 ⊢ ;;420 ∈ ℂ |
| 15 | 14 | mullidi 11266 | . 2 ⊢ (1 · ;;420) = ;;420 |
| 16 | 7nn0 12548 | . . 3 ⊢ 7 ∈ ℕ0 | |
| 17 | 6nn0 12547 | . . 3 ⊢ 6 ∈ ℕ0 | |
| 18 | eqid 2737 | . . 3 ⊢ ;60 = ;60 | |
| 19 | 7cn 12360 | . . . . 5 ⊢ 7 ∈ ℂ | |
| 20 | 6cn 12357 | . . . . 5 ⊢ 6 ∈ ℂ | |
| 21 | 7t6e42 12846 | . . . . 5 ⊢ (7 · 6) = ;42 | |
| 22 | 19, 20, 21 | mulcomli 11270 | . . . 4 ⊢ (6 · 7) = ;42 |
| 23 | 2cn 12341 | . . . . 5 ⊢ 2 ∈ ℂ | |
| 24 | 23 | addridi 11448 | . . . 4 ⊢ (2 + 0) = 2 |
| 25 | 5, 10, 12, 22, 24 | decaddi 12793 | . . 3 ⊢ ((6 · 7) + 0) = ;42 |
| 26 | 0cn 11253 | . . . 4 ⊢ 0 ∈ ℂ | |
| 27 | 19 | mul01i 11451 | . . . . 5 ⊢ (7 · 0) = 0 |
| 28 | 12 | dec0h 12755 | . . . . . 6 ⊢ 0 = ;00 |
| 29 | 28 | eqcomi 2746 | . . . . 5 ⊢ ;00 = 0 |
| 30 | 27, 29 | eqtr4i 2768 | . . . 4 ⊢ (7 · 0) = ;00 |
| 31 | 19, 26, 30 | mulcomli 11270 | . . 3 ⊢ (0 · 7) = ;00 |
| 32 | 16, 17, 12, 18, 12, 12, 25, 31 | decmul1c 12798 | . 2 ⊢ (;60 · 7) = ;;420 |
| 33 | 2, 3, 4, 8, 9, 15, 32 | lcmeprodgcdi 42008 | 1 ⊢ (;60 lcm 7) = ;;420 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 (class class class)co 7431 0cc0 11155 1c1 11156 · cmul 11160 2c2 12321 4c4 12323 6c6 12325 7c7 12326 ;cdc 12733 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-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 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-fz 13548 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 df-prm 16709 |
| This theorem is referenced by: lcm7un 42020 |
| Copyright terms: Public domain | W3C validator |