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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 12545 | . . . 4 ⊢ 4 ∈ ℕ0 | |
| 2 | 2nn 12339 | . . . 4 ⊢ 2 ∈ ℕ | |
| 3 | 1, 2 | decnncl 12753 | . . 3 ⊢ ;42 ∈ ℕ | 
| 4 | 3 | decnncl2 12757 | . 2 ⊢ ;;420 ∈ ℕ | 
| 5 | 8nn 12361 | . 2 ⊢ 8 ∈ ℕ | |
| 6 | 4nn 12349 | . 2 ⊢ 4 ∈ ℕ | |
| 7 | 8nn0 12549 | . . . 4 ⊢ 8 ∈ ℕ0 | |
| 8 | 7, 6 | decnncl 12753 | . . 3 ⊢ ;84 ∈ ℕ | 
| 9 | 8 | decnncl2 12757 | . 2 ⊢ ;;840 ∈ ℕ | 
| 10 | 420gcd8e4 42007 | . 2 ⊢ (;;420 gcd 8) = 4 | |
| 11 | eqid 2737 | . 2 ⊢ (4 · ;;840) = (4 · ;;840) | |
| 12 | 4, 5 | mulcomnni 41988 | . . . . 5 ⊢ (;;420 · 8) = (8 · ;;420) | 
| 13 | 4t2e8 12434 | . . . . . 6 ⊢ (4 · 2) = 8 | |
| 14 | 13 | oveq1i 7441 | . . . . 5 ⊢ ((4 · 2) · ;;420) = (8 · ;;420) | 
| 15 | 12, 14 | eqtr4i 2768 | . . . 4 ⊢ (;;420 · 8) = ((4 · 2) · ;;420) | 
| 16 | 6, 2, 4 | mulassnni 41987 | . . . 4 ⊢ ((4 · 2) · ;;420) = (4 · (2 · ;;420)) | 
| 17 | 15, 16 | eqtri 2765 | . . 3 ⊢ (;;420 · 8) = (4 · (2 · ;;420)) | 
| 18 | 2 | nnnn0i 12534 | . . . . 5 ⊢ 2 ∈ ℕ0 | 
| 19 | 3 | nnnn0i 12534 | . . . . 5 ⊢ ;42 ∈ ℕ0 | 
| 20 | 0nn0 12541 | . . . . 5 ⊢ 0 ∈ ℕ0 | |
| 21 | eqid 2737 | . . . . 5 ⊢ ;;420 = ;;420 | |
| 22 | eqid 2737 | . . . . . . 7 ⊢ ;42 = ;42 | |
| 23 | 4cn 12351 | . . . . . . . . . 10 ⊢ 4 ∈ ℂ | |
| 24 | 2cn 12341 | . . . . . . . . . 10 ⊢ 2 ∈ ℂ | |
| 25 | 23, 24, 13 | mulcomli 11270 | . . . . . . . . 9 ⊢ (2 · 4) = 8 | 
| 26 | 25 | oveq1i 7441 | . . . . . . . 8 ⊢ ((2 · 4) + 0) = (8 + 0) | 
| 27 | 8cn 12363 | . . . . . . . . 9 ⊢ 8 ∈ ℂ | |
| 28 | 27 | addridi 11448 | . . . . . . . 8 ⊢ (8 + 0) = 8 | 
| 29 | 26, 28 | eqtri 2765 | . . . . . . 7 ⊢ ((2 · 4) + 0) = 8 | 
| 30 | 2t2e4 12430 | . . . . . . . 8 ⊢ (2 · 2) = 4 | |
| 31 | 1 | dec0h 12755 | . . . . . . . . 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 12799 | . . . . . 6 ⊢ (2 · ;42) = ;84 | 
| 35 | 23 | addridi 11448 | . . . . . 6 ⊢ (4 + 0) = 4 | 
| 36 | 7, 1, 20, 34, 35 | decaddi 12793 | . . . . 5 ⊢ ((2 · ;42) + 0) = ;84 | 
| 37 | 2t0e0 12435 | . . . . . 6 ⊢ (2 · 0) = 0 | |
| 38 | 20 | dec0h 12755 | . . . . . . 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 12799 | . . . 4 ⊢ (2 · ;;420) = ;;840 | 
| 42 | 41 | oveq2i 7442 | . . 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 42008 | 1 ⊢ (;;420 lcm 8) = ;;840 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 (class class class)co 7431 0cc0 11155 + caddc 11158 · cmul 11160 2c2 12321 4c4 12323 8c8 12327 ;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-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 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-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 | 
| This theorem is referenced by: lcm8un 42021 | 
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