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Mirrors > Home > MPE Home > Th. List > Mathboxes > 420gcd8e4 | Structured version Visualization version GIF version |
Description: The gcd of 420 and 8 is 4. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
420gcd8e4 | ⊢ (;;420 gcd 8) = 4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 8nn 12338 | . . . 4 ⊢ 8 ∈ ℕ | |
2 | 4nn 12326 | . . . 4 ⊢ 4 ∈ ℕ | |
3 | 5nn0 12523 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
4 | 2nn 12316 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | 3, 4 | decnncl 12728 | . . . . 5 ⊢ ;52 ∈ ℕ |
6 | 5 | nnzi 12617 | . . . 4 ⊢ ;52 ∈ ℤ |
7 | 1, 2, 6 | gcdaddmzz2nncomi 41466 | . . 3 ⊢ (8 gcd 4) = (8 gcd ((;52 · 8) + 4)) |
8 | 4nn0 12522 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
9 | 1nn0 12519 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12723 | . . . . . 6 ⊢ ;41 ∈ ℕ0 |
11 | 6nn0 12524 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
12 | 0nn0 12518 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
13 | eqid 2728 | . . . . . 6 ⊢ ;;416 = ;;416 | |
14 | 8 | dec0h 12730 | . . . . . 6 ⊢ 4 = ;04 |
15 | 1p1e2 12368 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
16 | 10 | nn0cni 12515 | . . . . . . . 8 ⊢ ;41 ∈ ℂ |
17 | 16 | addridi 11432 | . . . . . . 7 ⊢ (;41 + 0) = ;41 |
18 | 8, 9, 15, 17 | decsuc 12739 | . . . . . 6 ⊢ ((;41 + 0) + 1) = ;42 |
19 | 6p4e10 12780 | . . . . . 6 ⊢ (6 + 4) = ;10 | |
20 | 10, 11, 12, 8, 13, 14, 18, 19 | decaddc2 12764 | . . . . 5 ⊢ (;;416 + 4) = ;;420 |
21 | 8nn0 12526 | . . . . . . . 8 ⊢ 8 ∈ ℕ0 | |
22 | 2nn0 12520 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
23 | eqid 2728 | . . . . . . . 8 ⊢ ;52 = ;52 | |
24 | 0p1e1 12365 | . . . . . . . . 9 ⊢ (0 + 1) = 1 | |
25 | 8t5e40 12826 | . . . . . . . . 9 ⊢ (8 · 5) = ;40 | |
26 | 8, 12, 24, 25 | decsuc 12739 | . . . . . . . 8 ⊢ ((8 · 5) + 1) = ;41 |
27 | 8t2e16 12823 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
28 | 21, 3, 22, 23, 11, 9, 26, 27 | decmul2c 12774 | . . . . . . 7 ⊢ (8 · ;52) = ;;416 |
29 | 1, 5 | mulcomnni 41458 | . . . . . . 7 ⊢ (8 · ;52) = (;52 · 8) |
30 | 28, 29 | eqtr3i 2758 | . . . . . 6 ⊢ ;;416 = (;52 · 8) |
31 | 30 | oveq1i 7430 | . . . . 5 ⊢ (;;416 + 4) = ((;52 · 8) + 4) |
32 | 20, 31 | eqtr3i 2758 | . . . 4 ⊢ ;;420 = ((;52 · 8) + 4) |
33 | 32 | oveq2i 7431 | . . 3 ⊢ (8 gcd ;;420) = (8 gcd ((;52 · 8) + 4)) |
34 | 7, 33 | eqtr4i 2759 | . 2 ⊢ (8 gcd 4) = (8 gcd ;;420) |
35 | 1, 2 | gcdcomnni 41459 | . . 3 ⊢ (8 gcd 4) = (4 gcd 8) |
36 | 4t2e8 12411 | . . . . 5 ⊢ (4 · 2) = 8 | |
37 | 36 | oveq2i 7431 | . . . 4 ⊢ (4 gcd (4 · 2)) = (4 gcd 8) |
38 | 2, 4 | gcdmultiplei 41464 | . . . 4 ⊢ (4 gcd (4 · 2)) = 4 |
39 | 37, 38 | eqtr3i 2758 | . . 3 ⊢ (4 gcd 8) = 4 |
40 | 35, 39 | eqtri 2756 | . 2 ⊢ (8 gcd 4) = 4 |
41 | 8, 4 | decnncl 12728 | . . . 4 ⊢ ;42 ∈ ℕ |
42 | 41 | decnncl2 12732 | . . 3 ⊢ ;;420 ∈ ℕ |
43 | 1, 42 | gcdcomnni 41459 | . 2 ⊢ (8 gcd ;;420) = (;;420 gcd 8) |
44 | 34, 40, 43 | 3eqtr3ri 2765 | 1 ⊢ (;;420 gcd 8) = 4 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7420 0cc0 11139 1c1 11140 + caddc 11142 · cmul 11144 2c2 12298 4c4 12300 5c5 12301 6c6 12302 8c8 12304 ;cdc 12708 gcd cgcd 16469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9466 df-inf 9467 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-dvds 16232 df-gcd 16470 |
This theorem is referenced by: 420lcm8e840 41482 |
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