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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 12337 | . . . 4 ⊢ 8 ∈ ℕ | |
2 | 4nn 12325 | . . . 4 ⊢ 4 ∈ ℕ | |
3 | 5nn0 12522 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
4 | 2nn 12315 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | 3, 4 | decnncl 12727 | . . . . 5 ⊢ ;52 ∈ ℕ |
6 | 5 | nnzi 12616 | . . . 4 ⊢ ;52 ∈ ℤ |
7 | 1, 2, 6 | gcdaddmzz2nncomi 41522 | . . 3 ⊢ (8 gcd 4) = (8 gcd ((;52 · 8) + 4)) |
8 | 4nn0 12521 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
9 | 1nn0 12518 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12722 | . . . . . 6 ⊢ ;41 ∈ ℕ0 |
11 | 6nn0 12523 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
12 | 0nn0 12517 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
13 | eqid 2725 | . . . . . 6 ⊢ ;;416 = ;;416 | |
14 | 8 | dec0h 12729 | . . . . . 6 ⊢ 4 = ;04 |
15 | 1p1e2 12367 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
16 | 10 | nn0cni 12514 | . . . . . . . 8 ⊢ ;41 ∈ ℂ |
17 | 16 | addridi 11431 | . . . . . . 7 ⊢ (;41 + 0) = ;41 |
18 | 8, 9, 15, 17 | decsuc 12738 | . . . . . 6 ⊢ ((;41 + 0) + 1) = ;42 |
19 | 6p4e10 12779 | . . . . . 6 ⊢ (6 + 4) = ;10 | |
20 | 10, 11, 12, 8, 13, 14, 18, 19 | decaddc2 12763 | . . . . 5 ⊢ (;;416 + 4) = ;;420 |
21 | 8nn0 12525 | . . . . . . . 8 ⊢ 8 ∈ ℕ0 | |
22 | 2nn0 12519 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
23 | eqid 2725 | . . . . . . . 8 ⊢ ;52 = ;52 | |
24 | 0p1e1 12364 | . . . . . . . . 9 ⊢ (0 + 1) = 1 | |
25 | 8t5e40 12825 | . . . . . . . . 9 ⊢ (8 · 5) = ;40 | |
26 | 8, 12, 24, 25 | decsuc 12738 | . . . . . . . 8 ⊢ ((8 · 5) + 1) = ;41 |
27 | 8t2e16 12822 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
28 | 21, 3, 22, 23, 11, 9, 26, 27 | decmul2c 12773 | . . . . . . 7 ⊢ (8 · ;52) = ;;416 |
29 | 1, 5 | mulcomnni 41514 | . . . . . . 7 ⊢ (8 · ;52) = (;52 · 8) |
30 | 28, 29 | eqtr3i 2755 | . . . . . 6 ⊢ ;;416 = (;52 · 8) |
31 | 30 | oveq1i 7426 | . . . . 5 ⊢ (;;416 + 4) = ((;52 · 8) + 4) |
32 | 20, 31 | eqtr3i 2755 | . . . 4 ⊢ ;;420 = ((;52 · 8) + 4) |
33 | 32 | oveq2i 7427 | . . 3 ⊢ (8 gcd ;;420) = (8 gcd ((;52 · 8) + 4)) |
34 | 7, 33 | eqtr4i 2756 | . 2 ⊢ (8 gcd 4) = (8 gcd ;;420) |
35 | 1, 2 | gcdcomnni 41515 | . . 3 ⊢ (8 gcd 4) = (4 gcd 8) |
36 | 4t2e8 12410 | . . . . 5 ⊢ (4 · 2) = 8 | |
37 | 36 | oveq2i 7427 | . . . 4 ⊢ (4 gcd (4 · 2)) = (4 gcd 8) |
38 | 2, 4 | gcdmultiplei 41520 | . . . 4 ⊢ (4 gcd (4 · 2)) = 4 |
39 | 37, 38 | eqtr3i 2755 | . . 3 ⊢ (4 gcd 8) = 4 |
40 | 35, 39 | eqtri 2753 | . 2 ⊢ (8 gcd 4) = 4 |
41 | 8, 4 | decnncl 12727 | . . . 4 ⊢ ;42 ∈ ℕ |
42 | 41 | decnncl2 12731 | . . 3 ⊢ ;;420 ∈ ℕ |
43 | 1, 42 | gcdcomnni 41515 | . 2 ⊢ (8 gcd ;;420) = (;;420 gcd 8) |
44 | 34, 40, 43 | 3eqtr3ri 2762 | 1 ⊢ (;;420 gcd 8) = 4 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 · cmul 11143 2c2 12297 4c4 12299 5c5 12300 6c6 12301 8c8 12303 ;cdc 12707 gcd cgcd 16468 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-gcd 16469 |
This theorem is referenced by: 420lcm8e840 41538 |
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