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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 12255 | . . . 4 ⊢ 8 ∈ ℕ | |
2 | 4nn 12243 | . . . 4 ⊢ 4 ∈ ℕ | |
3 | 5nn0 12440 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
4 | 2nn 12233 | . . . . . 6 ⊢ 2 ∈ ℕ | |
5 | 3, 4 | decnncl 12645 | . . . . 5 ⊢ ;52 ∈ ℕ |
6 | 5 | nnzi 12534 | . . . 4 ⊢ ;52 ∈ ℤ |
7 | 1, 2, 6 | gcdaddmzz2nncomi 40482 | . . 3 ⊢ (8 gcd 4) = (8 gcd ((;52 · 8) + 4)) |
8 | 4nn0 12439 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
9 | 1nn0 12436 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
10 | 8, 9 | deccl 12640 | . . . . . 6 ⊢ ;41 ∈ ℕ0 |
11 | 6nn0 12441 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
12 | 0nn0 12435 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
13 | eqid 2737 | . . . . . 6 ⊢ ;;416 = ;;416 | |
14 | 8 | dec0h 12647 | . . . . . 6 ⊢ 4 = ;04 |
15 | 1p1e2 12285 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
16 | 10 | nn0cni 12432 | . . . . . . . 8 ⊢ ;41 ∈ ℂ |
17 | 16 | addid1i 11349 | . . . . . . 7 ⊢ (;41 + 0) = ;41 |
18 | 8, 9, 15, 17 | decsuc 12656 | . . . . . 6 ⊢ ((;41 + 0) + 1) = ;42 |
19 | 6p4e10 12697 | . . . . . 6 ⊢ (6 + 4) = ;10 | |
20 | 10, 11, 12, 8, 13, 14, 18, 19 | decaddc2 12681 | . . . . 5 ⊢ (;;416 + 4) = ;;420 |
21 | 8nn0 12443 | . . . . . . . 8 ⊢ 8 ∈ ℕ0 | |
22 | 2nn0 12437 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
23 | eqid 2737 | . . . . . . . 8 ⊢ ;52 = ;52 | |
24 | 0p1e1 12282 | . . . . . . . . 9 ⊢ (0 + 1) = 1 | |
25 | 8t5e40 12743 | . . . . . . . . 9 ⊢ (8 · 5) = ;40 | |
26 | 8, 12, 24, 25 | decsuc 12656 | . . . . . . . 8 ⊢ ((8 · 5) + 1) = ;41 |
27 | 8t2e16 12740 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
28 | 21, 3, 22, 23, 11, 9, 26, 27 | decmul2c 12691 | . . . . . . 7 ⊢ (8 · ;52) = ;;416 |
29 | 1, 5 | mulcomnni 40474 | . . . . . . 7 ⊢ (8 · ;52) = (;52 · 8) |
30 | 28, 29 | eqtr3i 2767 | . . . . . 6 ⊢ ;;416 = (;52 · 8) |
31 | 30 | oveq1i 7372 | . . . . 5 ⊢ (;;416 + 4) = ((;52 · 8) + 4) |
32 | 20, 31 | eqtr3i 2767 | . . . 4 ⊢ ;;420 = ((;52 · 8) + 4) |
33 | 32 | oveq2i 7373 | . . 3 ⊢ (8 gcd ;;420) = (8 gcd ((;52 · 8) + 4)) |
34 | 7, 33 | eqtr4i 2768 | . 2 ⊢ (8 gcd 4) = (8 gcd ;;420) |
35 | 1, 2 | gcdcomnni 40475 | . . 3 ⊢ (8 gcd 4) = (4 gcd 8) |
36 | 4t2e8 12328 | . . . . 5 ⊢ (4 · 2) = 8 | |
37 | 36 | oveq2i 7373 | . . . 4 ⊢ (4 gcd (4 · 2)) = (4 gcd 8) |
38 | 2, 4 | gcdmultiplei 40480 | . . . 4 ⊢ (4 gcd (4 · 2)) = 4 |
39 | 37, 38 | eqtr3i 2767 | . . 3 ⊢ (4 gcd 8) = 4 |
40 | 35, 39 | eqtri 2765 | . 2 ⊢ (8 gcd 4) = 4 |
41 | 8, 4 | decnncl 12645 | . . . 4 ⊢ ;42 ∈ ℕ |
42 | 41 | decnncl2 12649 | . . 3 ⊢ ;;420 ∈ ℕ |
43 | 1, 42 | gcdcomnni 40475 | . 2 ⊢ (8 gcd ;;420) = (;;420 gcd 8) |
44 | 34, 40, 43 | 3eqtr3ri 2774 | 1 ⊢ (;;420 gcd 8) = 4 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 (class class class)co 7362 0cc0 11058 1c1 11059 + caddc 11061 · cmul 11063 2c2 12215 4c4 12217 5c5 12218 6c6 12219 8c8 12221 ;cdc 12625 gcd cgcd 16381 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-pre-sup 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-rp 12923 df-seq 13914 df-exp 13975 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-dvds 16144 df-gcd 16382 |
This theorem is referenced by: 420lcm8e840 40497 |
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