| Mathbox for metakunt |
< Previous
Next >
Nearby theorems |
||
| 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 12361 | . . . 4 ⊢ 8 ∈ ℕ | |
| 2 | 4nn 12349 | . . . 4 ⊢ 4 ∈ ℕ | |
| 3 | 5nn0 12546 | . . . . . 6 ⊢ 5 ∈ ℕ0 | |
| 4 | 2nn 12339 | . . . . . 6 ⊢ 2 ∈ ℕ | |
| 5 | 3, 4 | decnncl 12753 | . . . . 5 ⊢ ;52 ∈ ℕ |
| 6 | 5 | nnzi 12641 | . . . 4 ⊢ ;52 ∈ ℤ |
| 7 | 1, 2, 6 | gcdaddmzz2nncomi 41996 | . . 3 ⊢ (8 gcd 4) = (8 gcd ((;52 · 8) + 4)) |
| 8 | 4nn0 12545 | . . . . . . 7 ⊢ 4 ∈ ℕ0 | |
| 9 | 1nn0 12542 | . . . . . . 7 ⊢ 1 ∈ ℕ0 | |
| 10 | 8, 9 | deccl 12748 | . . . . . 6 ⊢ ;41 ∈ ℕ0 |
| 11 | 6nn0 12547 | . . . . . 6 ⊢ 6 ∈ ℕ0 | |
| 12 | 0nn0 12541 | . . . . . 6 ⊢ 0 ∈ ℕ0 | |
| 13 | eqid 2737 | . . . . . 6 ⊢ ;;416 = ;;416 | |
| 14 | 8 | dec0h 12755 | . . . . . 6 ⊢ 4 = ;04 |
| 15 | 1p1e2 12391 | . . . . . . 7 ⊢ (1 + 1) = 2 | |
| 16 | 10 | nn0cni 12538 | . . . . . . . 8 ⊢ ;41 ∈ ℂ |
| 17 | 16 | addridi 11448 | . . . . . . 7 ⊢ (;41 + 0) = ;41 |
| 18 | 8, 9, 15, 17 | decsuc 12764 | . . . . . 6 ⊢ ((;41 + 0) + 1) = ;42 |
| 19 | 6p4e10 12805 | . . . . . 6 ⊢ (6 + 4) = ;10 | |
| 20 | 10, 11, 12, 8, 13, 14, 18, 19 | decaddc2 12789 | . . . . 5 ⊢ (;;416 + 4) = ;;420 |
| 21 | 8nn0 12549 | . . . . . . . 8 ⊢ 8 ∈ ℕ0 | |
| 22 | 2nn0 12543 | . . . . . . . 8 ⊢ 2 ∈ ℕ0 | |
| 23 | eqid 2737 | . . . . . . . 8 ⊢ ;52 = ;52 | |
| 24 | 0p1e1 12388 | . . . . . . . . 9 ⊢ (0 + 1) = 1 | |
| 25 | 8t5e40 12851 | . . . . . . . . 9 ⊢ (8 · 5) = ;40 | |
| 26 | 8, 12, 24, 25 | decsuc 12764 | . . . . . . . 8 ⊢ ((8 · 5) + 1) = ;41 |
| 27 | 8t2e16 12848 | . . . . . . . 8 ⊢ (8 · 2) = ;16 | |
| 28 | 21, 3, 22, 23, 11, 9, 26, 27 | decmul2c 12799 | . . . . . . 7 ⊢ (8 · ;52) = ;;416 |
| 29 | 1, 5 | mulcomnni 41988 | . . . . . . 7 ⊢ (8 · ;52) = (;52 · 8) |
| 30 | 28, 29 | eqtr3i 2767 | . . . . . 6 ⊢ ;;416 = (;52 · 8) |
| 31 | 30 | oveq1i 7441 | . . . . 5 ⊢ (;;416 + 4) = ((;52 · 8) + 4) |
| 32 | 20, 31 | eqtr3i 2767 | . . . 4 ⊢ ;;420 = ((;52 · 8) + 4) |
| 33 | 32 | oveq2i 7442 | . . 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 41989 | . . 3 ⊢ (8 gcd 4) = (4 gcd 8) |
| 36 | 4t2e8 12434 | . . . . 5 ⊢ (4 · 2) = 8 | |
| 37 | 36 | oveq2i 7442 | . . . 4 ⊢ (4 gcd (4 · 2)) = (4 gcd 8) |
| 38 | 2, 4 | gcdmultiplei 41994 | . . . 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 12753 | . . . 4 ⊢ ;42 ∈ ℕ |
| 42 | 41 | decnncl2 12757 | . . 3 ⊢ ;;420 ∈ ℕ |
| 43 | 1, 42 | gcdcomnni 41989 | . 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 1540 (class class class)co 7431 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 2c2 12321 4c4 12323 5c5 12324 6c6 12325 8c8 12327 ;cdc 12733 gcd cgcd 16531 |
| 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-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 |
| This theorem is referenced by: 420lcm8e840 42012 |
| Copyright terms: Public domain | W3C validator |