![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ex-gcd | Structured version Visualization version GIF version |
Description: Example for df-gcd 16432. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-gcd | ⊢ (-6 gcd 9) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 12297 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 12582 | . . 3 ⊢ 6 ∈ ℤ |
3 | 9nn 12306 | . . . 4 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 12582 | . . 3 ⊢ 9 ∈ ℤ |
5 | neggcd 16460 | . . 3 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (-6 gcd 9) = (6 gcd 9)) | |
6 | 2, 4, 5 | mp2an 691 | . 2 ⊢ (-6 gcd 9) = (6 gcd 9) |
7 | 6cn 12299 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 3cn 12289 | . . . . . 6 ⊢ 3 ∈ ℂ | |
9 | 6p3e9 12368 | . . . . . 6 ⊢ (6 + 3) = 9 | |
10 | 7, 8, 9 | addcomli 11402 | . . . . 5 ⊢ (3 + 6) = 9 |
11 | 10 | eqcomi 2742 | . . . 4 ⊢ 9 = (3 + 6) |
12 | 11 | oveq2i 7415 | . . 3 ⊢ (6 gcd 9) = (6 gcd (3 + 6)) |
13 | 3z 12591 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | gcdcom 16450 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (3 gcd 6)) | |
15 | 2, 13, 14 | mp2an 691 | . . . . 5 ⊢ (6 gcd 3) = (3 gcd 6) |
16 | 3p3e6 12360 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
17 | 16 | eqcomi 2742 | . . . . . 6 ⊢ 6 = (3 + 3) |
18 | 17 | oveq2i 7415 | . . . . 5 ⊢ (3 gcd 6) = (3 gcd (3 + 3)) |
19 | 15, 18 | eqtri 2761 | . . . 4 ⊢ (6 gcd 3) = (3 gcd (3 + 3)) |
20 | gcdadd 16463 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (6 gcd (3 + 6))) | |
21 | 2, 13, 20 | mp2an 691 | . . . 4 ⊢ (6 gcd 3) = (6 gcd (3 + 6)) |
22 | gcdid 16464 | . . . . . 6 ⊢ (3 ∈ ℤ → (3 gcd 3) = (abs‘3)) | |
23 | 13, 22 | ax-mp 5 | . . . . 5 ⊢ (3 gcd 3) = (abs‘3) |
24 | gcdadd 16463 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 3 ∈ ℤ) → (3 gcd 3) = (3 gcd (3 + 3))) | |
25 | 13, 13, 24 | mp2an 691 | . . . . 5 ⊢ (3 gcd 3) = (3 gcd (3 + 3)) |
26 | 3re 12288 | . . . . . 6 ⊢ 3 ∈ ℝ | |
27 | 0re 11212 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
28 | 3pos 12313 | . . . . . . 7 ⊢ 0 < 3 | |
29 | 27, 26, 28 | ltleii 11333 | . . . . . 6 ⊢ 0 ≤ 3 |
30 | absid 15239 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → (abs‘3) = 3) | |
31 | 26, 29, 30 | mp2an 691 | . . . . 5 ⊢ (abs‘3) = 3 |
32 | 23, 25, 31 | 3eqtr3i 2769 | . . . 4 ⊢ (3 gcd (3 + 3)) = 3 |
33 | 19, 21, 32 | 3eqtr3i 2769 | . . 3 ⊢ (6 gcd (3 + 6)) = 3 |
34 | 12, 33 | eqtri 2761 | . 2 ⊢ (6 gcd 9) = 3 |
35 | 6, 34 | eqtri 2761 | 1 ⊢ (-6 gcd 9) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2107 class class class wbr 5147 ‘cfv 6540 (class class class)co 7404 ℝcr 11105 0cc0 11106 + caddc 11109 ≤ cle 11245 -cneg 11441 3c3 12264 6c6 12267 9c9 12270 ℤcz 12554 abscabs 15177 gcd cgcd 16431 |
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 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-dvds 16194 df-gcd 16432 |
This theorem is referenced by: ex-lcm 29691 |
Copyright terms: Public domain | W3C validator |