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| Mirrors > Home > MPE Home > Th. List > ex-gcd | Structured version Visualization version GIF version | ||
| Description: Example for df-gcd 16532. (Contributed by AV, 5-Sep-2021.) |
| Ref | Expression |
|---|---|
| ex-gcd | ⊢ (-6 gcd 9) = 3 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 6nn 12355 | . . . 4 ⊢ 6 ∈ ℕ | |
| 2 | 1 | nnzi 12641 | . . 3 ⊢ 6 ∈ ℤ |
| 3 | 9nn 12364 | . . . 4 ⊢ 9 ∈ ℕ | |
| 4 | 3 | nnzi 12641 | . . 3 ⊢ 9 ∈ ℤ |
| 5 | neggcd 16560 | . . 3 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (-6 gcd 9) = (6 gcd 9)) | |
| 6 | 2, 4, 5 | mp2an 692 | . 2 ⊢ (-6 gcd 9) = (6 gcd 9) |
| 7 | 6cn 12357 | . . . . . 6 ⊢ 6 ∈ ℂ | |
| 8 | 3cn 12347 | . . . . . 6 ⊢ 3 ∈ ℂ | |
| 9 | 6p3e9 12426 | . . . . . 6 ⊢ (6 + 3) = 9 | |
| 10 | 7, 8, 9 | addcomli 11453 | . . . . 5 ⊢ (3 + 6) = 9 |
| 11 | 10 | eqcomi 2746 | . . . 4 ⊢ 9 = (3 + 6) |
| 12 | 11 | oveq2i 7442 | . . 3 ⊢ (6 gcd 9) = (6 gcd (3 + 6)) |
| 13 | 3z 12650 | . . . . . 6 ⊢ 3 ∈ ℤ | |
| 14 | gcdcom 16550 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (3 gcd 6)) | |
| 15 | 2, 13, 14 | mp2an 692 | . . . . 5 ⊢ (6 gcd 3) = (3 gcd 6) |
| 16 | 3p3e6 12418 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
| 17 | 16 | eqcomi 2746 | . . . . . 6 ⊢ 6 = (3 + 3) |
| 18 | 17 | oveq2i 7442 | . . . . 5 ⊢ (3 gcd 6) = (3 gcd (3 + 3)) |
| 19 | 15, 18 | eqtri 2765 | . . . 4 ⊢ (6 gcd 3) = (3 gcd (3 + 3)) |
| 20 | gcdadd 16563 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (6 gcd (3 + 6))) | |
| 21 | 2, 13, 20 | mp2an 692 | . . . 4 ⊢ (6 gcd 3) = (6 gcd (3 + 6)) |
| 22 | gcdid 16564 | . . . . . 6 ⊢ (3 ∈ ℤ → (3 gcd 3) = (abs‘3)) | |
| 23 | 13, 22 | ax-mp 5 | . . . . 5 ⊢ (3 gcd 3) = (abs‘3) |
| 24 | gcdadd 16563 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 3 ∈ ℤ) → (3 gcd 3) = (3 gcd (3 + 3))) | |
| 25 | 13, 13, 24 | mp2an 692 | . . . . 5 ⊢ (3 gcd 3) = (3 gcd (3 + 3)) |
| 26 | 3re 12346 | . . . . . 6 ⊢ 3 ∈ ℝ | |
| 27 | 0re 11263 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
| 28 | 3pos 12371 | . . . . . . 7 ⊢ 0 < 3 | |
| 29 | 27, 26, 28 | ltleii 11384 | . . . . . 6 ⊢ 0 ≤ 3 |
| 30 | absid 15335 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → (abs‘3) = 3) | |
| 31 | 26, 29, 30 | mp2an 692 | . . . . 5 ⊢ (abs‘3) = 3 |
| 32 | 23, 25, 31 | 3eqtr3i 2773 | . . . 4 ⊢ (3 gcd (3 + 3)) = 3 |
| 33 | 19, 21, 32 | 3eqtr3i 2773 | . . 3 ⊢ (6 gcd (3 + 6)) = 3 |
| 34 | 12, 33 | eqtri 2765 | . 2 ⊢ (6 gcd 9) = 3 |
| 35 | 6, 34 | eqtri 2765 | 1 ⊢ (-6 gcd 9) = 3 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 0cc0 11155 + caddc 11158 ≤ cle 11296 -cneg 11493 3c3 12322 6c6 12325 9c9 12328 ℤcz 12613 abscabs 15273 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-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: ex-lcm 30477 |
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