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Mirrors > Home > ILE Home > Th. List > ex-gcd | GIF version |
Description: Example for df-gcd 10339. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-gcd | ⊢ (-6 gcd 9) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 8197 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 8372 | . . 3 ⊢ 6 ∈ ℤ |
3 | 9nn 8200 | . . . 4 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 8372 | . . 3 ⊢ 9 ∈ ℤ |
5 | neggcd 10374 | . . 3 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (-6 gcd 9) = (6 gcd 9)) | |
6 | 2, 4, 5 | mp2an 416 | . 2 ⊢ (-6 gcd 9) = (6 gcd 9) |
7 | 6cn 8121 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 3cn 8114 | . . . . . 6 ⊢ 3 ∈ ℂ | |
9 | 6p3e9 8182 | . . . . . 6 ⊢ (6 + 3) = 9 | |
10 | 7, 8, 9 | addcomli 7253 | . . . . 5 ⊢ (3 + 6) = 9 |
11 | 10 | eqcomi 2085 | . . . 4 ⊢ 9 = (3 + 6) |
12 | 11 | oveq2i 5543 | . . 3 ⊢ (6 gcd 9) = (6 gcd (3 + 6)) |
13 | 3z 8380 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | gcdcom 10365 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (3 gcd 6)) | |
15 | 2, 13, 14 | mp2an 416 | . . . . 5 ⊢ (6 gcd 3) = (3 gcd 6) |
16 | 3p3e6 8174 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
17 | 16 | eqcomi 2085 | . . . . . 6 ⊢ 6 = (3 + 3) |
18 | 17 | oveq2i 5543 | . . . . 5 ⊢ (3 gcd 6) = (3 gcd (3 + 3)) |
19 | 15, 18 | eqtri 2101 | . . . 4 ⊢ (6 gcd 3) = (3 gcd (3 + 3)) |
20 | gcdadd 10376 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (6 gcd (3 + 6))) | |
21 | 2, 13, 20 | mp2an 416 | . . . 4 ⊢ (6 gcd 3) = (6 gcd (3 + 6)) |
22 | gcdid 10377 | . . . . . 6 ⊢ (3 ∈ ℤ → (3 gcd 3) = (abs‘3)) | |
23 | 13, 22 | ax-mp 7 | . . . . 5 ⊢ (3 gcd 3) = (abs‘3) |
24 | gcdadd 10376 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 3 ∈ ℤ) → (3 gcd 3) = (3 gcd (3 + 3))) | |
25 | 13, 13, 24 | mp2an 416 | . . . . 5 ⊢ (3 gcd 3) = (3 gcd (3 + 3)) |
26 | 3re 8113 | . . . . . 6 ⊢ 3 ∈ ℝ | |
27 | 0re 7119 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
28 | 3pos 8133 | . . . . . . 7 ⊢ 0 < 3 | |
29 | 27, 26, 28 | ltleii 7213 | . . . . . 6 ⊢ 0 ≤ 3 |
30 | absid 9957 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → (abs‘3) = 3) | |
31 | 26, 29, 30 | mp2an 416 | . . . . 5 ⊢ (abs‘3) = 3 |
32 | 23, 25, 31 | 3eqtr3i 2109 | . . . 4 ⊢ (3 gcd (3 + 3)) = 3 |
33 | 19, 21, 32 | 3eqtr3i 2109 | . . 3 ⊢ (6 gcd (3 + 6)) = 3 |
34 | 12, 33 | eqtri 2101 | . 2 ⊢ (6 gcd 9) = 3 |
35 | 6, 34 | eqtri 2101 | 1 ⊢ (-6 gcd 9) = 3 |
Colors of variables: wff set class |
Syntax hints: = wceq 1284 ∈ wcel 1433 class class class wbr 3785 ‘cfv 4922 (class class class)co 5532 ℝcr 6980 0cc0 6981 + caddc 6984 ≤ cle 7154 -cneg 7280 3c3 8090 6c6 8093 9c9 8096 ℤcz 8351 abscabs 9883 gcd cgcd 10338 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-coll 3893 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-iinf 4329 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094 ax-arch 7095 ax-caucvg 7096 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-csb 2909 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-if 3352 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-iun 3680 df-br 3786 df-opab 3840 df-mpt 3841 df-tr 3876 df-id 4048 df-po 4051 df-iso 4052 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-1st 5787 df-2nd 5788 df-recs 5943 df-frec 6001 df-sup 6397 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-inn 8040 df-2 8098 df-3 8099 df-4 8100 df-5 8101 df-6 8102 df-7 8103 df-8 8104 df-9 8105 df-n0 8289 df-z 8352 df-uz 8620 df-q 8705 df-rp 8735 df-fz 9030 df-fzo 9153 df-fl 9274 df-mod 9325 df-iseq 9432 df-iexp 9476 df-cj 9729 df-re 9730 df-im 9731 df-rsqrt 9884 df-abs 9885 df-dvds 10196 df-gcd 10339 |
This theorem is referenced by: (None) |
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