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Mirrors > Home > ILE Home > Th. List > ex-gcd | GIF version |
Description: Example for df-gcd 11861. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-gcd | ⊢ (-6 gcd 9) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 9013 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 9203 | . . 3 ⊢ 6 ∈ ℤ |
3 | 9nn 9016 | . . . 4 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 9203 | . . 3 ⊢ 9 ∈ ℤ |
5 | neggcd 11901 | . . 3 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (-6 gcd 9) = (6 gcd 9)) | |
6 | 2, 4, 5 | mp2an 423 | . 2 ⊢ (-6 gcd 9) = (6 gcd 9) |
7 | 6cn 8930 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 3cn 8923 | . . . . . 6 ⊢ 3 ∈ ℂ | |
9 | 6p3e9 8998 | . . . . . 6 ⊢ (6 + 3) = 9 | |
10 | 7, 8, 9 | addcomli 8034 | . . . . 5 ⊢ (3 + 6) = 9 |
11 | 10 | eqcomi 2168 | . . . 4 ⊢ 9 = (3 + 6) |
12 | 11 | oveq2i 5847 | . . 3 ⊢ (6 gcd 9) = (6 gcd (3 + 6)) |
13 | 3z 9211 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | gcdcom 11891 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (3 gcd 6)) | |
15 | 2, 13, 14 | mp2an 423 | . . . . 5 ⊢ (6 gcd 3) = (3 gcd 6) |
16 | 3p3e6 8990 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
17 | 16 | eqcomi 2168 | . . . . . 6 ⊢ 6 = (3 + 3) |
18 | 17 | oveq2i 5847 | . . . . 5 ⊢ (3 gcd 6) = (3 gcd (3 + 3)) |
19 | 15, 18 | eqtri 2185 | . . . 4 ⊢ (6 gcd 3) = (3 gcd (3 + 3)) |
20 | gcdadd 11903 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (6 gcd (3 + 6))) | |
21 | 2, 13, 20 | mp2an 423 | . . . 4 ⊢ (6 gcd 3) = (6 gcd (3 + 6)) |
22 | gcdid 11904 | . . . . . 6 ⊢ (3 ∈ ℤ → (3 gcd 3) = (abs‘3)) | |
23 | 13, 22 | ax-mp 5 | . . . . 5 ⊢ (3 gcd 3) = (abs‘3) |
24 | gcdadd 11903 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 3 ∈ ℤ) → (3 gcd 3) = (3 gcd (3 + 3))) | |
25 | 13, 13, 24 | mp2an 423 | . . . . 5 ⊢ (3 gcd 3) = (3 gcd (3 + 3)) |
26 | 3re 8922 | . . . . . 6 ⊢ 3 ∈ ℝ | |
27 | 0re 7890 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
28 | 3pos 8942 | . . . . . . 7 ⊢ 0 < 3 | |
29 | 27, 26, 28 | ltleii 7992 | . . . . . 6 ⊢ 0 ≤ 3 |
30 | absid 10999 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → (abs‘3) = 3) | |
31 | 26, 29, 30 | mp2an 423 | . . . . 5 ⊢ (abs‘3) = 3 |
32 | 23, 25, 31 | 3eqtr3i 2193 | . . . 4 ⊢ (3 gcd (3 + 3)) = 3 |
33 | 19, 21, 32 | 3eqtr3i 2193 | . . 3 ⊢ (6 gcd (3 + 6)) = 3 |
34 | 12, 33 | eqtri 2185 | . 2 ⊢ (6 gcd 9) = 3 |
35 | 6, 34 | eqtri 2185 | 1 ⊢ (-6 gcd 9) = 3 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 class class class wbr 3976 ‘cfv 5182 (class class class)co 5836 ℝcr 7743 0cc0 7744 + caddc 7747 ≤ cle 7925 -cneg 8061 3c3 8900 6c6 8903 9c9 8906 ℤcz 9182 abscabs 10925 gcd cgcd 11860 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862 ax-arch 7863 ax-caucvg 7864 |
This theorem depends on definitions: df-bi 116 df-stab 821 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-if 3516 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-tr 4075 df-id 4265 df-po 4268 df-iso 4269 df-iord 4338 df-on 4340 df-ilim 4341 df-suc 4343 df-iom 4562 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-1st 6100 df-2nd 6101 df-recs 6264 df-frec 6350 df-sup 6940 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-n0 9106 df-z 9183 df-uz 9458 df-q 9549 df-rp 9581 df-fz 9936 df-fzo 10068 df-fl 10195 df-mod 10248 df-seqfrec 10371 df-exp 10445 df-cj 10770 df-re 10771 df-im 10772 df-rsqrt 10926 df-abs 10927 df-dvds 11714 df-gcd 11861 |
This theorem is referenced by: (None) |
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