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Mirrors > Home > ILE Home > Th. List > ex-gcd | GIF version |
Description: Example for df-gcd 11672. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-gcd | ⊢ (-6 gcd 9) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 8909 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 9099 | . . 3 ⊢ 6 ∈ ℤ |
3 | 9nn 8912 | . . . 4 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 9099 | . . 3 ⊢ 9 ∈ ℤ |
5 | neggcd 11707 | . . 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 8826 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 3cn 8819 | . . . . . 6 ⊢ 3 ∈ ℂ | |
9 | 6p3e9 8894 | . . . . . 6 ⊢ (6 + 3) = 9 | |
10 | 7, 8, 9 | addcomli 7931 | . . . . 5 ⊢ (3 + 6) = 9 |
11 | 10 | eqcomi 2144 | . . . 4 ⊢ 9 = (3 + 6) |
12 | 11 | oveq2i 5793 | . . 3 ⊢ (6 gcd 9) = (6 gcd (3 + 6)) |
13 | 3z 9107 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | gcdcom 11698 | . . . . . 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 8886 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
17 | 16 | eqcomi 2144 | . . . . . 6 ⊢ 6 = (3 + 3) |
18 | 17 | oveq2i 5793 | . . . . 5 ⊢ (3 gcd 6) = (3 gcd (3 + 3)) |
19 | 15, 18 | eqtri 2161 | . . . 4 ⊢ (6 gcd 3) = (3 gcd (3 + 3)) |
20 | gcdadd 11709 | . . . . 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 11710 | . . . . . 6 ⊢ (3 ∈ ℤ → (3 gcd 3) = (abs‘3)) | |
23 | 13, 22 | ax-mp 5 | . . . . 5 ⊢ (3 gcd 3) = (abs‘3) |
24 | gcdadd 11709 | . . . . . 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 8818 | . . . . . 6 ⊢ 3 ∈ ℝ | |
27 | 0re 7790 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
28 | 3pos 8838 | . . . . . . 7 ⊢ 0 < 3 | |
29 | 27, 26, 28 | ltleii 7890 | . . . . . 6 ⊢ 0 ≤ 3 |
30 | absid 10875 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → (abs‘3) = 3) | |
31 | 26, 29, 30 | mp2an 423 | . . . . 5 ⊢ (abs‘3) = 3 |
32 | 23, 25, 31 | 3eqtr3i 2169 | . . . 4 ⊢ (3 gcd (3 + 3)) = 3 |
33 | 19, 21, 32 | 3eqtr3i 2169 | . . 3 ⊢ (6 gcd (3 + 6)) = 3 |
34 | 12, 33 | eqtri 2161 | . 2 ⊢ (6 gcd 9) = 3 |
35 | 6, 34 | eqtri 2161 | 1 ⊢ (-6 gcd 9) = 3 |
Colors of variables: wff set class |
Syntax hints: = wceq 1332 ∈ wcel 1481 class class class wbr 3937 ‘cfv 5131 (class class class)co 5782 ℝcr 7643 0cc0 7644 + caddc 7647 ≤ cle 7825 -cneg 7958 3c3 8796 6c6 8799 9c9 8802 ℤcz 9078 abscabs 10801 gcd cgcd 11671 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-sup 6879 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-5 8806 df-6 8807 df-7 8808 df-8 8809 df-9 8810 df-n0 9002 df-z 9079 df-uz 9351 df-q 9439 df-rp 9471 df-fz 9822 df-fzo 9951 df-fl 10074 df-mod 10127 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803 df-dvds 11530 df-gcd 11672 |
This theorem is referenced by: (None) |
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