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Mirrors > Home > MPE Home > Th. List > ex-gcd | Structured version Visualization version GIF version |
Description: Example for df-gcd 15055. (Contributed by AV, 5-Sep-2021.) |
Ref | Expression |
---|---|
ex-gcd | ⊢ (-6 gcd 9) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 6nn 11066 | . . . 4 ⊢ 6 ∈ ℕ | |
2 | 1 | nnzi 11278 | . . 3 ⊢ 6 ∈ ℤ |
3 | 9nn 11069 | . . . 4 ⊢ 9 ∈ ℕ | |
4 | 3 | nnzi 11278 | . . 3 ⊢ 9 ∈ ℤ |
5 | neggcd 15082 | . . 3 ⊢ ((6 ∈ ℤ ∧ 9 ∈ ℤ) → (-6 gcd 9) = (6 gcd 9)) | |
6 | 2, 4, 5 | mp2an 704 | . 2 ⊢ (-6 gcd 9) = (6 gcd 9) |
7 | 6cn 10979 | . . . . . 6 ⊢ 6 ∈ ℂ | |
8 | 3cn 10972 | . . . . . 6 ⊢ 3 ∈ ℂ | |
9 | 6p3e9 11047 | . . . . . 6 ⊢ (6 + 3) = 9 | |
10 | 7, 8, 9 | addcomli 10107 | . . . . 5 ⊢ (3 + 6) = 9 |
11 | 10 | eqcomi 2619 | . . . 4 ⊢ 9 = (3 + 6) |
12 | 11 | oveq2i 6560 | . . 3 ⊢ (6 gcd 9) = (6 gcd (3 + 6)) |
13 | 3z 11287 | . . . . . 6 ⊢ 3 ∈ ℤ | |
14 | gcdcom 15073 | . . . . . 6 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (3 gcd 6)) | |
15 | 2, 13, 14 | mp2an 704 | . . . . 5 ⊢ (6 gcd 3) = (3 gcd 6) |
16 | 3p3e6 11038 | . . . . . . 7 ⊢ (3 + 3) = 6 | |
17 | 16 | eqcomi 2619 | . . . . . 6 ⊢ 6 = (3 + 3) |
18 | 17 | oveq2i 6560 | . . . . 5 ⊢ (3 gcd 6) = (3 gcd (3 + 3)) |
19 | 15, 18 | eqtri 2632 | . . . 4 ⊢ (6 gcd 3) = (3 gcd (3 + 3)) |
20 | gcdadd 15085 | . . . . 5 ⊢ ((6 ∈ ℤ ∧ 3 ∈ ℤ) → (6 gcd 3) = (6 gcd (3 + 6))) | |
21 | 2, 13, 20 | mp2an 704 | . . . 4 ⊢ (6 gcd 3) = (6 gcd (3 + 6)) |
22 | gcdid 15086 | . . . . . 6 ⊢ (3 ∈ ℤ → (3 gcd 3) = (abs‘3)) | |
23 | 13, 22 | ax-mp 5 | . . . . 5 ⊢ (3 gcd 3) = (abs‘3) |
24 | gcdadd 15085 | . . . . . 6 ⊢ ((3 ∈ ℤ ∧ 3 ∈ ℤ) → (3 gcd 3) = (3 gcd (3 + 3))) | |
25 | 13, 13, 24 | mp2an 704 | . . . . 5 ⊢ (3 gcd 3) = (3 gcd (3 + 3)) |
26 | 3re 10971 | . . . . . 6 ⊢ 3 ∈ ℝ | |
27 | 0re 9919 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
28 | 3pos 10991 | . . . . . . 7 ⊢ 0 < 3 | |
29 | 27, 26, 28 | ltleii 10039 | . . . . . 6 ⊢ 0 ≤ 3 |
30 | absid 13884 | . . . . . 6 ⊢ ((3 ∈ ℝ ∧ 0 ≤ 3) → (abs‘3) = 3) | |
31 | 26, 29, 30 | mp2an 704 | . . . . 5 ⊢ (abs‘3) = 3 |
32 | 23, 25, 31 | 3eqtr3i 2640 | . . . 4 ⊢ (3 gcd (3 + 3)) = 3 |
33 | 19, 21, 32 | 3eqtr3i 2640 | . . 3 ⊢ (6 gcd (3 + 6)) = 3 |
34 | 12, 33 | eqtri 2632 | . 2 ⊢ (6 gcd 9) = 3 |
35 | 6, 34 | eqtri 2632 | 1 ⊢ (-6 gcd 9) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 + caddc 9818 ≤ cle 9954 -cneg 10146 3c3 10948 6c6 10951 9c9 10954 ℤcz 11254 abscabs 13822 gcd cgcd 15054 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-dvds 14822 df-gcd 15055 |
This theorem is referenced by: ex-lcm 26707 |
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