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Mirrors > Home > MPE Home > Th. List > Mathboxes > neggcdnni | Structured version Visualization version GIF version |
Description: Negation invariance for gcd. (Contributed by metakunt, 25-Apr-2024.) |
Ref | Expression |
---|---|
neggcdnni.1 | ⊢ 𝑀 ∈ ℕ |
neggcdnni.2 | ⊢ 𝑁 ∈ ℕ |
Ref | Expression |
---|---|
neggcdnni | ⊢ (-𝑀 gcd 𝑁) = (𝑀 gcd 𝑁) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neggcdnni.1 | . . . 4 ⊢ 𝑀 ∈ ℕ | |
2 | 1 | nnzi 11985 | . . 3 ⊢ 𝑀 ∈ ℤ |
3 | neggcdnni.2 | . . . 4 ⊢ 𝑁 ∈ ℕ | |
4 | 3 | nnzi 11985 | . . 3 ⊢ 𝑁 ∈ ℤ |
5 | 2, 4 | pm3.2i 473 | . 2 ⊢ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) |
6 | neggcd 15849 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 gcd 𝑁) = (𝑀 gcd 𝑁)) | |
7 | 5, 6 | ax-mp 5 | 1 ⊢ (-𝑀 gcd 𝑁) = (𝑀 gcd 𝑁) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 (class class class)co 7133 -cneg 10849 ℕcn 11616 ℤcz 11960 gcd cgcd 15821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5179 ax-nul 5186 ax-pow 5242 ax-pr 5306 ax-un 7439 ax-cnex 10571 ax-resscn 10572 ax-1cn 10573 ax-icn 10574 ax-addcl 10575 ax-addrcl 10576 ax-mulcl 10577 ax-mulrcl 10578 ax-mulcom 10579 ax-addass 10580 ax-mulass 10581 ax-distr 10582 ax-i2m1 10583 ax-1ne0 10584 ax-1rid 10585 ax-rnegex 10586 ax-rrecex 10587 ax-cnre 10588 ax-pre-lttri 10589 ax-pre-lttrn 10590 ax-pre-ltadd 10591 ax-pre-mulgt0 10592 ax-pre-sup 10593 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3007 df-nel 3111 df-ral 3130 df-rex 3131 df-reu 3132 df-rmo 3133 df-rab 3134 df-v 3475 df-sbc 3753 df-csb 3861 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3932 df-nul 4270 df-if 4444 df-pw 4517 df-sn 4544 df-pr 4546 df-tp 4548 df-op 4550 df-uni 4815 df-iun 4897 df-br 5043 df-opab 5105 df-mpt 5123 df-tr 5149 df-id 5436 df-eprel 5441 df-po 5450 df-so 5451 df-fr 5490 df-we 5492 df-xp 5537 df-rel 5538 df-cnv 5539 df-co 5540 df-dm 5541 df-rn 5542 df-res 5543 df-ima 5544 df-pred 6124 df-ord 6170 df-on 6171 df-lim 6172 df-suc 6173 df-iota 6290 df-fun 6333 df-fn 6334 df-f 6335 df-f1 6336 df-fo 6337 df-f1o 6338 df-fv 6339 df-riota 7091 df-ov 7136 df-oprab 7137 df-mpo 7138 df-om 7559 df-2nd 7668 df-wrecs 7925 df-recs 7986 df-rdg 8024 df-er 8267 df-en 8488 df-dom 8489 df-sdom 8490 df-sup 8884 df-inf 8885 df-pnf 10655 df-mnf 10656 df-xr 10657 df-ltxr 10658 df-le 10659 df-sub 10850 df-neg 10851 df-div 11276 df-nn 11617 df-2 11679 df-3 11680 df-n0 11877 df-z 11961 df-uz 12223 df-rp 12369 df-seq 13354 df-exp 13415 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-abs 14575 df-dvds 15588 df-gcd 15822 |
This theorem is referenced by: (None) |
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