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Theorem gcdneg 16451

Description: Negating one operand of the gcd operator does not alter the result. (Contributed by Paul Chapman, 21-Mar-2011.)

**Assertion**
Ref	Expression
gcdneg	⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁))

**Proof of Theorem gcdneg**
Step	Hyp	Ref	Expression
1		oveq12 7362	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
2	1	adantl 481	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (0 gcd 0))
3		zcn 12494	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
4	3	negeq0d 11485	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ↔ -𝑁 = 0))
5	4	anbi2d 630	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑀 = 0 ∧ -𝑁 = 0)))
6	5	adantl 481	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑀 = 0 ∧ -𝑁 = 0)))
7		oveq12 7362	. . . . . 6 ⊢ ((𝑀 = 0 ∧ -𝑁 = 0) → (𝑀 gcd -𝑁) = (0 gcd 0))
8	6, 7	biimtrdi 253	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd -𝑁) = (0 gcd 0)))
9	8	imp 406	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd -𝑁) = (0 gcd 0))
10	2, 9	eqtr4d 2767	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
11		gcddvds 16432	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
12		gcdcl 16435	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
13	12	nn0zd 12515	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℤ)
14		dvdsnegb 16202	. . . . . . . . 9 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ -𝑁))
15	13, 14	sylancom 588	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ -𝑁))
16	15	anbi2d 630	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁) ↔ ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁)))
17	11, 16	mpbid 232	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁))
18	6	notbid 318	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ 𝑁 = 0) ↔ ¬ (𝑀 = 0 ∧ -𝑁 = 0)))
19		simpl 482	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ)
20		znegcl 12528	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
21	20	adantl 481	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → -𝑁 ∈ ℤ)
22		dvdslegcd 16433	. . . . . . . . . 10 ⊢ ((((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ -𝑁 = 0)) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁)))
23	22	ex 412	. . . . . . . . 9 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ -𝑁 = 0) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁))))
24	13, 19, 21, 23	syl3anc 1373	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ -𝑁 = 0) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁))))
25	18, 24	sylbid 240	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁))))
26	25	com12 32	. . . . . 6 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁))))
27	17, 26	mpdi 45	. . . . 5 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁)))
28	27	impcom 407	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁))
29		gcddvds 16432	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ -𝑁))
30	20, 29	sylan2 593	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ -𝑁))
31		gcdcl 16435	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℕ₀)
32	31	nn0zd 12515	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℤ)
33	20, 32	sylan2 593	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℤ)
34		dvdsnegb 16202	. . . . . . . . 9 ⊢ (((𝑀 gcd -𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑁 ↔ (𝑀 gcd -𝑁) ∥ -𝑁))
35	33, 34	sylancom 588	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑁 ↔ (𝑀 gcd -𝑁) ∥ -𝑁))
36	35	anbi2d 630	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ 𝑁) ↔ ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ -𝑁)))
37	30, 36	mpbird 257	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ 𝑁))
38		simpr 484	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
39		dvdslegcd 16433	. . . . . . . . 9 ⊢ ((((𝑀 gcd -𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ 𝑁) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁)))
40	39	ex 412	. . . . . . . 8 ⊢ (((𝑀 gcd -𝑁) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ 𝑁) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁))))
41	33, 19, 38, 40	syl3anc 1373	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ 𝑁 = 0) → (((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ 𝑁) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁))))
42	41	com12 32	. . . . . 6 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ 𝑁) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁))))
43	37, 42	mpdi 45	. . . . 5 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁)))
44	43	impcom 407	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁))
45	13	zred 12598	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℝ)
46	33	zred 12598	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℝ)
47	45, 46	letri3d 11276	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) = (𝑀 gcd -𝑁) ↔ ((𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁) ∧ (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁))))
48	47	adantr 480	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → ((𝑀 gcd 𝑁) = (𝑀 gcd -𝑁) ↔ ((𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁) ∧ (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁))))
49	28, 44, 48	mpbir2and 713	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
50	10, 49	pm2.61dan 812	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
51	50	eqcomd 2735	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 class class class wbr 5095 (class class class)co 7353 0cc0 11028 ≤ cle 11169 -cneg 11366 ℤcz 12489 ∥ cdvds 16181 gcd cgcd 16423

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4862 df-iun 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8632 df-en 8880 df-dom 8881 df-sdom 8882 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-n0 12403 df-z 12490 df-uz 12754 df-rp 12912 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-dvds 16182 df-gcd 16424

This theorem is referenced by: neggcd 16452 odinv 19458 divnumden2 32773 gcdnegnni 41962

W3C validator