gcdneg - Intuitionistic Logic Explorer

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Theorem gcdneg 11915

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 5851	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
2	1	adantl 275	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (0 gcd 0))
3		zcn 9196	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
4	3	negeq0d 8201	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ↔ -𝑁 = 0))
5	4	anbi2d 460	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑀 = 0 ∧ -𝑁 = 0)))
6	5	adantl 275	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑀 = 0 ∧ -𝑁 = 0)))
7		oveq12 5851	. . . . . 6 ⊢ ((𝑀 = 0 ∧ -𝑁 = 0) → (𝑀 gcd -𝑁) = (0 gcd 0))
8	6, 7	syl6bi 162	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd -𝑁) = (0 gcd 0)))
9	8	imp 123	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd -𝑁) = (0 gcd 0))
10	2, 9	eqtr4d 2201	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
11		gcddvds 11896	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
12		gcdcl 11899	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
13	12	nn0zd 9311	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℤ)
14		dvdsnegb 11748	. . . . . . . . 9 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ -𝑁))
15	13, 14	sylancom 417	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ -𝑁))
16	15	anbi2d 460	. . . . . . 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 146	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁))
18	6	notbid 657	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ 𝑁 = 0) ↔ ¬ (𝑀 = 0 ∧ -𝑁 = 0)))
19		simpl 108	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ)
20		znegcl 9222	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
21	20	adantl 275	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → -𝑁 ∈ ℤ)
22		dvdslegcd 11897	. . . . . . . . . 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 114	. . . . . . . . 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 1228	. . . . . . . 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 149	. . . . . . 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 30	. . . . . 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 43	. . . . 5 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁)))
28	27	impcom 124	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁))
29		gcddvds 11896	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ -𝑁))
30	20, 29	sylan2 284	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ -𝑁))
31		gcdcl 11899	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℕ₀)
32	31	nn0zd 9311	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℤ)
33	20, 32	sylan2 284	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℤ)
34		dvdsnegb 11748	. . . . . . . . 9 ⊢ (((𝑀 gcd -𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑁 ↔ (𝑀 gcd -𝑁) ∥ -𝑁))
35	33, 34	sylancom 417	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑁 ↔ (𝑀 gcd -𝑁) ∥ -𝑁))
36	35	anbi2d 460	. . . . . . 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 166	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ 𝑁))
38		simpr 109	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
39		dvdslegcd 11897	. . . . . . . . 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 114	. . . . . . . 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 1228	. . . . . . 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 30	. . . . . 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 43	. . . . 5 ⊢ (¬ (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁)))
44	43	impcom 124	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁))
45	13	zred 9313	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℝ)
46	33	zred 9313	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℝ)
47	45, 46	letri3d 8014	. . . . 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 274	. . . 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 934	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
50		gcdmndc 11877	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∧ 𝑁 = 0))
51		exmiddc 826	. . . 4 ⊢ (DECID (𝑀 = 0 ∧ 𝑁 = 0) → ((𝑀 = 0 ∧ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∧ 𝑁 = 0)))
52	50, 51	syl 14	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∧ 𝑁 = 0) ∨ ¬ (𝑀 = 0 ∧ 𝑁 = 0)))
53	10, 49, 52	mpjaodan 788	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
54	53	eqcomd 2171	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ↔ wb 104 ∨ wo 698 DECID wdc 824 ∧ w3a 968 = wceq 1343 ∈ wcel 2136 class class class wbr 3982 (class class class)co 5842 0cc0 7753 ≤ cle 7934 -cneg 8070 ℤcz 9191 ∥ cdvds 11727 gcd cgcd 11875

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 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-coll 4097 ax-sep 4100 ax-nul 4108 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-iinf 4565 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871 ax-arch 7872 ax-caucvg 7873

This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 969 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-csb 3046 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-nul 3410 df-if 3521 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-int 3825 df-iun 3868 df-br 3983 df-opab 4044 df-mpt 4045 df-tr 4081 df-id 4271 df-po 4274 df-iso 4275 df-iord 4344 df-on 4346 df-ilim 4347 df-suc 4349 df-iom 4568 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-f1 5193 df-fo 5194 df-f1o 5195 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-1st 6108 df-2nd 6109 df-recs 6273 df-frec 6359 df-sup 6949 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-inn 8858 df-2 8916 df-3 8917 df-4 8918 df-n0 9115 df-z 9192 df-uz 9467 df-q 9558 df-rp 9590 df-fz 9945 df-fzo 10078 df-fl 10205 df-mod 10258 df-seqfrec 10381 df-exp 10455 df-cj 10784 df-re 10785 df-im 10786 df-rsqrt 10940 df-abs 10941 df-dvds 11728 df-gcd 11876

This theorem is referenced by: neggcd 11916 gcdabs 11921

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