gcdneg - Intuitionistic Logic Explorer

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Theorem gcdneg 12555

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 6027	. . . . 5 ⊢ ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd 𝑁) = (0 gcd 0))
2	1	adantl 277	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (0 gcd 0))
3		zcn 9484	. . . . . . . . 9 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℂ)
4	3	negeq0d 8482	. . . . . . . 8 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ↔ -𝑁 = 0))
5	4	anbi2d 464	. . . . . . 7 ⊢ (𝑁 ∈ ℤ → ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑀 = 0 ∧ -𝑁 = 0)))
6	5	adantl 277	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∧ 𝑁 = 0) ↔ (𝑀 = 0 ∧ -𝑁 = 0)))
7		oveq12 6027	. . . . . 6 ⊢ ((𝑀 = 0 ∧ -𝑁 = 0) → (𝑀 gcd -𝑁) = (0 gcd 0))
8	6, 7	biimtrdi 163	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 = 0 ∧ 𝑁 = 0) → (𝑀 gcd -𝑁) = (0 gcd 0)))
9	8	imp 124	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd -𝑁) = (0 gcd 0))
10	2, 9	eqtr4d 2267	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
11		gcddvds 12536	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ 𝑁))
12		gcdcl 12539	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℕ₀)
13	12	nn0zd 9600	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℤ)
14		dvdsnegb 12371	. . . . . . . . 9 ⊢ (((𝑀 gcd 𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ -𝑁))
15	13, 14	sylancom 420	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑁 ↔ (𝑀 gcd 𝑁) ∥ -𝑁))
16	15	anbi2d 464	. . . . . . 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 147	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd 𝑁) ∥ 𝑀 ∧ (𝑀 gcd 𝑁) ∥ -𝑁))
18	6	notbid 673	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬ (𝑀 = 0 ∧ 𝑁 = 0) ↔ ¬ (𝑀 = 0 ∧ -𝑁 = 0)))
19		simpl 109	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ)
20		znegcl 9510	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℤ → -𝑁 ∈ ℤ)
21	20	adantl 277	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → -𝑁 ∈ ℤ)
22		dvdslegcd 12537	. . . . . . . . . 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 115	. . . . . . . . 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 1273	. . . . . . . 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 150	. . . . . . 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 125	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) ≤ (𝑀 gcd -𝑁))
29		gcddvds 12536	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ -𝑁))
30	20, 29	sylan2 286	. . . . . . 7 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ -𝑁))
31		gcdcl 12539	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℕ₀)
32	31	nn0zd 9600	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ -𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℤ)
33	20, 32	sylan2 286	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℤ)
34		dvdsnegb 12371	. . . . . . . . 9 ⊢ (((𝑀 gcd -𝑁) ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑁 ↔ (𝑀 gcd -𝑁) ∥ -𝑁))
35	33, 34	sylancom 420	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑁 ↔ (𝑀 gcd -𝑁) ∥ -𝑁))
36	35	anbi2d 464	. . . . . . 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 167	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑀 gcd -𝑁) ∥ 𝑀 ∧ (𝑀 gcd -𝑁) ∥ 𝑁))
38		simpr 110	. . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑁 ∈ ℤ)
39		dvdslegcd 12537	. . . . . . . . 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 115	. . . . . . . 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 1273	. . . . . . 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 125	. . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd -𝑁) ≤ (𝑀 gcd 𝑁))
45	13	zred 9602	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) ∈ ℝ)
46	33	zred 9602	. . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) ∈ ℝ)
47	45, 46	letri3d 8295	. . . . 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 276	. . . 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 952	. . 3 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬ (𝑀 = 0 ∧ 𝑁 = 0)) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
50		gcdmndc 12528	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID (𝑀 = 0 ∧ 𝑁 = 0))
51		exmiddc 843	. . . 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 805	. 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd 𝑁) = (𝑀 gcd -𝑁))
54	53	eqcomd 2237	1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 DECID wdc 841 ∧ w3a 1004 = wceq 1397 ∈ wcel 2202 class class class wbr 4088 (class class class)co 6018 0cc0 8032 ≤ cle 8215 -cneg 8351 ℤcz 9479 ∥ cdvds 12350 gcd cgcd 12526

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 ax-pre-mulext 8150 ax-arch 8151 ax-caucvg 8152

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-po 4393 df-iso 4394 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-sup 7183 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 df-div 8853 df-inn 9144 df-2 9202 df-3 9203 df-4 9204 df-n0 9403 df-z 9480 df-uz 9756 df-q 9854 df-rp 9889 df-fz 10244 df-fzo 10378 df-fl 10531 df-mod 10586 df-seqfrec 10711 df-exp 10802 df-cj 11404 df-re 11405 df-im 11406 df-rsqrt 11560 df-abs 11561 df-dvds 12351 df-gcd 12527

This theorem is referenced by: neggcd 12556 gcdabs 12561

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