Theorem gcdneg 12678

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 e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) = ( M gcd N ) )

Proof of Theorem gcdneg

Step	Hyp	Ref	Expression
1		oveq12 6059	. . . . 5 |- ( ( M = 0 /\ N = 0 ) -> ( M gcd N ) = ( 0 gcd 0 ) )
2	1	adantl 277	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( 0 gcd 0 ) )
3		zcn 9582	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
4	3	negeq0d 8576	. . . . . . . 8 |- ( N e. ZZ -> ( N = 0 <-> -u N = 0 ) )
5	4	anbi2d 464	. . . . . . 7 |- ( N e. ZZ -> ( ( M = 0 /\ N = 0 ) <-> ( M = 0 /\ -u N = 0 ) ) )
6	5	adantl 277	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) <-> ( M = 0 /\ -u N = 0 ) ) )
7		oveq12 6059	. . . . . 6 |- ( ( M = 0 /\ -u N = 0 ) -> ( M gcd -u N ) = ( 0 gcd 0 ) )
8	6, 7	biimtrdi 163	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) -> ( M gcd -u N ) = ( 0 gcd 0 ) ) )
9	8	imp 124	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd -u N ) = ( 0 gcd 0 ) )
10	2, 9	eqtr4d 2268	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd -u N ) )
11		gcddvds 12659	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
12		gcdcl 12662	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
13	12	nn0zd 9698	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
14		dvdsnegb 12494	. . . . . . . . 9 |- ( ( ( M gcd N ) e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || N <-> ( M gcd N ) || -u N ) )
15	13, 14	sylancom 420	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || N <-> ( M gcd N ) || -u N ) )
16	15	anbi2d 464	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || N ) <-> ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) ) )
17	11, 16	mpbid 147	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) )
18	6	notbid 673	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) <-> -. ( M = 0 /\ -u N = 0 ) ) )
19		simpl 109	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> M e. ZZ )
20		znegcl 9608	. . . . . . . . . 10 |- ( N e. ZZ -> -u N e. ZZ )
21	20	adantl 277	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> -u N e. ZZ )
22		dvdslegcd 12660	. . . . . . . . . 10 |- ( ( ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ -u N e. ZZ ) /\ -. ( M = 0 /\ -u N = 0 ) ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) )
23	22	ex 115	. . . . . . . . 9 |- ( ( ( M gcd N ) e. ZZ /\ M e. ZZ /\ -u N e. ZZ ) -> ( -. ( M = 0 /\ -u N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) ) )
24	13, 19, 21, 23	syl3anc 1274	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ -u N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) ) )
25	18, 24	sylbid 150	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) ) )
26	25	com12 30	. . . . . 6 |- ( -. ( M = 0 /\ N = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd N ) || M /\ ( M gcd N ) || -u N ) -> ( M gcd N ) <_ ( M gcd -u N ) ) ) )
27	17, 26	mpdi 43	. . . . 5 |- ( -. ( M = 0 /\ N = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) <_ ( M gcd -u N ) ) )
28	27	impcom 125	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) <_ ( M gcd -u N ) )
29		gcddvds 12659	. . . . . . . 8 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || -u N ) )
30	20, 29	sylan2 286	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || -u N ) )
31		gcdcl 12662	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M gcd -u N ) e. NN0 )
32	31	nn0zd 9698	. . . . . . . . . 10 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M gcd -u N ) e. ZZ )
33	20, 32	sylan2 286	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) e. ZZ )
34		dvdsnegb 12494	. . . . . . . . 9 |- ( ( ( M gcd -u N ) e. ZZ /\ N e. ZZ ) -> ( ( M gcd -u N ) || N <-> ( M gcd -u N ) || -u N ) )
35	33, 34	sylancom 420	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd -u N ) || N <-> ( M gcd -u N ) || -u N ) )
36	35	anbi2d 464	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) <-> ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || -u N ) ) )
37	30, 36	mpbird 167	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) )
38		simpr 110	. . . . . . . 8 |- ( ( M e. ZZ /\ N e. ZZ ) -> N e. ZZ )
39		dvdslegcd 12660	. . . . . . . . 9 |- ( ( ( ( M gcd -u N ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) -> ( M gcd -u N ) <_ ( M gcd N ) ) )
40	39	ex 115	. . . . . . . 8 |- ( ( ( M gcd -u N ) e. ZZ /\ M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) -> ( M gcd -u N ) <_ ( M gcd N ) ) ) )
41	33, 19, 38, 40	syl3anc 1274	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( -. ( M = 0 /\ N = 0 ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) -> ( M gcd -u N ) <_ ( M gcd N ) ) ) )
42	41	com12 30	. . . . . 6 |- ( -. ( M = 0 /\ N = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( ( ( M gcd -u N ) || M /\ ( M gcd -u N ) || N ) -> ( M gcd -u N ) <_ ( M gcd N ) ) ) )
43	37, 42	mpdi 43	. . . . 5 |- ( -. ( M = 0 /\ N = 0 ) -> ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) <_ ( M gcd N ) ) )
44	43	impcom 125	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd -u N ) <_ ( M gcd N ) )
45	13	zred 9700	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
46	33	zred 9700	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) e. RR )
47	45, 46	letri3d 8389	. . . . 5 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) = ( M gcd -u N ) <-> ( ( M gcd N ) <_ ( M gcd -u N ) /\ ( M gcd -u N ) <_ ( M gcd N ) ) ) )
48	47	adantr 276	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( ( M gcd N ) = ( M gcd -u N ) <-> ( ( M gcd N ) <_ ( M gcd -u N ) /\ ( M gcd -u N ) <_ ( M gcd N ) ) ) )
49	28, 44, 48	mpbir2and 953	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd -u N ) )
50		gcdmndc 12651	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
51		exmiddc 844	. . . 4 |- (DECID ( M = 0 /\ N = 0 ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
52	50, 51	syl 14	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
53	10, 49, 52	mpjaodan 806	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd -u N ) )
54	53	eqcomd 2238	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) = ( M gcd N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2203   class class class wbr 4109  (class class class)co 6050   0cc0 8127   <_ cle 8309   -ucneg 8445   ZZcz 9577   || cdvds 12473   gcd cgcd 12649

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-sup 7275  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-fz 10343  df-fzo 10477  df-fl 10630  df-mod 10685  df-seqfrec 10810  df-exp 10901  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-dvds 12474  df-gcd 12650

This theorem is referenced by:  neggcd  12679  gcdabs  12684