Theorem gcdneg 12418

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 5976	. . . . 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 9412	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
4	3	negeq0d 8410	. . . . . . . 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 5976	. . . . . 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 2243	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd -u N ) )
11		gcddvds 12399	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
12		gcdcl 12402	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
13	12	nn0zd 9528	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
14		dvdsnegb 12234	. . . . . . . . 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 669	. . . . . . . 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 9438	. . . . . . . . . 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 12400	. . . . . . . . . 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 1250	. . . . . . . 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 12399	. . . . . . . 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 12402	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M gcd -u N ) e. NN0 )
32	31	nn0zd 9528	. . . . . . . . . 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 12234	. . . . . . . . 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 12400	. . . . . . . . 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 1250	. . . . . . 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 9530	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
46	33	zred 9530	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) e. RR )
47	45, 46	letri3d 8223	. . . . 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 947	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd -u N ) )
50		gcdmndc 12391	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
51		exmiddc 838	. . . 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 800	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = ( M gcd -u N ) )
54	53	eqcomd 2213	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 710  DECID wdc 836   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   0cc0 7960   <_ cle 8143   -ucneg 8279   ZZcz 9407   || cdvds 12213   gcd cgcd 12389

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078  ax-arch 8079  ax-caucvg 8080

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-sup 7112  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-2 9130  df-3 9131  df-4 9132  df-n0 9331  df-z 9408  df-uz 9684  df-q 9776  df-rp 9811  df-fz 10166  df-fzo 10300  df-fl 10450  df-mod 10505  df-seqfrec 10630  df-exp 10721  df-cj 11268  df-re 11269  df-im 11270  df-rsqrt 11424  df-abs 11425  df-dvds 12214  df-gcd 12390

This theorem is referenced by:  neggcd  12419  gcdabs  12424