Theorem gcdneg 12616

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 6037	. . . . 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 9528	. . . . . . . . 9 |- ( N e. ZZ -> N e. CC )
4	3	negeq0d 8524	. . . . . . . 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 6037	. . . . . 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 2267	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M gcd N ) = ( M gcd -u N ) )
11		gcddvds 12597	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M gcd N ) || M /\ ( M gcd N ) || N ) )
12		gcdcl 12600	. . . . . . . . . 10 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. NN0 )
13	12	nn0zd 9644	. . . . . . . . 9 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. ZZ )
14		dvdsnegb 12432	. . . . . . . . 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 9554	. . . . . . . . . 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 12598	. . . . . . . . . 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 12597	. . . . . . . 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 12600	. . . . . . . . . . 11 |- ( ( M e. ZZ /\ -u N e. ZZ ) -> ( M gcd -u N ) e. NN0 )
32	31	nn0zd 9644	. . . . . . . . . 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 12432	. . . . . . . . 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 12598	. . . . . . . . 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 9646	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) e. RR )
46	33	zred 9646	. . . . . 6 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd -u N ) e. RR )
47	45, 46	letri3d 8337	. . . . 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 12589	. . . 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 2237	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 2202   class class class wbr 4093  (class class class)co 6028   0cc0 8075   <_ cle 8257   -ucneg 8393   ZZcz 9523   || cdvds 12411   gcd cgcd 12587

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-sup 7226  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-q 9898  df-rp 9933  df-fz 10289  df-fzo 10423  df-fl 10576  df-mod 10631  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622  df-dvds 12412  df-gcd 12588

This theorem is referenced by:  neggcd  12617  gcdabs  12622