Theorem gcdval 11954

Description: The value of the gcd operator. ( M gcd N ) is the greatest common divisor of M and N. If M and N are both 0, the result is defined conventionally as 0. (Contributed by Paul Chapman, 21-Mar-2011.) (Revised by Mario Carneiro, 10-Nov-2013.)

Assertion

Ref	Expression
gcdval	|- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )

Distinct variable groups:   n, M   n, N

Proof of Theorem gcdval

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M = 0 /\ N = 0 ) )
2	1	iftrued 3541	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) = 0 )
3		0nn0 9189	. . . 4 |- 0 e. NN0
4	2, 3	eqeltrdi 2268	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) e. NN0 )
5		simpr 110	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> -. ( M = 0 /\ N = 0 ) )
6	5	iffalsed 3544	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
7		gcdsupcl 11953	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) e. NN )
8	6, 7	eqeltrd 2254	. . . 4 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) e. NN )
9	8	nnnn0d 9227	. . 3 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) e. NN0 )
10		gcdmndc 11939	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
11		exmiddc 836	. . . 4 |- (DECID ( M = 0 /\ N = 0 ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
12	10, 11	syl 14	. . 3 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( ( M = 0 /\ N = 0 ) \/ -. ( M = 0 /\ N = 0 ) ) )
13	4, 9, 12	mpjaodan 798	. 2 |- ( ( M e. ZZ /\ N e. ZZ ) -> if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) e. NN0 )
14		eqeq1 2184	. . . . 5 |- ( x = M -> ( x = 0 <-> M = 0 ) )
15	14	anbi1d 465	. . . 4 |- ( x = M -> ( ( x = 0 /\ y = 0 ) <-> ( M = 0 /\ y = 0 ) ) )
16		breq2 4007	. . . . . . 7 |- ( x = M -> ( n || x <-> n || M ) )
17	16	anbi1d 465	. . . . . 6 |- ( x = M -> ( ( n || x /\ n || y ) <-> ( n || M /\ n || y ) ) )
18	17	rabbidv 2726	. . . . 5 |- ( x = M -> { n e. ZZ | ( n || x /\ n || y ) } = { n e. ZZ | ( n || M /\ n || y ) } )
19	18	supeq1d 6985	. . . 4 |- ( x = M -> sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) = sup ( { n e. ZZ | ( n || M /\ n || y ) } , RR , < ) )
20	15, 19	ifbieq2d 3558	. . 3 |- ( x = M -> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) = if ( ( M = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || y ) } , RR , < ) ) )
21		eqeq1 2184	. . . . 5 |- ( y = N -> ( y = 0 <-> N = 0 ) )
22	21	anbi2d 464	. . . 4 |- ( y = N -> ( ( M = 0 /\ y = 0 ) <-> ( M = 0 /\ N = 0 ) ) )
23		breq2 4007	. . . . . . 7 |- ( y = N -> ( n || y <-> n || N ) )
24	23	anbi2d 464	. . . . . 6 |- ( y = N -> ( ( n || M /\ n || y ) <-> ( n || M /\ n || N ) ) )
25	24	rabbidv 2726	. . . . 5 |- ( y = N -> { n e. ZZ | ( n || M /\ n || y ) } = { n e. ZZ | ( n || M /\ n || N ) } )
26	25	supeq1d 6985	. . . 4 |- ( y = N -> sup ( { n e. ZZ | ( n || M /\ n || y ) } , RR , < ) = sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) )
27	22, 26	ifbieq2d 3558	. . 3 |- ( y = N -> if ( ( M = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || y ) } , RR , < ) ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )
28		df-gcd 11938	. . 3 |- gcd = ( x e. ZZ , y e. ZZ |-> if ( ( x = 0 /\ y = 0 ) , 0 , sup ( { n e. ZZ | ( n || x /\ n || y ) } , RR , < ) ) )
29	20, 27, 28	ovmpog 6008	. 2 |- ( ( M e. ZZ /\ N e. ZZ /\ if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) e. NN0 ) -> ( M gcd N ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )
30	13, 29	mpd3an3 1338	1 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M gcd N ) = if ( ( M = 0 /\ N = 0 ) , 0 , sup ( { n e. ZZ | ( n || M /\ n || N ) } , RR , < ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 708  DECID wdc 834   = wceq 1353   e. wcel 2148   {crab 2459   ifcif 3534   class class class wbr 4003  (class class class)co 5874   supcsup 6980   RRcr 7809   0cc0 7810   < clt 7990   NNcn 8917   NN0cn0 9174   ZZcz 9251   || cdvds 11789   gcd cgcd 11937

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-sup 6982  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-fz 10007  df-fzo 10140  df-fl 10267  df-mod 10320  df-seqfrec 10443  df-exp 10517  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-dvds 11790  df-gcd 11938

This theorem is referenced by:  gcd0val  11955  gcdn0val  11956  gcdf  11967  gcdcom  11968  dfgcd2  12009  gcdass  12010