Theorem gcdval 11914

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 109	. . . . 5 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ ( M = 0 /\ N = 0 ) ) -> ( M = 0 /\ N = 0 ) )
2	1	iftrued 3533	. . . 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 9150	. . . 4 |- 0 e. NN0
4	2, 3	eqeltrdi 2261	. . 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 109	. . . . . 6 |- ( ( ( M e. ZZ /\ N e. ZZ ) /\ -. ( M = 0 /\ N = 0 ) ) -> -. ( M = 0 /\ N = 0 ) )
6	5	iffalsed 3536	. . . . 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 11913	. . . . 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 2247	. . . 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 9188	. . 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 11899	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
11		exmiddc 831	. . . 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 793	. 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 2177	. . . . 5 |- ( x = M -> ( x = 0 <-> M = 0 ) )
15	14	anbi1d 462	. . . 4 |- ( x = M -> ( ( x = 0 /\ y = 0 ) <-> ( M = 0 /\ y = 0 ) ) )
16		breq2 3993	. . . . . . 7 |- ( x = M -> ( n || x <-> n || M ) )
17	16	anbi1d 462	. . . . . 6 |- ( x = M -> ( ( n || x /\ n || y ) <-> ( n || M /\ n || y ) ) )
18	17	rabbidv 2719	. . . . 5 |- ( x = M -> { n e. ZZ | ( n || x /\ n || y ) } = { n e. ZZ | ( n || M /\ n || y ) } )
19	18	supeq1d 6964	. . . 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 3550	. . 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 2177	. . . . 5 |- ( y = N -> ( y = 0 <-> N = 0 ) )
22	21	anbi2d 461	. . . 4 |- ( y = N -> ( ( M = 0 /\ y = 0 ) <-> ( M = 0 /\ N = 0 ) ) )
23		breq2 3993	. . . . . . 7 |- ( y = N -> ( n || y <-> n || N ) )
24	23	anbi2d 461	. . . . . 6 |- ( y = N -> ( ( n || M /\ n || y ) <-> ( n || M /\ n || N ) ) )
25	24	rabbidv 2719	. . . . 5 |- ( y = N -> { n e. ZZ | ( n || M /\ n || y ) } = { n e. ZZ | ( n || M /\ n || N ) } )
26	25	supeq1d 6964	. . . 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 3550	. . 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 11898	. . 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 5987	. 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 1333	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 103   \/ wo 703  DECID wdc 829   = wceq 1348   e. wcel 2141   {crab 2452   ifcif 3526   class class class wbr 3989  (class class class)co 5853   supcsup 6959   RRcr 7773   0cc0 7774   < clt 7954   NNcn 8878   NN0cn0 9135   ZZcz 9212   || cdvds 11749   gcd cgcd 11897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-frec 6370  df-sup 6961  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-fz 9966  df-fzo 10099  df-fl 10226  df-mod 10279  df-seqfrec 10402  df-exp 10476  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-dvds 11750  df-gcd 11898

This theorem is referenced by:  gcd0val  11915  gcdn0val  11916  gcdf  11927  gcdcom  11928  dfgcd2  11969  gcdass  11970