Theorem gcdval 12529

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 3612	. . . 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 9416	. . . 4 |- 0 e. NN0
4	2, 3	eqeltrdi 2322	. . 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 3615	. . . . 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 12528	. . . . 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 2308	. . . 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 9454	. . 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 12525	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
11		exmiddc 843	. . . 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 805	. 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 2238	. . . . 5 |- ( x = M -> ( x = 0 <-> M = 0 ) )
15	14	anbi1d 465	. . . 4 |- ( x = M -> ( ( x = 0 /\ y = 0 ) <-> ( M = 0 /\ y = 0 ) ) )
16		breq2 4092	. . . . . . 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 2791	. . . . 5 |- ( x = M -> { n e. ZZ | ( n || x /\ n || y ) } = { n e. ZZ | ( n || M /\ n || y ) } )
19	18	supeq1d 7185	. . . 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 3630	. . 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 2238	. . . . 5 |- ( y = N -> ( y = 0 <-> N = 0 ) )
22	21	anbi2d 464	. . . 4 |- ( y = N -> ( ( M = 0 /\ y = 0 ) <-> ( M = 0 /\ N = 0 ) ) )
23		breq2 4092	. . . . . . 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 2791	. . . . 5 |- ( y = N -> { n e. ZZ | ( n || M /\ n || y ) } = { n e. ZZ | ( n || M /\ n || N ) } )
26	25	supeq1d 7185	. . . 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 3630	. . 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 12524	. . 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 6155	. 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 1374	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 715  DECID wdc 841   = wceq 1397   e. wcel 2202   {crab 2514   ifcif 3605   class class class wbr 4088  (class class class)co 6017   supcsup 7180   RRcr 8030   0cc0 8031   < clt 8213   NNcn 9142   NN0cn0 9401   ZZcz 9478   || cdvds 12347   gcd cgcd 12523

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-frec 6556  df-sup 7182  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-fz 10243  df-fzo 10377  df-fl 10529  df-mod 10584  df-seqfrec 10709  df-exp 10800  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-dvds 12348  df-gcd 12524

This theorem is referenced by:  gcd0val  12530  gcdn0val  12531  gcdf  12542  gcdcom  12543  dfgcd2  12584  gcdass  12585