Theorem gcdval 12593

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 3616	. . . 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 9459	. . . 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 3619	. . . . 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 12592	. . . . 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 9499	. . 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 12589	. . . 4 |- ( ( M e. ZZ /\ N e. ZZ ) -> DECID ( M = 0 /\ N = 0 ) )
11		exmiddc 844	. . . 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 806	. 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 4097	. . . . . . 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 2792	. . . . 5 |- ( x = M -> { n e. ZZ | ( n || x /\ n || y ) } = { n e. ZZ | ( n || M /\ n || y ) } )
19	18	supeq1d 7229	. . . 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 3634	. . 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 4097	. . . . . . 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 2792	. . . . 5 |- ( y = N -> { n e. ZZ | ( n || M /\ n || y ) } = { n e. ZZ | ( n || M /\ n || N ) } )
26	25	supeq1d 7229	. . . 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 3634	. . 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 12588	. . 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 6166	. 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 1375	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 716  DECID wdc 842   = wceq 1398   e. wcel 2202   {crab 2515   ifcif 3607   class class class wbr 4093  (class class class)co 6028   supcsup 7224   RRcr 8074   0cc0 8075   < clt 8256   NNcn 9185   NN0cn0 9444   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:  gcd0val  12594  gcdn0val  12595  gcdf  12606  gcdcom  12607  dfgcd2  12648  gcdass  12649