Theorem eucalgcvga 11728

Description: Once Euclid's Algorithm halts after N steps, the second element of the state remains 0 . (Contributed by Paul Chapman, 22-Jun-2011.) (Revised by Mario Carneiro, 29-May-2014.)

Hypotheses

Ref	Expression
eucalgval.1	|- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )
eucalg.2	|- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
eucalgcvga.3	|- N = ( 2nd ` A )

Assertion

Ref	Expression
eucalgcvga	|- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` N ) -> ( 2nd ` ( R ` K ) ) = 0 ) )

Distinct variable groups:   x, y, N   x, A, y   x, R

Allowed substitution hints:   R( y)   E( x, y)   K( x, y)

Proof of Theorem eucalgcvga

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eucalgcvga.3	. . . . . . 7 |- N = ( 2nd ` A )
2		xp2nd 6057	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
3	1, 2	eqeltrid 2224	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> N e. NN0 )
4		eluznn0 9386	. . . . . 6 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) ) -> K e. NN0 )
5	3, 4	sylan 281	. . . . 5 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> K e. NN0 )
6		nn0uz 9353	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
7		eucalg.2	. . . . . . 7 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
8		0zd 9059	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> 0 e. ZZ )
9		id 19	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> A e. ( NN0 X. NN0 ) )
10		eucalgval.1	. . . . . . . . 9 |- E = ( x e. NN0 , y e. NN0 |-> if ( y = 0 , <. x , y >. , <. y , ( x mod y ) >. ) )
11	10	eucalgf 11725	. . . . . . . 8 |- E : ( NN0 X. NN0 ) --> ( NN0 X. NN0 )
12	11	a1i 9	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> E : ( NN0 X. NN0 ) --> ( NN0 X. NN0 ) )
13	6, 7, 8, 9, 12	algrf 11715	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
14	13	ffvelrnda 5548	. . . . 5 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. NN0 ) -> ( R ` K ) e. ( NN0 X. NN0 ) )
15	5, 14	syldan 280	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( R ` K ) e. ( NN0 X. NN0 ) )
16		fvres 5438	. . . 4 |- ( ( R ` K ) e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( R ` K ) ) = ( 2nd ` ( R ` K ) ) )
17	15, 16	syl 14	. . 3 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( R ` K ) ) = ( 2nd ` ( R ` K ) ) )
18		simpl 108	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> A e. ( NN0 X. NN0 ) )
19		fvres 5438	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( 2nd ` A ) )
20	19, 1	syl6eqr 2188	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = N )
21	20	fveq2d 5418	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) = ( ZZ>= ` N ) )
22	21	eleq2d 2207	. . . . 5 |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) <-> K e. ( ZZ>= ` N ) ) )
23	22	biimpar 295	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> K e. ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) )
24		f2ndres 6051	. . . . 5 |- ( 2nd |` ( NN0 X. NN0 ) ) : ( NN0 X. NN0 ) --> NN0
25	10	eucalglt 11727	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` z ) ) =/= 0 -> ( 2nd ` ( E ` z ) ) < ( 2nd ` z ) ) )
26	11	ffvelrni 5547	. . . . . . . 8 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
27		fvres 5438	. . . . . . . 8 |- ( ( E ` z ) e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( 2nd ` ( E ` z ) ) )
28	26, 27	syl 14	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( 2nd ` ( E ` z ) ) )
29	28	neeq1d 2324	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) =/= 0 <-> ( 2nd ` ( E ` z ) ) =/= 0 ) )
30		fvres 5438	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) = ( 2nd ` z ) )
31	28, 30	breq12d 3937	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) < ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) <-> ( 2nd ` ( E ` z ) ) < ( 2nd ` z ) ) )
32	25, 29, 31	3imtr4d 202	. . . . 5 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) =/= 0 -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) < ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) ) )
33		eqid 2137	. . . . 5 |- ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( ( 2nd |` ( NN0 X. NN0 ) ) ` A )
34	11, 7, 24, 32, 33	algcvga 11721	. . . 4 |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( R ` K ) ) = 0 ) )
35	18, 23, 34	sylc 62	. . 3 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( R ` K ) ) = 0 )
36	17, 35	eqtr3d 2172	. 2 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( 2nd ` ( R ` K ) ) = 0 )
37	36	ex 114	1 |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` N ) -> ( 2nd ` ( R ` K ) ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480   =/= wne 2306   ifcif 3469   {csn 3522   <.cop 3525   class class class wbr 3924   X. cxp 4532   |` cres 4536   o. ccom 4538   -->wf 5114   ` cfv 5118  (class class class)co 5767   e. cmpo 5769   1stc1st 6029   2ndc2nd 6030   0cc0 7613   < clt 7793   NN0cn0 8970   ZZ>=cuz 9319   mod cmo 10088   seqcseq 10211

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731  ax-arch 7732

This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-q 9405  df-rp 9435  df-fl 10036  df-mod 10089  df-seqfrec 10212

This theorem is referenced by:  eucalg  11729