Theorem eucalgcvga 11742

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 6064	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
3	1, 2	eqeltrid 2226	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> N e. NN0 )
4		eluznn0 9396	. . . . . 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 9363	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
7		eucalg.2	. . . . . . 7 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
8		0zd 9069	. . . . . . 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 11739	. . . . . . . 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 11729	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
14	13	ffvelrnda 5555	. . . . 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 5445	. . . 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 5445	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( 2nd ` A ) )
20	19, 1	syl6eqr 2190	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = N )
21	20	fveq2d 5425	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) = ( ZZ>= ` N ) )
22	21	eleq2d 2209	. . . . 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 6058	. . . . 5 |- ( 2nd |` ( NN0 X. NN0 ) ) : ( NN0 X. NN0 ) --> NN0
25	10	eucalglt 11741	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` z ) ) =/= 0 -> ( 2nd ` ( E ` z ) ) < ( 2nd ` z ) ) )
26	11	ffvelrni 5554	. . . . . . . 8 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
27		fvres 5445	. . . . . . . 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 2326	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) =/= 0 <-> ( 2nd ` ( E ` z ) ) =/= 0 ) )
30		fvres 5445	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) = ( 2nd ` z ) )
31	28, 30	breq12d 3942	. . . . . 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 2139	. . . . 5 |- ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( ( 2nd |` ( NN0 X. NN0 ) ) ` A )
34	11, 7, 24, 32, 33	algcvga 11735	. . . 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 2174	. 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 2308   ifcif 3474   {csn 3527   <.cop 3530   class class class wbr 3929   X. cxp 4537   |` cres 4541   o. ccom 4543   -->wf 5119   ` cfv 5123  (class class class)co 5774   e. cmpo 5776   1stc1st 6036   2ndc2nd 6037   0cc0 7623   < clt 7803   NN0cn0 8980   ZZ>=cuz 9329   mod cmo 10098   seqcseq 10221

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 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7714  ax-resscn 7715  ax-1cn 7716  ax-1re 7717  ax-icn 7718  ax-addcl 7719  ax-addrcl 7720  ax-mulcl 7721  ax-mulrcl 7722  ax-addcom 7723  ax-mulcom 7724  ax-addass 7725  ax-mulass 7726  ax-distr 7727  ax-i2m1 7728  ax-0lt1 7729  ax-1rid 7730  ax-0id 7731  ax-rnegex 7732  ax-precex 7733  ax-cnre 7734  ax-pre-ltirr 7735  ax-pre-ltwlin 7736  ax-pre-lttrn 7737  ax-pre-apti 7738  ax-pre-ltadd 7739  ax-pre-mulgt0 7740  ax-pre-mulext 7741  ax-arch 7742

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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7805  df-mnf 7806  df-xr 7807  df-ltxr 7808  df-le 7809  df-sub 7938  df-neg 7939  df-reap 8340  df-ap 8347  df-div 8436  df-inn 8724  df-n0 8981  df-z 9058  df-uz 9330  df-q 9415  df-rp 9445  df-fl 10046  df-mod 10099  df-seqfrec 10222

This theorem is referenced by:  eucalg  11743