Theorem eucalgcvga 11990

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 6134	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
3	1, 2	eqeltrid 2253	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> N e. NN0 )
4		eluznn0 9537	. . . . . 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 9500	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
7		eucalg.2	. . . . . . 7 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
8		0zd 9203	. . . . . . 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 11987	. . . . . . . 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 11977	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
14	13	ffvelrnda 5620	. . . . 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 5510	. . . 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 5510	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( 2nd ` A ) )
20	19, 1	eqtr4di 2217	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = N )
21	20	fveq2d 5490	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) = ( ZZ>= ` N ) )
22	21	eleq2d 2236	. . . . 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 6128	. . . . 5 |- ( 2nd |` ( NN0 X. NN0 ) ) : ( NN0 X. NN0 ) --> NN0
25	10	eucalglt 11989	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` z ) ) =/= 0 -> ( 2nd ` ( E ` z ) ) < ( 2nd ` z ) ) )
26	11	ffvelrni 5619	. . . . . . . 8 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
27		fvres 5510	. . . . . . . 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 2354	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) =/= 0 <-> ( 2nd ` ( E ` z ) ) =/= 0 ) )
30		fvres 5510	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) = ( 2nd ` z ) )
31	28, 30	breq12d 3995	. . . . . 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 2165	. . . . 5 |- ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( ( 2nd |` ( NN0 X. NN0 ) ) ` A )
34	11, 7, 24, 32, 33	algcvga 11983	. . . 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 2200	. 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 1343   e. wcel 2136   =/= wne 2336   ifcif 3520   {csn 3576   <.cop 3579   class class class wbr 3982   X. cxp 4602   |` cres 4606   o. ccom 4608   -->wf 5184   ` cfv 5188  (class class class)co 5842   e. cmpo 5844   1stc1st 6106   2ndc2nd 6107   0cc0 7753   < clt 7933   NN0cn0 9114   ZZ>=cuz 9466   mod cmo 10257   seqcseq 10380

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-fl 10205  df-mod 10258  df-seqfrec 10381

This theorem is referenced by:  eucalg  11991