Theorem eucalgcvga 12836

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 6400	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
3	1, 2	eqeltrid 2325	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> N e. NN0 )
4		eluznn0 9999	. . . . . 6 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) ) -> K e. NN0 )
5	3, 4	sylan 283	. . . . 5 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> K e. NN0 )
6		nn0uz 9957	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
7		eucalg.2	. . . . . . 7 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) )
8		0zd 9656	. . . . . . 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 12833	. . . . . . . 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 12823	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
14	13	ffvelcdmda 5843	. . . . 5 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. NN0 ) -> ( R ` K ) e. ( NN0 X. NN0 ) )
15	5, 14	syldan 282	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( R ` K ) e. ( NN0 X. NN0 ) )
16		fvres 5719	. . . 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 109	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> A e. ( NN0 X. NN0 ) )
19		fvres 5719	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( 2nd ` A ) )
20	19, 1	eqtr4di 2289	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = N )
21	20	fveq2d 5699	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) = ( ZZ>= ` N ) )
22	21	eleq2d 2308	. . . . 5 |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) <-> K e. ( ZZ>= ` N ) ) )
23	22	biimpar 297	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> K e. ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) )
24		f2ndres 6394	. . . . 5 |- ( 2nd |` ( NN0 X. NN0 ) ) : ( NN0 X. NN0 ) --> NN0
25	10	eucalglt 12835	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` z ) ) =/= 0 -> ( 2nd ` ( E ` z ) ) < ( 2nd ` z ) ) )
26	11	ffvelcdmi 5842	. . . . . . . 8 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
27		fvres 5719	. . . . . . . 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 2438	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) =/= 0 <-> ( 2nd ` ( E ` z ) ) =/= 0 ) )
30		fvres 5719	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) = ( 2nd ` z ) )
31	28, 30	breq12d 4143	. . . . . 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 203	. . . . 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 2238	. . . . 5 |- ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( ( 2nd |` ( NN0 X. NN0 ) ) ` A )
34	11, 7, 24, 32, 33	algcvga 12829	. . . 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 2273	. 2 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( 2nd ` ( R ` K ) ) = 0 )
37	36	ex 115	1 |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` N ) -> ( 2nd ` ( R ` K ) ) = 0 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   =/= wne 2420   ifcif 3638   {csn 3709   <.cop 3712   class class class wbr 4130   X. cxp 4772   |` cres 4776   o. ccom 4778   -->wf 5373   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   1stc1st 6372   2ndc2nd 6373   0cc0 8179   < clt 8360   NN0cn0 9563   ZZ>=cuz 9921   mod cmo 10759   seqcseq 10884

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-fl 10705  df-mod 10760  df-seqfrec 10885

This theorem is used by:  eucalg  12837