Theorem eucialgcvga 10440

Description: Once Euclid's Algorithm halts after N steps, the second element of the state remains 0 . (Contributed by Jim Kingdon, 11-Jan-2022.)

Hypotheses

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

Assertion

Ref	Expression
eucialgcvga	|- ( 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 eucialgcvga

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eucialgcvga.3	. . . . . . 7 |- N = ( 2nd ` A )
2		xp2nd 5813	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( 2nd ` A ) e. NN0 )
3	1, 2	syl5eqel 2165	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> N e. NN0 )
4		eluznn0 8686	. . . . . 6 |- ( ( N e. NN0 /\ K e. ( ZZ>= ` N ) ) -> K e. NN0 )
5	3, 4	sylan 277	. . . . 5 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> K e. NN0 )
6		nn0uz 8653	. . . . . . 7 |- NN0 = ( ZZ>= ` 0 )
7		eucialg.2	. . . . . . 7 |- R = seq 0 ( ( E o. 1st ) , ( NN0 X. { A } ) , ( NN0 X. NN0 ) )
8		0zd 8363	. . . . . . 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 10437	. . . . . . . 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		nn0ex 8294	. . . . . . . . 9 |- NN0 e. _V
14	13, 13	xpex 4471	. . . . . . . 8 |- ( NN0 X. NN0 ) e. _V
15	14	a1i 9	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( NN0 X. NN0 ) e. _V )
16	6, 7, 8, 9, 12, 15	ialgrf 10427	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> R : NN0 --> ( NN0 X. NN0 ) )
17	16	ffvelrnda 5323	. . . . 5 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. NN0 ) -> ( R ` K ) e. ( NN0 X. NN0 ) )
18	5, 17	syldan 276	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( R ` K ) e. ( NN0 X. NN0 ) )
19		fvres 5219	. . . 4 |- ( ( R ` K ) e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( R ` K ) ) = ( 2nd ` ( R ` K ) ) )
20	18, 19	syl 14	. . 3 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( R ` K ) ) = ( 2nd ` ( R ` K ) ) )
21		simpl 107	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> A e. ( NN0 X. NN0 ) )
22		fvres 5219	. . . . . . . 8 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( 2nd ` A ) )
23	22, 1	syl6eqr 2131	. . . . . . 7 |- ( A e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = N )
24	23	fveq2d 5202	. . . . . 6 |- ( A e. ( NN0 X. NN0 ) -> ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) = ( ZZ>= ` N ) )
25	24	eleq2d 2148	. . . . 5 |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) <-> K e. ( ZZ>= ` N ) ) )
26	25	biimpar 291	. . . 4 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> K e. ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) )
27		f2ndres 5807	. . . . 5 |- ( 2nd |` ( NN0 X. NN0 ) ) : ( NN0 X. NN0 ) --> NN0
28	10	eucalglt 10439	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd ` ( E ` z ) ) =/= 0 -> ( 2nd ` ( E ` z ) ) < ( 2nd ` z ) ) )
29	11	ffvelrni 5322	. . . . . . . 8 |- ( z e. ( NN0 X. NN0 ) -> ( E ` z ) e. ( NN0 X. NN0 ) )
30		fvres 5219	. . . . . . . 8 |- ( ( E ` z ) e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( 2nd ` ( E ` z ) ) )
31	29, 30	syl 14	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) = ( 2nd ` ( E ` z ) ) )
32	31	neeq1d 2263	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) =/= 0 <-> ( 2nd ` ( E ` z ) ) =/= 0 ) )
33		fvres 5219	. . . . . . 7 |- ( z e. ( NN0 X. NN0 ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) = ( 2nd ` z ) )
34	31, 33	breq12d 3798	. . . . . 6 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) < ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) <-> ( 2nd ` ( E ` z ) ) < ( 2nd ` z ) ) )
35	28, 32, 34	3imtr4d 201	. . . . 5 |- ( z e. ( NN0 X. NN0 ) -> ( ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) =/= 0 -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( E ` z ) ) < ( ( 2nd |` ( NN0 X. NN0 ) ) ` z ) ) )
36		eqid 2081	. . . . 5 |- ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) = ( ( 2nd |` ( NN0 X. NN0 ) ) ` A )
37	11, 7, 27, 35, 36, 14	ialgcvga 10433	. . . 4 |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` ( ( 2nd |` ( NN0 X. NN0 ) ) ` A ) ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( R ` K ) ) = 0 ) )
38	21, 26, 37	sylc 61	. . 3 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( ( 2nd |` ( NN0 X. NN0 ) ) ` ( R ` K ) ) = 0 )
39	20, 38	eqtr3d 2115	. 2 |- ( ( A e. ( NN0 X. NN0 ) /\ K e. ( ZZ>= ` N ) ) -> ( 2nd ` ( R ` K ) ) = 0 )
40	39	ex 113	1 |- ( A e. ( NN0 X. NN0 ) -> ( K e. ( ZZ>= ` N ) -> ( 2nd ` ( R ` K ) ) = 0 ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   =/= wne 2245   _Vcvv 2601   ifcif 3351   {csn 3398   <.cop 3401   class class class wbr 3785   X. cxp 4361   |` cres 4365   o. ccom 4367   -->wf 4918   ` cfv 4922  (class class class)co 5532   |-> cmpt2 5534   1stc1st 5785   2ndc2nd 5786   0cc0 6981   < clt 7153   NN0cn0 8288   ZZ>=cuz 8619   mod cmo 9324   seqcseq 9431

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-n0 8289  df-z 8352  df-uz 8620  df-q 8705  df-rp 8735  df-fl 9274  df-mod 9325  df-iseq 9432

This theorem is referenced by:  eucialg  10441