Theorem eucalgcvga 13754

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

Hypotheses

Ref	Expression
eucalg.1	|- S = (NN0 X. NN0)
eucalg.2	|- E = {<.x, y>. | (x e. (NN0 X. NN0) /\ y = if((2nd` x) = 0, x, <.(2nd` x), ( mod ` x)>.))}
eucalg.3	|- R = ((E o. (1st |` (S X. S))) seq0 (NN0 X. {A}))
eucalgcvga.4	|- N = (2nd` A)

Assertion

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

Distinct variable group:   x,y

Proof of Theorem eucalgcvga

Step	Hyp	Ref	Expression
1		simpl 346	. . . . 5 |- ((A e. (NN0 X. NN0) /\ K e. (ZZ>=` N)) -> A e. (NN0 X. NN0))
2		eluznn0 13661	. . . . . 6 |- ((N e. NN0 /\ K e. (ZZ>=` N)) -> K e. NN0)
3		xp2nd 10156	. . . . . . 7 |- (A e. (NN0 X. NN0) -> (2nd` A) e. NN0)
4		eucalgcvga.4	. . . . . . 7 |- N = (2nd` A)
5	3, 4	syl5eqel 1975	. . . . . 6 |- (A e. (NN0 X. NN0) -> N e. NN0)
6	2, 5	sylan 497	. . . . 5 |- ((A e. (NN0 X. NN0) /\ K e. (ZZ>=` N)) -> K e. NN0)
7		ffvelrn 4787	. . . . . . 7 |- ((R:NN0-->(NN0 X. NN0) /\ K e. NN0) -> (R` K) e. (NN0 X. NN0))
8		nn0ex 7314	. . . . . . . . 9 |- NN0 e. _V
9	8, 8	xpex 4096	. . . . . . . 8 |- (NN0 X. NN0) e. _V
10		eucalg.2	. . . . . . . . 9 |- E = {<.x, y>. | (x e. (NN0 X. NN0) /\ y = if((2nd` x) = 0, x, <.(2nd` x), ( mod ` x)>.))}
11	10	eucalgf 13751	. . . . . . . 8 |- E:(NN0 X. NN0)-->(NN0 X. NN0)
12		eucalg.3	. . . . . . . . 9 |- R = ((E o. (1st |` (S X. S))) seq0 (NN0 X. {A}))
13		eucalg.1	. . . . . . . . . . . . 13 |- S = (NN0 X. NN0)
14	13, 13	xpeq12i 4023	. . . . . . . . . . . 12 |- (S X. S) = ((NN0 X. NN0) X. (NN0 X. NN0))
15	14	reseq2i 13591	. . . . . . . . . . 11 |- (1st |` (S X. S)) = (1st |` ((NN0 X. NN0) X. (NN0 X. NN0)))
16	15	coeq2i 4126	. . . . . . . . . 10 |- (E o. (1st |` (S X. S))) = (E o. (1st |` ((NN0 X. NN0) X. (NN0 X. NN0))))
17	16	opreq1i 4892	. . . . . . . . 9 |- ((E o. (1st |` (S X. S))) seq0 (NN0 X. {A})) = ((E o. (1st |` ((NN0 X. NN0) X. (NN0 X. NN0)))) seq0 (NN0 X. {A}))
18	12, 17	eqtri 1908	. . . . . . . 8 |- R = ((E o. (1st |` ((NN0 X. NN0) X. (NN0 X. NN0)))) seq0 (NN0 X. {A}))
19	9, 11, 18	algrf 13739	. . . . . . 7 |- (A e. (NN0 X. NN0) -> R:NN0-->(NN0 X. NN0))
20	7, 19	sylan 497	. . . . . 6 |- ((A e. (NN0 X. NN0) /\ K e. NN0) -> (R` K) e. (NN0 X. NN0))
21		fvres 4691	. . . . . 6 |- ((R` K) e. (NN0 X. NN0) -> ((2nd |` (NN0 X. NN0))` (R` K)) = (2nd` (R` K)))
22	20, 21	syl 12	. . . . 5 |- ((A e. (NN0 X. NN0) /\ K e. NN0) -> ((2nd |` (NN0 X. NN0))` (R` K)) = (2nd` (R` K)))
23	1, 6, 22	syl11anc 524	. . . 4 |- ((A e. (NN0 X. NN0) /\ K e. (ZZ>=` N)) -> ((2nd |` (NN0 X. NN0))` (R` K)) = (2nd` (R` K)))
24		fvres 4691	. . . . . . . . 9 |- (A e. (NN0 X. NN0) -> ((2nd |` (NN0 X. NN0))` A) = (2nd` A))
25	24, 4	syl6eqr 1946	. . . . . . . 8 |- (A e. (NN0 X. NN0) -> ((2nd |` (NN0 X. NN0))` A) = N)
26	25	fveq2d 4685	. . . . . . 7 |- (A e. (NN0 X. NN0) -> (ZZ>=` ((2nd |` (NN0 X. NN0))` A)) = (ZZ>=` N))
27	26	eleq2d 1964	. . . . . 6 |- (A e. (NN0 X. NN0) -> (K e. (ZZ>=` ((2nd |` (NN0 X. NN0))` A)) <-> K e. (ZZ>=` N)))
28		f2ndres 5035	. . . . . . 7 |- (2nd |` (NN0 X. NN0)):(NN0 X. NN0)-->NN0
29	10	eucalglt 13753	. . . . . . . 8 |- (z e. (NN0 X. NN0) -> ((2nd` (E` z)) =/= 0 -> (2nd` (E` z)) < (2nd` z)))
30	11	ffvelrni 4788	. . . . . . . . . . 11 |- (z e. (NN0 X. NN0) -> (E` z) e. (NN0 X. NN0))
31		fvres 4691	. . . . . . . . . . 11 |- ((E` z) e. (NN0 X. NN0) -> ((2nd |` (NN0 X. NN0))` (E` z)) = (2nd` (E` z)))
32	30, 31	syl 12	. . . . . . . . . 10 |- (z e. (NN0 X. NN0) -> ((2nd |` (NN0 X. NN0))` (E` z)) = (2nd` (E` z)))
33	32	neeq1d 2028	. . . . . . . . 9 |- (z e. (NN0 X. NN0) -> (((2nd |` (NN0 X. NN0))` (E` z)) =/= 0 <-> (2nd` (E` z)) =/= 0))
34		fvres 4691	. . . . . . . . . 10 |- (z e. (NN0 X. NN0) -> ((2nd |` (NN0 X. NN0))` z) = (2nd` z))
35	32, 34	breq12d 3351	. . . . . . . . 9 |- (z e. (NN0 X. NN0) -> (((2nd |` (NN0 X. NN0))` (E` z)) < ((2nd |` (NN0 X. NN0))` z) <-> (2nd` (E` z)) < (2nd` z)))
36	33, 35	imbi12d 688	. . . . . . . 8 |- (z e. (NN0 X. NN0) -> ((((2nd |` (NN0 X. NN0))` (E` z)) =/= 0 -> ((2nd |` (NN0 X. NN0))` (E` z)) < ((2nd |` (NN0 X. NN0))` z)) <-> ((2nd` (E` z)) =/= 0 -> (2nd` (E` z)) < (2nd` z))))
37	29, 36	mpbird 213	. . . . . . 7 |- (z e. (NN0 X. NN0) -> (((2nd |` (NN0 X. NN0))` (E` z)) =/= 0 -> ((2nd |` (NN0 X. NN0))` (E` z)) < ((2nd |` (NN0 X. NN0))` z)))
38		eqid 1884	. . . . . . 7 |- ((2nd |` (NN0 X. NN0))` A) = ((2nd |` (NN0 X. NN0))` A)
39	9, 11, 18, 28, 37, 38	algcvga 13747	. . . . . 6 |- (A e. (NN0 X. NN0) -> (K e. (ZZ>=` ((2nd |` (NN0 X. NN0))` A)) -> ((2nd |` (NN0 X. NN0))` (R` K)) = 0))
40	27, 39	sylbird 222	. . . . 5 |- (A e. (NN0 X. NN0) -> (K e. (ZZ>=` N) -> ((2nd |` (NN0 X. NN0))` (R` K)) = 0))
41	40	imp 377	. . . 4 |- ((A e. (NN0 X. NN0) /\ K e. (ZZ>=` N)) -> ((2nd |` (NN0 X. NN0))` (R` K)) = 0)
42	23, 41	eqtr3d 1927	. . 3 |- ((A e. (NN0 X. NN0) /\ K e. (ZZ>=` N)) -> (2nd` (R` K)) = 0)
43	13	eleq2i 1961	. . 3 |- (A e. S <-> A e. (NN0 X. NN0))
44	42, 43	sylanb 498	. 2 |- ((A e. S /\ K e. (ZZ>=` N)) -> (2nd` (R` K)) = 0)
45	44	ex 402	1 |- (A e. S -> (K e. (ZZ>=` N) -> (2nd` (R` K)) = 0))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300   =/= wne 2017  ifcif 2982  {csn 3044  <.cop 3046   class class class wbr 3338  {copab 3395   X. cxp 3984   |` cres 3988   o. ccom 3990  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  2ndc2nd 5019  0cc0 6386  NN0cn0 6450   < clt 6653   mod cmo 7499  ZZ>=cuz 7586   seq0 cseq0 7775

This theorem is referenced by:  eucalg 13755  mulgcdlem4 13759

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-rp 7232  df-n0 7309  df-z 7345  df-fl 7463  df-mod 7500  df-uz 7587  df-seq1 7721  df-shft 7754  df-seqz 7776  df-seq0 7777

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