Theorem seq1lem1 6674

Description: We prove by induction that the first member of the ordered pair value of the internal sequence of seq1 equals its index.

Hypothesis

Ref	Expression
seq1lem1.1	|- G = (rec({<.x, y>. | y = (x + 1)}, 1) |` om)

Assertion

Ref	Expression
seq1lem1	|- (A e. NN -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` A)) = A)

Distinct variable groups:   x,y   z,w   w,B   x,C

Proof of Theorem seq1lem1

Step	Hyp	Ref	Expression
1		fveq2 3835	. . . 4 |- (v = 1 -> ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v) = ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` 1))
2	1	fveq2d 3839	. . 3 |- (v = 1 -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v)) = (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` 1)))
3		id 59	. . 3 |- (v = 1 -> v = 1)
4	2, 3	eqeq12d 1532	. 2 |- (v = 1 -> ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v)) = v <-> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` 1)) = 1))
5		fveq2 3835	. . . 4 |- (v = u -> ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v) = ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u))
6	5	fveq2d 3839	. . 3 |- (v = u -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v)) = (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)))
7		id 59	. . 3 |- (v = u -> v = u)
8	6, 7	eqeq12d 1532	. 2 |- (v = u -> ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v)) = v <-> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) = u))
9		fveq2 3835	. . . 4 |- (v = (u + 1) -> ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v) = ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1)))
10	9	fveq2d 3839	. . 3 |- (v = (u + 1) -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v)) = (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1))))
11		id 59	. . 3 |- (v = (u + 1) -> v = (u + 1))
12	10, 11	eqeq12d 1532	. 2 |- (v = (u + 1) -> ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v)) = v <-> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1))) = (u + 1)))
13		fveq2 3835	. . . 4 |- (v = A -> ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v) = ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` A))
14	13	fveq2d 3839	. . 3 |- (v = A -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v)) = (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` A)))
15		id 59	. . 3 |- (v = A -> v = A)
16	14, 15	eqeq12d 1532	. 2 |- (v = A -> ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` v)) = v <-> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` A)) = A))
17		opex 2858	. . . . 5 |- <.1, C>. e. V
18		1z 6327	. . . . . 6 |- 1 e. ZZ
19		seq1lem1.1	. . . . . 6 |- G = (rec({<.x, y>. | y = (x + 1)}, 1) |` om)
20	18, 19	uzrdginii 6667	. . . . 5 |- (<.1, C>. e. V -> ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` 1) = <.1, C>.)
21	17, 20	ax-mp 7	. . . 4 |- ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` 1) = <.1, C>.
22	21	fveq2i 3838	. . 3 |- (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` 1)) = (1st` <.1, C>.)
23	18	elisseti 1864	. . . 4 |- 1 e. V
24	23	op1st 4146	. . 3 |- (1st` <.1, C>.) = 1
25	22, 24	eqtri 1538	. 2 |- (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` 1)) = 1
26		nnzrab 6325	. . . . . . . . 9 |- NN = {v e. ZZ | 1 <_ v}
27	26	eleq2i 1581	. . . . . . . 8 |- (u e. NN <-> u e. {v e. ZZ | 1 <_ v})
28	18, 19	uzrdgsuci 6668	. . . . . . . 8 |- (u e. {v e. ZZ | 1 <_ v} -> ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1)) = ({<.z, w>. | w = <.((1st` z) + 1), B>.}` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)))
29	27, 28	sylbi 197	. . . . . . 7 |- (u e. NN -> ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1)) = ({<.z, w>. | w = <.((1st` z) + 1), B>.}` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)))
30		ax-17 1007	. . . . . . . . . 10 |- (w = <.((1st` z) + 1), B>. -> A.v w = <.((1st` z) + 1), B>.)
31		ax-17 1007	. . . . . . . . . . . 12 |- (w e. ((1st` v) + 1) -> A.z w e. ((1st` v) + 1))
32		visset 1859	. . . . . . . . . . . . 13 |- v e. V
33		ax-17 1007	. . . . . . . . . . . . 13 |- (w e. v -> A.z w e. v)
34	32, 33	hbcsb1 2076	. . . . . . . . . . . 12 |- (w e. [_v / z]_B -> A.z w e. [_v / z]_B)
35	31, 34	hbop 2561	. . . . . . . . . . 11 |- (w e. <.((1st` v) + 1), [_v / z]_B>. -> A.z w e. <.((1st` v) + 1), [_v / z]_B>.)
36	35	hbeleq 1610	. . . . . . . . . 10 |- (w = <.((1st` v) + 1), [_v / z]_B>. -> A.z w = <.((1st` v) + 1), [_v / z]_B>.)
37		fveq2 3835	. . . . . . . . . . . . 13 |- (z = v -> (1st` z) = (1st` v))
38	37	opreq1d 4033	. . . . . . . . . . . 12 |- (z = v -> ((1st` z) + 1) = ((1st` v) + 1))
39		csbeq1a 2057	. . . . . . . . . . . 12 |- (z = v -> B = [_v / z]_B)
40	38, 39	opeq12d 2560	. . . . . . . . . . 11 |- (z = v -> <.((1st` z) + 1), B>. = <.((1st` v) + 1), [_v / z]_B>.)
41	40	eqeq2d 1529	. . . . . . . . . 10 |- (z = v -> (w = <.((1st` z) + 1), B>. <-> w = <.((1st` v) + 1), [_v / z]_B>.))
42	30, 36, 41	cbvopab1 2748	. . . . . . . . 9 |- {<.z, w>. | w = <.((1st` z) + 1), B>.} = {<.v, w>. | w = <.((1st` v) + 1), [_v / z]_B>.}
43	42	fveq1i 3836	. . . . . . . 8 |- ({<.z, w>. | w = <.((1st` z) + 1), B>.}` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) = ({<.v, w>. | w = <.((1st` v) + 1), [_v / z]_B>.}` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u))
44		ax-17 1007	. . . . . . . . 9 |- (f e. ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) -> A.v f e. ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u))
45		ax-17 1007	. . . . . . . . 9 |- (f e. <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>. -> A.v f e. <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>.)
46		fvex 3843	. . . . . . . . 9 |- ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) e. V
47		opex 2858	. . . . . . . . 9 |- <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>. e. V
48		fveq2 3835	. . . . . . . . . . 11 |- (v = ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) -> (1st` v) = (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)))
49	48	opreq1d 4033	. . . . . . . . . 10 |- (v = ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) -> ((1st` v) + 1) = ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1))
50		csbeq1 2053	. . . . . . . . . 10 |- (v = ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) -> [_v / z]_B = [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B)
51	49, 50	opeq12d 2560	. . . . . . . . 9 |- (v = ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) -> <.((1st` v) + 1), [_v / z]_B>. = <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>.)
52	44, 45, 46, 47, 51	fvopabf 3900	. . . . . . . 8 |- ({<.v, w>. | w = <.((1st` v) + 1), [_v / z]_B>.}` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) = <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>.
53	43, 52	eqtri 1538	. . . . . . 7 |- ({<.z, w>. | w = <.((1st` z) + 1), B>.}` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) = <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>.
54	29, 53	syl6eq 1566	. . . . . 6 |- (u e. NN -> ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1)) = <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>.)
55	54	fveq2d 3839	. . . . 5 |- (u e. NN -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1))) = (1st` <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>.))
56		oprex 4041	. . . . . 6 |- ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1) e. V
57	56	op1st 4146	. . . . 5 |- (1st` <.((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1), [_((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u) / z]_B>.) = ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1)
58	55, 57	syl6eq 1566	. . . 4 |- (u e. NN -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1))) = ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1))
59		opreq1 4026	. . . 4 |- ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) = u -> ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) + 1) = (u + 1))
60	58, 59	sylan9eq 1570	. . 3 |- ((u e. NN /\ (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) = u) -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1))) = (u + 1))
61	60	ex 371	. 2 |- (u e. NN -> ((1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` u)) = u -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` (u + 1))) = (u + 1)))
62	4, 8, 12, 16, 25, 61	nnind 6082	1 |- (A e. NN -> (1st` ((rec({<.z, w>. | w = <.((1st` z) + 1), B>.}, <.1, C>.) o. `'G)` A)) = A)

Syntax hints:   -> wi 3   = wceq 992   e. wcel 994  {crab 1694  Vcvv 1857  [_csb 2051  <.cop 2469   class class class wbr 2692  {copab 2740  omcom 3218  `'ccnv 3250   |` cres 3253   o. ccom 3255  ` cfv 3263  (class class class)co 4021  1stc1st 4138  reccrdg 4232  1c1 5389   + caddc 5391   <_ cle 5449  NNcn 5450  ZZcz 5452

This theorem is referenced by:  seq1suclem 6681

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-nul 2784  ax-pow 2818  ax-pr 2855  ax-un 3089  ax-inf2 4770

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 782  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-reu 1697  df-rab 1698  df-v 1858  df-sbc 1987  df-csb 2052  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-pss 2107  df-nul 2333  df-if 2416  df-pw 2459  df-sn 2470  df-pr 2471  df-tp 2473  df-op 2474  df-uni 2570  df-int 2601  df-iun 2635  df-br 2693  df-opab 2741  df-tr 2755  df-eprel 2910  df-id 2913  df-po 2918  df-so 2929  df-fr 2947  df-we 2962  df-ord 2978  df-on 2979  df-lim 2980  df-suc 2981  df-om 3219  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-fv 3279  df-opr 4023  df-oprab 4024  df-1st 4140  df-2nd 4141  df-rdg 4233  df-1o 4269  df-oadd 4271  df-omul 4272  df-er 4401  df-ec 4403  df-qs 4406  df-en 4509  df-dom 4510  df-sdom 4511  df-ni 5154  df-pli 5155  df-mi 5156  df-lti 5157  df-plpq 5189  df-mpq 5190  df-enq 5191  df-nq 5192  df-plq 5193  df-mq 5194  df-rq 5195  df-ltq 5196  df-1q 5197  df-np 5240  df-1p 5241  df-plp 5242  df-mp 5243  df-ltp 5244  df-plpr 5318  df-mpr 5319  df-enr 5320  df-nr 5321  df-plr 5322  df-mr 5323  df-ltr 5324  df-0r 5325  df-1r 5326  df-m1r 5327  df-c 5394  df-0 5395  df-1 5396  df-i 5397  df-r 5398  df-plus 5399  df-mul 5400  df-lt 5401  df-sub 5510  df-neg 5512  df-pnf 5641  df-mnf 5642  df-xr 5643  df-ltxr 5644  df-le 5645  df-n 6070  df-n0 6268  df-z 6304

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