Theorem seq1eq2 15817

Description: Equality of series based on equality of finitely many terms.

Hypotheses

Ref	Expression
seq1eq2.1	|- S e. _V
seq1eq2.2	|- F e. _V
seq1eq2.3	|- G e. _V

Assertion

Ref	Expression
seq1eq2	|- ((N e. NN /\ A.k e. (1...N)(F` k) = (G` k)) -> ((S seq1 F)` N) = ((S seq1 G)` N))

Distinct variable groups:   k,N   k,F   k,G

Proof of Theorem seq1eq2

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (l = 1 -> (1...l) = (1...1))
2	1	raleqdv 2269	. . . 4 |- (l = 1 -> (A.k e. (1...l)(F` k) = (G` k) <-> A.k e. (1...1)(F` k) = (G` k)))
3		fveq2 4681	. . . . 5 |- (l = 1 -> ((S seq1 F)` l) = ((S seq1 F)` 1))
4		fveq2 4681	. . . . 5 |- (l = 1 -> ((S seq1 G)` l) = ((S seq1 G)` 1))
5	3, 4	eqeq12d 1899	. . . 4 |- (l = 1 -> (((S seq1 F)` l) = ((S seq1 G)` l) <-> ((S seq1 F)` 1) = ((S seq1 G)` 1)))
6	2, 5	imbi12d 688	. . 3 |- (l = 1 -> ((A.k e. (1...l)(F` k) = (G` k) -> ((S seq1 F)` l) = ((S seq1 G)` l)) <-> (A.k e. (1...1)(F` k) = (G` k) -> ((S seq1 F)` 1) = ((S seq1 G)` 1))))
7		opreq2 4890	. . . . 5 |- (l = j -> (1...l) = (1...j))
8	7	raleqdv 2269	. . . 4 |- (l = j -> (A.k e. (1...l)(F` k) = (G` k) <-> A.k e. (1...j)(F` k) = (G` k)))
9		fveq2 4681	. . . . 5 |- (l = j -> ((S seq1 F)` l) = ((S seq1 F)` j))
10		fveq2 4681	. . . . 5 |- (l = j -> ((S seq1 G)` l) = ((S seq1 G)` j))
11	9, 10	eqeq12d 1899	. . . 4 |- (l = j -> (((S seq1 F)` l) = ((S seq1 G)` l) <-> ((S seq1 F)` j) = ((S seq1 G)` j)))
12	8, 11	imbi12d 688	. . 3 |- (l = j -> ((A.k e. (1...l)(F` k) = (G` k) -> ((S seq1 F)` l) = ((S seq1 G)` l)) <-> (A.k e. (1...j)(F` k) = (G` k) -> ((S seq1 F)` j) = ((S seq1 G)` j))))
13		opreq2 4890	. . . . 5 |- (l = (j + 1) -> (1...l) = (1...(j + 1)))
14	13	raleqdv 2269	. . . 4 |- (l = (j + 1) -> (A.k e. (1...l)(F` k) = (G` k) <-> A.k e. (1...(j + 1))(F` k) = (G` k)))
15		fveq2 4681	. . . . 5 |- (l = (j + 1) -> ((S seq1 F)` l) = ((S seq1 F)` (j + 1)))
16		fveq2 4681	. . . . 5 |- (l = (j + 1) -> ((S seq1 G)` l) = ((S seq1 G)` (j + 1)))
17	15, 16	eqeq12d 1899	. . . 4 |- (l = (j + 1) -> (((S seq1 F)` l) = ((S seq1 G)` l) <-> ((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1))))
18	14, 17	imbi12d 688	. . 3 |- (l = (j + 1) -> ((A.k e. (1...l)(F` k) = (G` k) -> ((S seq1 F)` l) = ((S seq1 G)` l)) <-> (A.k e. (1...(j + 1))(F` k) = (G` k) -> ((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1)))))
19		opreq2 4890	. . . . 5 |- (l = N -> (1...l) = (1...N))
20	19	raleqdv 2269	. . . 4 |- (l = N -> (A.k e. (1...l)(F` k) = (G` k) <-> A.k e. (1...N)(F` k) = (G` k)))
21		fveq2 4681	. . . . 5 |- (l = N -> ((S seq1 F)` l) = ((S seq1 F)` N))
22		fveq2 4681	. . . . 5 |- (l = N -> ((S seq1 G)` l) = ((S seq1 G)` N))
23	21, 22	eqeq12d 1899	. . . 4 |- (l = N -> (((S seq1 F)` l) = ((S seq1 G)` l) <-> ((S seq1 F)` N) = ((S seq1 G)` N)))
24	20, 23	imbi12d 688	. . 3 |- (l = N -> ((A.k e. (1...l)(F` k) = (G` k) -> ((S seq1 F)` l) = ((S seq1 G)` l)) <-> (A.k e. (1...N)(F` k) = (G` k) -> ((S seq1 F)` N) = ((S seq1 G)` N))))
25		1z 7368	. . . . . 6 |- 1 e. ZZ
26		fz1sbc 7696	. . . . . 6 |- (1 e. ZZ -> (A.k e. (1...1)(F` k) = (G` k) <-> [1 / k](F` k) = (G` k)))
27	25, 26	ax-mp 7	. . . . 5 |- (A.k e. (1...1)(F` k) = (G` k) <-> [1 / k](F` k) = (G` k))
28	25	elisseti 2301	. . . . . 6 |- 1 e. _V
29		fveq2 4681	. . . . . . 7 |- (k = 1 -> (F` k) = (F` 1))
30		fveq2 4681	. . . . . . 7 |- (k = 1 -> (G` k) = (G` 1))
31	29, 30	eqeq12d 1899	. . . . . 6 |- (k = 1 -> ((F` k) = (G` k) <-> (F` 1) = (G` 1)))
32	28, 31	sbcie 2485	. . . . 5 |- ([1 / k](F` k) = (G` k) <-> (F` 1) = (G` 1))
33	27, 32	bitri 190	. . . 4 |- (A.k e. (1...1)(F` k) = (G` k) <-> (F` 1) = (G` 1))
34		seq1eq2.1	. . . . . . 7 |- S e. _V
35		seq1eq2.2	. . . . . . 7 |- F e. _V
36	34, 35	seq11 7730	. . . . . 6 |- ((S seq1 F)` 1) = (F` 1)
37		seq1eq2.3	. . . . . . 7 |- G e. _V
38	34, 37	seq11 7730	. . . . . 6 |- ((S seq1 G)` 1) = (G` 1)
39	36, 38	eqeq12i 1897	. . . . 5 |- (((S seq1 F)` 1) = ((S seq1 G)` 1) <-> (F` 1) = (G` 1))
40	39	biimpri 169	. . . 4 |- ((F` 1) = (G` 1) -> ((S seq1 F)` 1) = ((S seq1 G)` 1))
41	33, 40	sylbi 216	. . 3 |- (A.k e. (1...1)(F` k) = (G` k) -> ((S seq1 F)` 1) = ((S seq1 G)` 1))
42		fzssp1 7679	. . . . . . 7 |- ((1 e. ZZ /\ j e. ZZ) -> (1...j) C_ (1...(j + 1)))
43		nnz 7362	. . . . . . 7 |- (j e. NN -> j e. ZZ)
44	42, 25, 43	sylancr 526	. . . . . 6 |- (j e. NN -> (1...j) C_ (1...(j + 1)))
45		ssralv 2672	. . . . . 6 |- ((1...j) C_ (1...(j + 1)) -> (A.k e. (1...(j + 1))(F` k) = (G` k) -> A.k e. (1...j)(F` k) = (G` k)))
46	44, 45	syl 12	. . . . 5 |- (j e. NN -> (A.k e. (1...(j + 1))(F` k) = (G` k) -> A.k e. (1...j)(F` k) = (G` k)))
47	46	imim1d 33	. . . 4 |- (j e. NN -> ((A.k e. (1...j)(F` k) = (G` k) -> ((S seq1 F)` j) = ((S seq1 G)` j)) -> (A.k e. (1...(j + 1))(F` k) = (G` k) -> ((S seq1 F)` j) = ((S seq1 G)` j))))
48		peano2nn 7118	. . . . . . 7 |- (j e. NN -> (j + 1) e. NN)
49		elnnuz 7609	. . . . . . . 8 |- ((j + 1) e. NN <-> (j + 1) e. (ZZ>=` 1))
50		eluzfz2 7659	. . . . . . . 8 |- ((j + 1) e. (ZZ>=` 1) -> (j + 1) e. (1...(j + 1)))
51	49, 50	sylbi 216	. . . . . . 7 |- ((j + 1) e. NN -> (j + 1) e. (1...(j + 1)))
52		fveq2 4681	. . . . . . . . 9 |- (k = (j + 1) -> (F` k) = (F` (j + 1)))
53		fveq2 4681	. . . . . . . . 9 |- (k = (j + 1) -> (G` k) = (G` (j + 1)))
54	52, 53	eqeq12d 1899	. . . . . . . 8 |- (k = (j + 1) -> ((F` k) = (G` k) <-> (F` (j + 1)) = (G` (j + 1))))
55	54	rcla4v 2376	. . . . . . 7 |- ((j + 1) e. (1...(j + 1)) -> (A.k e. (1...(j + 1))(F` k) = (G` k) -> (F` (j + 1)) = (G` (j + 1))))
56	48, 51, 55	3syl 24	. . . . . 6 |- (j e. NN -> (A.k e. (1...(j + 1))(F` k) = (G` k) -> (F` (j + 1)) = (G` (j + 1))))
57	34, 35	seq1p1 7731	. . . . . . . . . 10 |- (j e. NN -> ((S seq1 F)` (j + 1)) = (((S seq1 F)` j)S(F` (j + 1))))
58	34, 37	seq1p1 7731	. . . . . . . . . 10 |- (j e. NN -> ((S seq1 G)` (j + 1)) = (((S seq1 G)` j)S(G` (j + 1))))
59	57, 58	eqeq12d 1899	. . . . . . . . 9 |- (j e. NN -> (((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1)) <-> (((S seq1 F)` j)S(F` (j + 1))) = (((S seq1 G)` j)S(G` (j + 1)))))
60		opreq12 4891	. . . . . . . . 9 |- ((((S seq1 F)` j) = ((S seq1 G)` j) /\ (F` (j + 1)) = (G` (j + 1))) -> (((S seq1 F)` j)S(F` (j + 1))) = (((S seq1 G)` j)S(G` (j + 1))))
61	59, 60	syl5bir 227	. . . . . . . 8 |- (j e. NN -> ((((S seq1 F)` j) = ((S seq1 G)` j) /\ (F` (j + 1)) = (G` (j + 1))) -> ((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1))))
62	61	ancomsd 485	. . . . . . 7 |- (j e. NN -> (((F` (j + 1)) = (G` (j + 1)) /\ ((S seq1 F)` j) = ((S seq1 G)` j)) -> ((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1))))
63	62	exp3a 405	. . . . . 6 |- (j e. NN -> ((F` (j + 1)) = (G` (j + 1)) -> (((S seq1 F)` j) = ((S seq1 G)` j) -> ((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1)))))
64	56, 63	syld 30	. . . . 5 |- (j e. NN -> (A.k e. (1...(j + 1))(F` k) = (G` k) -> (((S seq1 F)` j) = ((S seq1 G)` j) -> ((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1)))))
65	64	a2d 16	. . . 4 |- (j e. NN -> ((A.k e. (1...(j + 1))(F` k) = (G` k) -> ((S seq1 F)` j) = ((S seq1 G)` j)) -> (A.k e. (1...(j + 1))(F` k) = (G` k) -> ((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1)))))
66	47, 65	syld 30	. . 3 |- (j e. NN -> ((A.k e. (1...j)(F` k) = (G` k) -> ((S seq1 F)` j) = ((S seq1 G)` j)) -> (A.k e. (1...(j + 1))(F` k) = (G` k) -> ((S seq1 F)` (j + 1)) = ((S seq1 G)` (j + 1)))))
67	6, 12, 18, 24, 41, 66	nnind 7120	. 2 |- (N e. NN -> (A.k e. (1...N)(F` k) = (G` k) -> ((S seq1 F)` N) = ((S seq1 G)` N)))
68	67	imp 377	1 |- ((N e. NN /\ A.k e. (1...N)(F` k) = (G` k)) -> ((S seq1 F)` N) = ((S seq1 G)` N))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  _Vcvv 2292   C_ wss 2593  ` cfv 3998  (class class class)co 4884  1c1 6387   + caddc 6389  NNcn 6449  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637   seq1 cseq1 7720

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-n 7108  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721

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