Theorem seqzfveq 7797

Description: Equality theorem for the recursive sequence builder.

Hypotheses

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

Assertion

Ref	Expression
seqzfveq	|- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(F` k) = (G` k)) -> ((<.M, S>. seq F)` N) = ((<.M, S>. seq G)` N))

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

Proof of Theorem seqzfveq

Step	Hyp	Ref	Expression
1		opreq2 4890	. . . . 5 |- (j = M -> (M...j) = (M...M))
2	1	raleqdv 2269	. . . 4 |- (j = M -> (A.k e. (M...j)(F` k) = (G` k) <-> A.k e. (M...M)(F` k) = (G` k)))
3		fveq2 4681	. . . . 5 |- (j = M -> ((<.M, S>. seq F)` j) = ((<.M, S>. seq F)` M))
4		fveq2 4681	. . . . 5 |- (j = M -> ((<.M, S>. seq G)` j) = ((<.M, S>. seq G)` M))
5	3, 4	eqeq12d 1899	. . . 4 |- (j = M -> (((<.M, S>. seq F)` j) = ((<.M, S>. seq G)` j) <-> ((<.M, S>. seq F)` M) = ((<.M, S>. seq G)` M)))
6	2, 5	imbi12d 688	. . 3 |- (j = M -> ((A.k e. (M...j)(F` k) = (G` k) -> ((<.M, S>. seq F)` j) = ((<.M, S>. seq G)` j)) <-> (A.k e. (M...M)(F` k) = (G` k) -> ((<.M, S>. seq F)` M) = ((<.M, S>. seq G)` M))))
7		opreq2 4890	. . . . 5 |- (j = m -> (M...j) = (M...m))
8	7	raleqdv 2269	. . . 4 |- (j = m -> (A.k e. (M...j)(F` k) = (G` k) <-> A.k e. (M...m)(F` k) = (G` k)))
9		fveq2 4681	. . . . 5 |- (j = m -> ((<.M, S>. seq F)` j) = ((<.M, S>. seq F)` m))
10		fveq2 4681	. . . . 5 |- (j = m -> ((<.M, S>. seq G)` j) = ((<.M, S>. seq G)` m))
11	9, 10	eqeq12d 1899	. . . 4 |- (j = m -> (((<.M, S>. seq F)` j) = ((<.M, S>. seq G)` j) <-> ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m)))
12	8, 11	imbi12d 688	. . 3 |- (j = m -> ((A.k e. (M...j)(F` k) = (G` k) -> ((<.M, S>. seq F)` j) = ((<.M, S>. seq G)` j)) <-> (A.k e. (M...m)(F` k) = (G` k) -> ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m))))
13		opreq2 4890	. . . . 5 |- (j = (m + 1) -> (M...j) = (M...(m + 1)))
14	13	raleqdv 2269	. . . 4 |- (j = (m + 1) -> (A.k e. (M...j)(F` k) = (G` k) <-> A.k e. (M...(m + 1))(F` k) = (G` k)))
15		fveq2 4681	. . . . 5 |- (j = (m + 1) -> ((<.M, S>. seq F)` j) = ((<.M, S>. seq F)` (m + 1)))
16		fveq2 4681	. . . . 5 |- (j = (m + 1) -> ((<.M, S>. seq G)` j) = ((<.M, S>. seq G)` (m + 1)))
17	15, 16	eqeq12d 1899	. . . 4 |- (j = (m + 1) -> (((<.M, S>. seq F)` j) = ((<.M, S>. seq G)` j) <-> ((<.M, S>. seq F)` (m + 1)) = ((<.M, S>. seq G)` (m + 1))))
18	14, 17	imbi12d 688	. . 3 |- (j = (m + 1) -> ((A.k e. (M...j)(F` k) = (G` k) -> ((<.M, S>. seq F)` j) = ((<.M, S>. seq G)` j)) <-> (A.k e. (M...(m + 1))(F` k) = (G` k) -> ((<.M, S>. seq F)` (m + 1)) = ((<.M, S>. seq G)` (m + 1)))))
19		opreq2 4890	. . . . 5 |- (j = N -> (M...j) = (M...N))
20	19	raleqdv 2269	. . . 4 |- (j = N -> (A.k e. (M...j)(F` k) = (G` k) <-> A.k e. (M...N)(F` k) = (G` k)))
21		fveq2 4681	. . . . 5 |- (j = N -> ((<.M, S>. seq F)` j) = ((<.M, S>. seq F)` N))
22		fveq2 4681	. . . . 5 |- (j = N -> ((<.M, S>. seq G)` j) = ((<.M, S>. seq G)` N))
23	21, 22	eqeq12d 1899	. . . 4 |- (j = N -> (((<.M, S>. seq F)` j) = ((<.M, S>. seq G)` j) <-> ((<.M, S>. seq F)` N) = ((<.M, S>. seq G)` N)))
24	20, 23	imbi12d 688	. . 3 |- (j = N -> ((A.k e. (M...j)(F` k) = (G` k) -> ((<.M, S>. seq F)` j) = ((<.M, S>. seq G)` j)) <-> (A.k e. (M...N)(F` k) = (G` k) -> ((<.M, S>. seq F)` N) = ((<.M, S>. seq G)` N))))
25		fveq2 4681	. . . . . . . 8 |- (k = M -> (F` k) = (F` M))
26		fveq2 4681	. . . . . . . 8 |- (k = M -> (G` k) = (G` M))
27	25, 26	eqeq12d 1899	. . . . . . 7 |- (k = M -> ((F` k) = (G` k) <-> (F` M) = (G` M)))
28	27	rcla4va 2378	. . . . . 6 |- ((M e. (M...M) /\ A.k e. (M...M)(F` k) = (G` k)) -> (F` M) = (G` M))
29		elfz3 7661	. . . . . 6 |- (M e. ZZ -> M e. (M...M))
30	28, 29	sylan 497	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(F` k) = (G` k)) -> (F` M) = (G` M))
31		seqzfveq.1	. . . . . . 7 |- S e. _V
32		seqzfveq.2	. . . . . . 7 |- F e. _V
33	31, 32	seqz1 7790	. . . . . 6 |- (M e. ZZ -> ((<.M, S>. seq F)` M) = (F` M))
34	33	adantr 425	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(F` k) = (G` k)) -> ((<.M, S>. seq F)` M) = (F` M))
35		seqzfveq.3	. . . . . . 7 |- G e. _V
36	31, 35	seqz1 7790	. . . . . 6 |- (M e. ZZ -> ((<.M, S>. seq G)` M) = (G` M))
37	36	adantr 425	. . . . 5 |- ((M e. ZZ /\ A.k e. (M...M)(F` k) = (G` k)) -> ((<.M, S>. seq G)` M) = (G` M))
38	30, 34, 37	3eqtr4d 1937	. . . 4 |- ((M e. ZZ /\ A.k e. (M...M)(F` k) = (G` k)) -> ((<.M, S>. seq F)` M) = ((<.M, S>. seq G)` M))
39	38	ex 402	. . 3 |- (M e. ZZ -> (A.k e. (M...M)(F` k) = (G` k) -> ((<.M, S>. seq F)` M) = ((<.M, S>. seq G)` M)))
40		eluzel2 7593	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> M e. ZZ)
41		eluzelz 7592	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> m e. ZZ)
42		fzssp1 7679	. . . . . . . . 9 |- ((M e. ZZ /\ m e. ZZ) -> (M...m) C_ (M...(m + 1)))
43	40, 41, 42	syl11anc 524	. . . . . . . 8 |- (m e. (ZZ>=` M) -> (M...m) C_ (M...(m + 1)))
44	43	sseld 2619	. . . . . . 7 |- (m e. (ZZ>=` M) -> (k e. (M...m) -> k e. (M...(m + 1))))
45	44	imim1d 33	. . . . . 6 |- (m e. (ZZ>=` M) -> ((k e. (M...(m + 1)) -> (F` k) = (G` k)) -> (k e. (M...m) -> (F` k) = (G` k))))
46	45	ralimdv2 2173	. . . . 5 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))(F` k) = (G` k) -> A.k e. (M...m)(F` k) = (G` k)))
47	46	imim1d 33	. . . 4 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)(F` k) = (G` k) -> ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m)) -> (A.k e. (M...(m + 1))(F` k) = (G` k) -> ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m))))
48		id 73	. . . . . . . 8 |- (((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m) -> ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m))
49		fveq2 4681	. . . . . . . . . . 11 |- (k = (m + 1) -> (F` k) = (F` (m + 1)))
50		fveq2 4681	. . . . . . . . . . 11 |- (k = (m + 1) -> (G` k) = (G` (m + 1)))
51	49, 50	eqeq12d 1899	. . . . . . . . . 10 |- (k = (m + 1) -> ((F` k) = (G` k) <-> (F` (m + 1)) = (G` (m + 1))))
52	51	rcla4va 2378	. . . . . . . . 9 |- (((m + 1) e. (M...(m + 1)) /\ A.k e. (M...(m + 1))(F` k) = (G` k)) -> (F` (m + 1)) = (G` (m + 1)))
53		peano2uz 7616	. . . . . . . . . 10 |- (m e. (ZZ>=` M) -> (m + 1) e. (ZZ>=` M))
54		eluzfz2 7659	. . . . . . . . . 10 |- ((m + 1) e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
55	53, 54	syl 12	. . . . . . . . 9 |- (m e. (ZZ>=` M) -> (m + 1) e. (M...(m + 1)))
56	52, 55	sylan 497	. . . . . . . 8 |- ((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(F` k) = (G` k)) -> (F` (m + 1)) = (G` (m + 1)))
57	48, 56	opreqan12rd 4903	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(F` k) = (G` k)) /\ ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m)) -> (((<.M, S>. seq F)` m)S(F` (m + 1))) = (((<.M, S>. seq G)` m)S(G` (m + 1))))
58	31, 32	seqzp1 7791	. . . . . . . 8 |- (m e. (ZZ>=` M) -> ((<.M, S>. seq F)` (m + 1)) = (((<.M, S>. seq F)` m)S(F` (m + 1))))
59	58	ad2antrr 440	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(F` k) = (G` k)) /\ ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m)) -> ((<.M, S>. seq F)` (m + 1)) = (((<.M, S>. seq F)` m)S(F` (m + 1))))
60	31, 35	seqzp1 7791	. . . . . . . 8 |- (m e. (ZZ>=` M) -> ((<.M, S>. seq G)` (m + 1)) = (((<.M, S>. seq G)` m)S(G` (m + 1))))
61	60	ad2antrr 440	. . . . . . 7 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(F` k) = (G` k)) /\ ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m)) -> ((<.M, S>. seq G)` (m + 1)) = (((<.M, S>. seq G)` m)S(G` (m + 1))))
62	57, 59, 61	3eqtr4d 1937	. . . . . 6 |- (((m e. (ZZ>=` M) /\ A.k e. (M...(m + 1))(F` k) = (G` k)) /\ ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m)) -> ((<.M, S>. seq F)` (m + 1)) = ((<.M, S>. seq G)` (m + 1)))
63	62	exp31 407	. . . . 5 |- (m e. (ZZ>=` M) -> (A.k e. (M...(m + 1))(F` k) = (G` k) -> (((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m) -> ((<.M, S>. seq F)` (m + 1)) = ((<.M, S>. seq G)` (m + 1)))))
64	63	a2d 16	. . . 4 |- (m e. (ZZ>=` M) -> ((A.k e. (M...(m + 1))(F` k) = (G` k) -> ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m)) -> (A.k e. (M...(m + 1))(F` k) = (G` k) -> ((<.M, S>. seq F)` (m + 1)) = ((<.M, S>. seq G)` (m + 1)))))
65	47, 64	syld 30	. . 3 |- (m e. (ZZ>=` M) -> ((A.k e. (M...m)(F` k) = (G` k) -> ((<.M, S>. seq F)` m) = ((<.M, S>. seq G)` m)) -> (A.k e. (M...(m + 1))(F` k) = (G` k) -> ((<.M, S>. seq F)` (m + 1)) = ((<.M, S>. seq G)` (m + 1)))))
66	6, 12, 18, 24, 39, 65	uzind4 7619	. 2 |- (N e. (ZZ>=` M) -> (A.k e. (M...N)(F` k) = (G` k) -> ((<.M, S>. seq F)` N) = ((<.M, S>. seq G)` N)))
67	66	imp 377	1 |- ((N e. (ZZ>=` M) /\ A.k e. (M...N)(F` k) = (G` k)) -> ((<.M, S>. seq F)` N) = ((<.M, S>. seq G)` N))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  <.cop 3046  ` cfv 3998  (class class class)co 4884  1c1 6387   + caddc 6389  ZZcz 6451  ZZ>=cuz 7586  ...cfz 7637   seq cseqz 7774

This theorem is referenced by:  seqzeq 7798  seqzresval 7802  seqzres 7803  seqzres2 7804  sumeq2 8245  seqzresval2 13616  prodeq2 14661

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  df-shft 7754  df-seqz 7776

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