Theorem cvgcmp3cetlem2 8450

Description: Lemma for cvgcmp3ce 8451.

Hypotheses

Ref	Expression
cvgcmp3cetlem2.1	|- A e. _V
cvgcmp3cetlem2.2	|- B e. NN
cvgcmp3cetlem2.3	|- (ph <-> ((F:NN-->RR /\ A.x e. NN 0 <_ (F` x)) /\ ( + seq1 F) ~~> A))

Assertion

Ref	Expression
cvgcmp3cetlem2	|- ((ph /\ ((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.x e. NN (B < x -> (abs` (G` x)) <_ (C x. (F` x)))))) -> E.y( + seq1 G) ~~> y)

Distinct variable groups:   x,y,B   x,F,y   x,G,y   x,C,y   ph,y

Proof of Theorem cvgcmp3cetlem2

Step	Hyp	Ref	Expression
1		fveq1 4680	. . . . . . . . . . 11 |- (F = if(ph, F, (NN X. {0})) -> (F` z) = (if(ph, F, (NN X. {0}))` z))
2	1	opreq2d 4898	. . . . . . . . . 10 |- (F = if(ph, F, (NN X. {0})) -> (C x. (F` z)) = (C x. (if(ph, F, (NN X. {0}))` z)))
3	2	breq2d 3350	. . . . . . . . 9 |- (F = if(ph, F, (NN X. {0})) -> ((abs` (G` z)) <_ (C x. (F` z)) <-> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))
4	3	imbi2d 674	. . . . . . . 8 |- (F = if(ph, F, (NN X. {0})) -> ((B < z -> (abs` (G` z)) <_ (C x. (F` z))) <-> (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z)))))
5	4	ralbidv 2123	. . . . . . 7 |- (F = if(ph, F, (NN X. {0})) -> (A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))) <-> A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z)))))
6	5	anbi2d 678	. . . . . 6 |- (F = if(ph, F, (NN X. {0})) -> ((G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z)))) <-> (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))))
7	6	anbi2d 678	. . . . 5 |- (F = if(ph, F, (NN X. {0})) -> (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))))) <-> ((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z)))))))
8	7	imbi1d 675	. . . 4 |- (F = if(ph, F, (NN X. {0})) -> ((((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))))) -> E.y( + seq1 G) ~~> y) <-> (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))) -> E.y( + seq1 G) ~~> y)))
9		cvgcmp3cetlem2.1	. . . . . 6 |- A e. _V
10		0re 6603	. . . . . . 7 |- 0 e. RR
11	10	elisseti 2301	. . . . . 6 |- 0 e. _V
12	9, 11	ifex 3031	. . . . 5 |- if(ph, A, 0) e. _V
13		cvgcmp3cetlem2.2	. . . . 5 |- B e. NN
14		feq1 4551	. . . . . . . . . . 11 |- (F = if(ph, F, (NN X. {0})) -> (F:NN-->RR <-> if(ph, F, (NN X. {0})):NN-->RR))
15	1	breq2d 3350	. . . . . . . . . . . 12 |- (F = if(ph, F, (NN X. {0})) -> (0 <_ (F` z) <-> 0 <_ (if(ph, F, (NN X. {0}))` z)))
16	15	ralbidv 2123	. . . . . . . . . . 11 |- (F = if(ph, F, (NN X. {0})) -> (A.z e. NN 0 <_ (F` z) <-> A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)))
17	14, 16	anbi12d 690	. . . . . . . . . 10 |- (F = if(ph, F, (NN X. {0})) -> ((F:NN-->RR /\ A.z e. NN 0 <_ (F` z)) <-> (if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z))))
18		opreq2 4890	. . . . . . . . . . 11 |- (F = if(ph, F, (NN X. {0})) -> ( + seq1 F) = ( + seq1 if(ph, F, (NN X. {0}))))
19	18	breq1d 3348	. . . . . . . . . 10 |- (F = if(ph, F, (NN X. {0})) -> (( + seq1 F) ~~> A <-> ( + seq1 if(ph, F, (NN X. {0}))) ~~> A))
20	17, 19	anbi12d 690	. . . . . . . . 9 |- (F = if(ph, F, (NN X. {0})) -> (((F:NN-->RR /\ A.z e. NN 0 <_ (F` z)) /\ ( + seq1 F) ~~> A) <-> ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> A)))
21		cvgcmp3cetlem2.3	. . . . . . . . . 10 |- (ph <-> ((F:NN-->RR /\ A.x e. NN 0 <_ (F` x)) /\ ( + seq1 F) ~~> A))
22		fveq2 4681	. . . . . . . . . . . . . 14 |- (x = z -> (F` x) = (F` z))
23	22	breq2d 3350	. . . . . . . . . . . . 13 |- (x = z -> (0 <_ (F` x) <-> 0 <_ (F` z)))
24	23	cbvralv 2280	. . . . . . . . . . . 12 |- (A.x e. NN 0 <_ (F` x) <-> A.z e. NN 0 <_ (F` z))
25	24	anbi2i 538	. . . . . . . . . . 11 |- ((F:NN-->RR /\ A.x e. NN 0 <_ (F` x)) <-> (F:NN-->RR /\ A.z e. NN 0 <_ (F` z)))
26	25	anbi1i 539	. . . . . . . . . 10 |- (((F:NN-->RR /\ A.x e. NN 0 <_ (F` x)) /\ ( + seq1 F) ~~> A) <-> ((F:NN-->RR /\ A.z e. NN 0 <_ (F` z)) /\ ( + seq1 F) ~~> A))
27	21, 26	bitri 190	. . . . . . . . 9 |- (ph <-> ((F:NN-->RR /\ A.z e. NN 0 <_ (F` z)) /\ ( + seq1 F) ~~> A))
28	20, 27	syl5bb 591	. . . . . . . 8 |- (F = if(ph, F, (NN X. {0})) -> (ph <-> ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> A)))
29		breq2 3342	. . . . . . . . 9 |- (A = if(ph, A, 0) -> (( + seq1 if(ph, F, (NN X. {0}))) ~~> A <-> ( + seq1 if(ph, F, (NN X. {0}))) ~~> if(ph, A, 0)))
30	29	anbi2d 678	. . . . . . . 8 |- (A = if(ph, A, 0) -> (((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> A) <-> ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> if(ph, A, 0))))
31		feq1 4551	. . . . . . . . . 10 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> ((NN X. {0}):NN-->RR <-> if(ph, F, (NN X. {0})):NN-->RR))
32		fveq1 4680	. . . . . . . . . . . 12 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> ((NN X. {0})` z) = (if(ph, F, (NN X. {0}))` z))
33	32	breq2d 3350	. . . . . . . . . . 11 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> (0 <_ ((NN X. {0})` z) <-> 0 <_ (if(ph, F, (NN X. {0}))` z)))
34	33	ralbidv 2123	. . . . . . . . . 10 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> (A.z e. NN 0 <_ ((NN X. {0})` z) <-> A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)))
35	31, 34	anbi12d 690	. . . . . . . . 9 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> (((NN X. {0}):NN-->RR /\ A.z e. NN 0 <_ ((NN X. {0})` z)) <-> (if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z))))
36		opreq2 4890	. . . . . . . . . 10 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> ( + seq1 (NN X. {0})) = ( + seq1 if(ph, F, (NN X. {0}))))
37	36	breq1d 3348	. . . . . . . . 9 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> (( + seq1 (NN X. {0})) ~~> 0 <-> ( + seq1 if(ph, F, (NN X. {0}))) ~~> 0))
38	35, 37	anbi12d 690	. . . . . . . 8 |- ((NN X. {0}) = if(ph, F, (NN X. {0})) -> ((((NN X. {0}):NN-->RR /\ A.z e. NN 0 <_ ((NN X. {0})` z)) /\ ( + seq1 (NN X. {0})) ~~> 0) <-> ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> 0)))
39		breq2 3342	. . . . . . . . 9 |- (0 = if(ph, A, 0) -> (( + seq1 if(ph, F, (NN X. {0}))) ~~> 0 <-> ( + seq1 if(ph, F, (NN X. {0}))) ~~> if(ph, A, 0)))
40	39	anbi2d 678	. . . . . . . 8 |- (0 = if(ph, A, 0) -> (((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> 0) <-> ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> if(ph, A, 0))))
41	11	fconst 4602	. . . . . . . . . . 11 |- (NN X. {0}):NN-->{0}
42		snssi 3129	. . . . . . . . . . . 12 |- (0 e. RR -> {0} C_ RR)
43	10, 42	ax-mp 7	. . . . . . . . . . 11 |- {0} C_ RR
44		fss 4571	. . . . . . . . . . 11 |- (((NN X. {0}):NN-->{0} /\ {0} C_ RR) -> (NN X. {0}):NN-->RR)
45	41, 43, 44	mp2an 761	. . . . . . . . . 10 |- (NN X. {0}):NN-->RR
46	11	fvconst2 4822	. . . . . . . . . . . 12 |- (z e. NN -> ((NN X. {0})` z) = 0)
47	10	leidi 6790	. . . . . . . . . . . 12 |- 0 <_ 0
48	46, 47	syl5breqr 3373	. . . . . . . . . . 11 |- (z e. NN -> 0 <_ ((NN X. {0})` z))
49	48	rgen 2159	. . . . . . . . . 10 |- A.z e. NN 0 <_ ((NN X. {0})` z)
50	45, 49	pm3.2i 307	. . . . . . . . 9 |- ((NN X. {0}):NN-->RR /\ A.z e. NN 0 <_ ((NN X. {0})` z))
51		ser1clim0 8433	. . . . . . . . 9 |- ( + seq1 (NN X. {0})) ~~> 0
52	50, 51	pm3.2i 307	. . . . . . . 8 |- (((NN X. {0}):NN-->RR /\ A.z e. NN 0 <_ ((NN X. {0})` z)) /\ ( + seq1 (NN X. {0})) ~~> 0)
53	28, 30, 38, 40, 52	elimhyp2v 3022	. . . . . . 7 |- ((if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)) /\ ( + seq1 if(ph, F, (NN X. {0}))) ~~> if(ph, A, 0))
54	53	simpli 347	. . . . . 6 |- (if(ph, F, (NN X. {0})):NN-->RR /\ A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z))
55	54	simpli 347	. . . . 5 |- if(ph, F, (NN X. {0})):NN-->RR
56	54	simpri 351	. . . . . 6 |- A.z e. NN 0 <_ (if(ph, F, (NN X. {0}))` z)
57	56	rspec 2158	. . . . 5 |- (z e. NN -> 0 <_ (if(ph, F, (NN X. {0}))` z))
58	53	simpri 351	. . . . 5 |- ( + seq1 if(ph, F, (NN X. {0}))) ~~> if(ph, A, 0)
59		biid 187	. . . . 5 |- (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))) <-> ((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))))
60	12, 13, 55, 57, 58, 59	cvgcmp3cetlem1 8449	. . . 4 |- (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (if(ph, F, (NN X. {0}))` z))))) -> E.y( + seq1 G) ~~> y)
61	8, 60	dedth 3011	. . 3 |- (ph -> (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))))) -> E.y( + seq1 G) ~~> y))
62		breq2 3342	. . . . . . 7 |- (x = z -> (B < x <-> B < z))
63		fveq2 4681	. . . . . . . . 9 |- (x = z -> (G` x) = (G` z))
64	63	fveq2d 4685	. . . . . . . 8 |- (x = z -> (abs` (G` x)) = (abs` (G` z)))
65	22	opreq2d 4898	. . . . . . . 8 |- (x = z -> (C x. (F` x)) = (C x. (F` z)))
66	64, 65	breq12d 3351	. . . . . . 7 |- (x = z -> ((abs` (G` x)) <_ (C x. (F` x)) <-> (abs` (G` z)) <_ (C x. (F` z))))
67	62, 66	imbi12d 688	. . . . . 6 |- (x = z -> ((B < x -> (abs` (G` x)) <_ (C x. (F` x))) <-> (B < z -> (abs` (G` z)) <_ (C x. (F` z)))))
68	67	cbvralv 2280	. . . . 5 |- (A.x e. NN (B < x -> (abs` (G` x)) <_ (C x. (F` x))) <-> A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))))
69	68	anbi2i 538	. . . 4 |- ((G:NN-->CC /\ A.x e. NN (B < x -> (abs` (G` x)) <_ (C x. (F` x)))) <-> (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z)))))
70	69	anbi2i 538	. . 3 |- (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.x e. NN (B < x -> (abs` (G` x)) <_ (C x. (F` x))))) <-> ((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.z e. NN (B < z -> (abs` (G` z)) <_ (C x. (F` z))))))
71	61, 70	syl5ib 223	. 2 |- (ph -> (((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.x e. NN (B < x -> (abs` (G` x)) <_ (C x. (F` x))))) -> E.y( + seq1 G) ~~> y))
72	71	imp 377	1 |- ((ph /\ ((C e. RR /\ 0 <_ C) /\ (G:NN-->CC /\ A.x e. NN (B < x -> (abs` (G` x)) <_ (C x. (F` x)))))) -> E.y( + seq1 G) ~~> y)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326  A.wral 2105  _Vcvv 2292   C_ wss 2593  ifcif 2982  {csn 3044   class class class wbr 3338   X. cxp 3984  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   <_ cle 6448  NNcn 6449   < clt 6653   seq1 cseq1 7720  abscabs 8000   ~~> cli 8234

This theorem is referenced by:  cvgcmp3ce 8451

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-sup 5664  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-2 7154  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-seq1 7721  df-shft 7754  df-seqz 7776  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235  df-sum 8240

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