Theorem caucvg3lem 8426

Description: Lemma for caucvg3i 8427.

Hypotheses

Ref	Expression
caucvg3lem.1	|- F:NN-->CC
caucvg3lem.2	|- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((F` y) - (F` w))) < z))
caucvg3lem.3	|- G Fn NN
caucvg3lem.4	|- (x e. NN -> (G` x) = (Re` (F` x)))
caucvg3lem.5	|- H Fn NN
caucvg3lem.6	|- (x e. NN -> (H` x) = (Im` (F` x)))
caucvg3lem.7	|- R Fn NN
caucvg3lem.8	|- (x e. NN -> (R` x) = (_i x. (H` x)))

Assertion

Ref	Expression
caucvg3lem	|- E.x e. CC F ~~> x

Distinct variable groups:   x,F   x,y,z,w,G   x,H,y,z,w   x,R

Proof of Theorem caucvg3lem

Step	Hyp	Ref	Expression
1		ffnfv 4801	. . . 4 |- (G:NN-->RR <-> (G Fn NN /\ A.x e. NN (G` x) e. RR))
2		caucvg3lem.3	. . . 4 |- G Fn NN
3		caucvg3lem.4	. . . . . 6 |- (x e. NN -> (G` x) = (Re` (F` x)))
4		caucvg3lem.1	. . . . . . . 8 |- F:NN-->CC
5	4	ffvelrni 4788	. . . . . . 7 |- (x e. NN -> (F` x) e. CC)
6		recl 8007	. . . . . . 7 |- ((F` x) e. CC -> (Re` (F` x)) e. RR)
7	5, 6	syl 12	. . . . . 6 |- (x e. NN -> (Re` (F` x)) e. RR)
8	3, 7	eqeltrd 1971	. . . . 5 |- (x e. NN -> (G` x) e. RR)
9	8	rgen 2159	. . . 4 |- A.x e. NN (G` x) e. RR
10	1, 2, 9	mpbir2an 800	. . 3 |- G:NN-->RR
11		caucvg3lem.2	. . . 4 |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((F` y) - (F` w))) < z))
12	4, 11, 2, 3	caurei 8179	. . 3 |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((G` y) - (G` w))) < z))
13	10, 12	caucvg2i 8425	. 2 |- E.v e. RR G ~~> v
14		ffnfv 4801	. . . . 5 |- (H:NN-->RR <-> (H Fn NN /\ A.x e. NN (H` x) e. RR))
15		caucvg3lem.5	. . . . 5 |- H Fn NN
16		caucvg3lem.6	. . . . . . 7 |- (x e. NN -> (H` x) = (Im` (F` x)))
17		imcl 8008	. . . . . . . 8 |- ((F` x) e. CC -> (Im` (F` x)) e. RR)
18	5, 17	syl 12	. . . . . . 7 |- (x e. NN -> (Im` (F` x)) e. RR)
19	16, 18	eqeltrd 1971	. . . . . 6 |- (x e. NN -> (H` x) e. RR)
20	19	rgen 2159	. . . . 5 |- A.x e. NN (H` x) e. RR
21	14, 15, 20	mpbir2an 800	. . . 4 |- H:NN-->RR
22	4, 11, 15, 16	cauimi 8180	. . . 4 |- A.z e. RR (0 < z -> E.w e. NN A.y e. NN (w < y -> (abs` ((H` y) - (H` w))) < z))
23	21, 22	caucvg2i 8425	. . 3 |- E.t e. RR H ~~> t
24		axicn 6423	. . . . . . 7 |- _i e. CC
25		1z 7368	. . . . . . . 8 |- 1 e. ZZ
26		elnnuz 7609	. . . . . . . . . 10 |- (x e. NN <-> x e. (ZZ>=` 1))
27	19	recnd 6468	. . . . . . . . . . 11 |- (x e. NN -> (H` x) e. CC)
28		caucvg3lem.8	. . . . . . . . . . 11 |- (x e. NN -> (R` x) = (_i x. (H` x)))
29	27, 28	jca 310	. . . . . . . . . 10 |- (x e. NN -> ((H` x) e. CC /\ (R` x) = (_i x. (H` x))))
30	26, 29	sylbir 218	. . . . . . . . 9 |- (x e. (ZZ>=` 1) -> ((H` x) e. CC /\ (R` x) = (_i x. (H` x))))
31	30	rgen 2159	. . . . . . . 8 |- A.x e. (ZZ>=` 1)((H` x) e. CC /\ (R` x) = (_i x. (H` x)))
32		nnex 7116	. . . . . . . . . 10 |- NN e. _V
33		fnex 4535	. . . . . . . . . 10 |- ((H Fn NN /\ NN e. _V) -> H e. _V)
34	15, 32, 33	mp2an 761	. . . . . . . . 9 |- H e. _V
35		caucvg3lem.7	. . . . . . . . . 10 |- R Fn NN
36		fnex 4535	. . . . . . . . . 10 |- ((R Fn NN /\ NN e. _V) -> R e. _V)
37	35, 32, 36	mp2an 761	. . . . . . . . 9 |- R e. _V
38		visset 2295	. . . . . . . . 9 |- t e. _V
39	34, 37, 38	climmulc2 8389	. . . . . . . 8 |- (((_i e. CC /\ H ~~> t) /\ (1 e. ZZ /\ A.x e. (ZZ>=` 1)((H` x) e. CC /\ (R` x) = (_i x. (H` x))))) -> R ~~> (_i x. t))
40	25, 31, 39	mpanr12 778	. . . . . . 7 |- ((_i e. CC /\ H ~~> t) -> R ~~> (_i x. t))
41	24, 40	mpan 759	. . . . . 6 |- (H ~~> t -> R ~~> (_i x. t))
42		oprex 4907	. . . . . . . 8 |- (_i x. t) e. _V
43		climcl 8238	. . . . . . . 8 |- (((_i x. t) e. _V /\ R ~~> (_i x. t)) -> (_i x. t) e. CC)
44	42, 43	mpan 759	. . . . . . 7 |- (R ~~> (_i x. t) -> (_i x. t) e. CC)
45		breq2 3342	. . . . . . . 8 |- (u = (_i x. t) -> (R ~~> u <-> R ~~> (_i x. t)))
46	45	rcla4ev 2381	. . . . . . 7 |- (((_i x. t) e. CC /\ R ~~> (_i x. t)) -> E.u e. CC R ~~> u)
47	44, 46	mpancom 769	. . . . . 6 |- (R ~~> (_i x. t) -> E.u e. CC R ~~> u)
48	41, 47	syl 12	. . . . 5 |- (H ~~> t -> E.u e. CC R ~~> u)
49	48	a1i 8	. . . 4 |- (t e. RR -> (H ~~> t -> E.u e. CC R ~~> u))
50	49	r19.23aiv 2211	. . 3 |- (E.t e. RR H ~~> t -> E.u e. CC R ~~> u)
51	23, 50	ax-mp 7	. 2 |- E.u e. CC R ~~> u
52	8	recnd 6468	. . . . . . . . . . . 12 |- (x e. NN -> (G` x) e. CC)
53		mulcl 6456	. . . . . . . . . . . . . 14 |- ((_i e. CC /\ (H` x) e. CC) -> (_i x. (H` x)) e. CC)
54	53, 24, 27	sylancr 526	. . . . . . . . . . . . 13 |- (x e. NN -> (_i x. (H` x)) e. CC)
55	28, 54	eqeltrd 1971	. . . . . . . . . . . 12 |- (x e. NN -> (R` x) e. CC)
56		replim 8011	. . . . . . . . . . . . . 14 |- ((F` x) e. CC -> (F` x) = ((Re` (F` x)) + (_i x. (Im` (F` x)))))
57	5, 56	syl 12	. . . . . . . . . . . . 13 |- (x e. NN -> (F` x) = ((Re` (F` x)) + (_i x. (Im` (F` x)))))
58	16	opreq2d 4898	. . . . . . . . . . . . . . 15 |- (x e. NN -> (_i x. (H` x)) = (_i x. (Im` (F` x))))
59	28, 58	eqtrd 1925	. . . . . . . . . . . . . 14 |- (x e. NN -> (R` x) = (_i x. (Im` (F` x))))
60	3, 59	opreq12d 4900	. . . . . . . . . . . . 13 |- (x e. NN -> ((G` x) + (R` x)) = ((Re` (F` x)) + (_i x. (Im` (F` x)))))
61	57, 60	eqtr4d 1928	. . . . . . . . . . . 12 |- (x e. NN -> (F` x) = ((G` x) + (R` x)))
62	52, 55, 61	3jca 1050	. . . . . . . . . . 11 |- (x e. NN -> ((G` x) e. CC /\ (R` x) e. CC /\ (F` x) = ((G` x) + (R` x))))
63	26, 62	sylbir 218	. . . . . . . . . 10 |- (x e. (ZZ>=` 1) -> ((G` x) e. CC /\ (R` x) e. CC /\ (F` x) = ((G` x) + (R` x))))
64	63	rgen 2159	. . . . . . . . 9 |- A.x e. (ZZ>=` 1)((G` x) e. CC /\ (R` x) e. CC /\ (F` x) = ((G` x) + (R` x)))
65		fnex 4535	. . . . . . . . . . 11 |- ((G Fn NN /\ NN e. _V) -> G e. _V)
66	2, 32, 65	mp2an 761	. . . . . . . . . 10 |- G e. _V
67		fex 4595	. . . . . . . . . . 11 |- ((F:NN-->CC /\ NN e. _V) -> F e. _V)
68	4, 32, 67	mp2an 761	. . . . . . . . . 10 |- F e. _V
69		visset 2295	. . . . . . . . . 10 |- v e. _V
70		visset 2295	. . . . . . . . . 10 |- u e. _V
71	66, 37, 68, 69, 70	climadd 8377	. . . . . . . . 9 |- (((G ~~> v /\ R ~~> u) /\ (1 e. ZZ /\ A.x e. (ZZ>=` 1)((G` x) e. CC /\ (R` x) e. CC /\ (F` x) = ((G` x) + (R` x))))) -> F ~~> (v + u))
72	25, 64, 71	mpanr12 778	. . . . . . . 8 |- ((G ~~> v /\ R ~~> u) -> F ~~> (v + u))
73		oprex 4907	. . . . . . . . . 10 |- (v + u) e. _V
74		climcl 8238	. . . . . . . . . 10 |- (((v + u) e. _V /\ F ~~> (v + u)) -> (v + u) e. CC)
75	73, 74	mpan 759	. . . . . . . . 9 |- (F ~~> (v + u) -> (v + u) e. CC)
76		breq2 3342	. . . . . . . . . 10 |- (x = (v + u) -> (F ~~> x <-> F ~~> (v + u)))
77	76	rcla4ev 2381	. . . . . . . . 9 |- (((v + u) e. CC /\ F ~~> (v + u)) -> E.x e. CC F ~~> x)
78	75, 77	mpancom 769	. . . . . . . 8 |- (F ~~> (v + u) -> E.x e. CC F ~~> x)
79	72, 78	syl 12	. . . . . . 7 |- ((G ~~> v /\ R ~~> u) -> E.x e. CC F ~~> x)
80	79	ex 402	. . . . . 6 |- (G ~~> v -> (R ~~> u -> E.x e. CC F ~~> x))
81	80	a1d 15	. . . . 5 |- (G ~~> v -> (u e. CC -> (R ~~> u -> E.x e. CC F ~~> x)))
82	81	r19.23adv 2215	. . . 4 |- (G ~~> v -> (E.u e. CC R ~~> u -> E.x e. CC F ~~> x))
83	82	a1i 8	. . 3 |- (v e. RR -> (G ~~> v -> (E.u e. CC R ~~> u -> E.x e. CC F ~~> x)))
84	83	r19.23aiv 2211	. 2 |- (E.v e. RR G ~~> v -> (E.u e. CC R ~~> u -> E.x e. CC F ~~> x))
85	13, 51, 84	mp2 54	1 |- E.x e. CC F ~~> x

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   class class class wbr 3338   Fn wfn 3993  -->wf 3994  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386  1c1 6387  _ici 6388   + caddc 6389   x. cmul 6391   - cmin 6445  NNcn 6449  ZZcz 6451   < clt 6653  ZZ>=cuz 7586  Recre 7997  Imcim 7998  abscabs 8000   ~~> cli 8234

This theorem is referenced by:  caucvg3i 8427

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-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-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004  df-clim 8235

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