Theorem efseq1ex 7248

Description: The series defining the exponential function converges.

Hypothesis

Ref	Expression
efcltlem.1	|- F = {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}

Assertion

Ref	Expression
efseq1ex	|- (A e. CC -> E.x( + seq1 F) ~~> x)

Distinct variable groups:   x,y,z,A   x,F

Proof of Theorem efseq1ex

Step	Hyp	Ref	Expression
1		opreq1 3953	. . . . . . . . . . . . 13 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> (A^y) = (if((A e. CC /\ A =/= 0), A, 1)^y))
2	1	opreq1d 3960	. . . . . . . . . . . 12 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> ((A^y) / (!` y)) = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))
3	2	eqeq2d 1478	. . . . . . . . . . 11 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> (z = ((A^y) / (!` y)) <-> z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y))))
4	3	anbi2d 614	. . . . . . . . . 10 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> ((y e. NN /\ z = ((A^y) / (!` y))) <-> (y e. NN /\ z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))))
5	4	opabbidv 2660	. . . . . . . . 9 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))} = {<.y, z>. | (y e. NN /\ z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))})
6	5	opreq2d 3961	. . . . . . . 8 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> ( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) = ( + seq1 {<.y, z>. | (y e. NN /\ z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))}))
7	6	breq1d 2619	. . . . . . 7 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> (( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) ~~> x <-> ( + seq1 {<.y, z>. | (y e. NN /\ z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))}) ~~> x))
8	7	exbidv 1274	. . . . . 6 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> (E.x( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) ~~> x <-> E.x( + seq1 {<.y, z>. | (y e. NN /\ z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))}) ~~> x))
9		eqid 1468	. . . . . . 7 |- {<.y, z>. | (y e. NN /\ z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))} = {<.y, z>. | (y e. NN /\ z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))}
10		eleq1 1526	. . . . . . . . . 10 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> (A e. CC <-> if((A e. CC /\ A =/= 0), A, 1) e. CC))
11		neeq1 1582	. . . . . . . . . 10 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> (A =/= 0 <-> if((A e. CC /\ A =/= 0), A, 1) =/= 0))
12	10, 11	anbi12d 626	. . . . . . . . 9 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> ((A e. CC /\ A =/= 0) <-> (if((A e. CC /\ A =/= 0), A, 1) e. CC /\ if((A e. CC /\ A =/= 0), A, 1) =/= 0)))
13		eleq1 1526	. . . . . . . . . 10 |- (1 = if((A e. CC /\ A =/= 0), A, 1) -> (1 e. CC <-> if((A e. CC /\ A =/= 0), A, 1) e. CC))
14		neeq1 1582	. . . . . . . . . 10 |- (1 = if((A e. CC /\ A =/= 0), A, 1) -> (1 =/= 0 <-> if((A e. CC /\ A =/= 0), A, 1) =/= 0))
15	13, 14	anbi12d 626	. . . . . . . . 9 |- (1 = if((A e. CC /\ A =/= 0), A, 1) -> ((1 e. CC /\ 1 =/= 0) <-> (if((A e. CC /\ A =/= 0), A, 1) e. CC /\ if((A e. CC /\ A =/= 0), A, 1) =/= 0)))
16		ax1cn 5241	. . . . . . . . . 10 |- 1 e. CC
17		ax1ne0 5252	. . . . . . . . . 10 |- 1 =/= 0
18	16, 17	pm3.2i 285	. . . . . . . . 9 |- (1 e. CC /\ 1 =/= 0)
19	12, 15, 18	elimhyp 2380	. . . . . . . 8 |- (if((A e. CC /\ A =/= 0), A, 1) e. CC /\ if((A e. CC /\ A =/= 0), A, 1) =/= 0)
20	19	pm3.26i 320	. . . . . . 7 |- if((A e. CC /\ A =/= 0), A, 1) e. CC
21	19	pm3.27i 324	. . . . . . 7 |- if((A e. CC /\ A =/= 0), A, 1) =/= 0
22	9, 20, 21	efcltlem1 7246	. . . . . 6 |- E.x( + seq1 {<.y, z>. | (y e. NN /\ z = ((if((A e. CC /\ A =/= 0), A, 1)^y) / (!` y)))}) ~~> x
23	8, 22	dedth 2373	. . . . 5 |- ((A e. CC /\ A =/= 0) -> E.x( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) ~~> x)
24		efcltlem.1	. . . . . . . 8 |- F = {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}
25	24	opreq2i 3957	. . . . . . 7 |- ( + seq1 F) = ( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))})
26	25	breq1i 2616	. . . . . 6 |- (( + seq1 F) ~~> x <-> ( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) ~~> x)
27	26	exbii 1047	. . . . 5 |- (E.x( + seq1 F) ~~> x <-> E.x( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) ~~> x)
28	23, 27	sylibr 200	. . . 4 |- ((A e. CC /\ A =/= 0) -> E.x( + seq1 F) ~~> x)
29	28	ex 373	. . 3 |- (A e. CC -> (A =/= 0 -> E.x( + seq1 F) ~~> x))
30		df-ne 1579	. . 3 |- (A =/= 0 <-> -. A = 0)
31	29, 30	syl5ibr 207	. 2 |- (A e. CC -> (-. A = 0 -> E.x( + seq1 F) ~~> x))
32	24	efcltlem2 7247	. . 3 |- (A = 0 -> ( + seq1 F) ~~> 0)
33		0cn 5300	. . . . 5 |- 0 e. CC
34	33	elisseti 1809	. . . 4 |- 0 e. V
35		breq2 2613	. . . 4 |- (x = 0 -> (( + seq1 F) ~~> x <-> ( + seq1 F) ~~> 0))
36	34, 35	cla4ev 1860	. . 3 |- (( + seq1 F) ~~> 0 -> E.x( + seq1 F) ~~> x)
37	32, 36	syl 10	. 2 |- (A = 0 -> E.x( + seq1 F) ~~> x)
38	31, 37	pm2.61d2 129	1 |- (A e. CC -> E.x( + seq1 F) ~~> x)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  ifcif 2351   class class class wbr 2609  {copab 2656  ` cfv 3172  (class class class)co 3948  CCcc 5204  0cc0 5206  1c1 5207   + caddc 5209   / cdiv 5266  NNcn 5268   seq1 cseq1 6244  ^cexp 6500  !cfa 6868   ~~> cli 6912

This theorem is referenced by:  dfef2 7249  efseq0ex 7253

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-nel 1580  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-en 4351  df-dom 4352  df-sdom 4353  df-sup 4548  df-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-ltr 5142  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-lt 5219  df-sub 5328  df-neg 5330  df-pnf 5459  df-mnf 5460  df-xr 5461  df-ltxr 5462  df-le 5463  df-div 5672  df-n 5873  df-2 5917  df-n0 6047  df-z 6083  df-fl 6172  df-seq1 6245  df-shft 6278  df-uz 6350  df-fz 6400  df-seqz 6465  df-seq0 6466  df-exp 6501  df-sqr 6600  df-re 6682  df-im 6683  df-cj 6684  df-abs 6685  df-fac 6869  df-clim 6913  df-sum 6918

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