Theorem efseq1ex 7429

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 4044	. . . . . . . . . . . . 13 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> (A^y) = (if((A e. CC /\ A =/= 0), A, 1)^y))
2	1	opreq1d 4051	. . . . . . . . . . . 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 1523	. . . . . . . . . . 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 618	. . . . . . . . . 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 2721	. . . . . . . . 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 4052	. . . . . . . 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 2679	. . . . . . 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 1312	. . . . . 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 1512	. . . . . . 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 1571	. . . . . . . . . 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 1627	. . . . . . . . . 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 630	. . . . . . . . 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 1571	. . . . . . . . . 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 1627	. . . . . . . . . 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 630	. . . . . . . . 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 5358	. . . . . . . . . 10 |- 1 e. CC
17		ax1ne0 5369	. . . . . . . . . 10 |- 1 =/= 0
18	16, 17	pm3.2i 283	. . . . . . . . 9 |- (1 e. CC /\ 1 =/= 0)
19	12, 15, 18	elimhyp 2435	. . . . . . . 8 |- (if((A e. CC /\ A =/= 0), A, 1) e. CC /\ if((A e. CC /\ A =/= 0), A, 1) =/= 0)
20	19	pm3.26i 318	. . . . . . 7 |- if((A e. CC /\ A =/= 0), A, 1) e. CC
21	19	pm3.27i 322	. . . . . . 7 |- if((A e. CC /\ A =/= 0), A, 1) =/= 0
22	9, 20, 21	efcltlem1 7427	. . . . . 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 2428	. . . . 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 4048	. . . . . . 7 |- ( + seq1 F) = ( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))})
26	25	breq1i 2676	. . . . . 6 |- (( + seq1 F) ~~> x <-> ( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) ~~> x)
27	26	exbii 1083	. . . . 5 |- (E.x( + seq1 F) ~~> x <-> E.x( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) ~~> x)
28	23, 27	sylibr 198	. . . 4 |- ((A e. CC /\ A =/= 0) -> E.x( + seq1 F) ~~> x)
29	28	ex 371	. . 3 |- (A e. CC -> (A =/= 0 -> E.x( + seq1 F) ~~> x))
30		df-ne 1624	. . 3 |- (A =/= 0 <-> -. A = 0)
31	29, 30	syl5ibr 205	. 2 |- (A e. CC -> (-. A = 0 -> E.x( + seq1 F) ~~> x))
32	24	efcltlem2 7428	. . 3 |- (A = 0 -> ( + seq1 F) ~~> 0)
33		0cn 5417	. . . . 5 |- 0 e. CC
34	33	elisseti 1856	. . . 4 |- 0 e. V
35		breq2 2673	. . . 4 |- (x = 0 -> (( + seq1 F) ~~> x <-> ( + seq1 F) ~~> 0))
36	34, 35	cla4ev 1907	. . 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 127	1 |- (A e. CC -> E.x( + seq1 F) ~~> x)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221   = wceq 988   e. wcel 990  E.wex 1012   =/= wne 1622  ifcif 2406   class class class wbr 2669  {copab 2717  ` cfv 3237  (class class class)co 4039  CCcc 5321  0cc0 5323  1c1 5324   + caddc 5326   / cdiv 5383  NNcn 5385   seq1 cseq1 6600  ^cexp 6691  !cfa 7054   ~~> cli 7097

This theorem is referenced by:  dfef2i 7430  efseq0ex 7434

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-nul 2761  ax-pow 2794  ax-pr 2832  ax-un 2920  ax-inf2 4711

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3or 779  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-nel 1625  df-ral 1687  df-rex 1688  df-reu 1689  df-rab 1690  df-v 1850  df-sbc 1979  df-csb 2044  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-pss 2099  df-nul 2325  df-if 2407  df-pw 2447  df-sn 2457  df-pr 2458  df-tp 2460  df-op 2461  df-uni 2552  df-int 2582  df-iun 2616  df-br 2670  df-opab 2718  df-tr 2732  df-eprel 2886  df-id 2889  df-po 2894  df-so 2904  df-fr 2972  df-we 2989  df-ord 3006  df-on 3007  df-lim 3008  df-suc 3009  df-om 3193  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-fv 3253  df-rdg 4008  df-opr 4041  df-oprab 4042  df-1st 4157  df-2nd 4158  df-1o 4217  df-oadd 4219  df-omul 4220  df-er 4345  df-ec 4347  df-qs 4350  df-en 4455  df-dom 4456  df-sdom 4457  df-sup 4658  df-ni 5089  df-pli 5090  df-mi 5091  df-lti 5092  df-plpq 5124  df-mpq 5125  df-enq 5126  df-nq 5127  df-plq 5128  df-mq 5129  df-rq 5130  df-ltq 5131  df-1q 5132  df-np 5175  df-1p 5176  df-plp 5177  df-mp 5178  df-ltp 5179  df-plpr 5253  df-mpr 5254  df-enr 5255  df-nr 5256  df-plr 5257  df-mr 5258  df-ltr 5259  df-0r 5260  df-1r 5261  df-m1r 5262  df-c 5329  df-0 5330  df-1 5331  df-i 5332  df-r 5333  df-plus 5334  df-mul 5335  df-lt 5336  df-sub 5445  df-neg 5447  df-pnf 5576  df-mnf 5577  df-xr 5578  df-ltxr 5579  df-le 5580  df-div 5789  df-n 6012  df-2 6058  df-n0 6210  df-z 6246  df-fl 6363  df-uz 6478  df-fz 6528  df-seq1 6601  df-shft 6634  df-seqz 6656  df-seq0 6657  df-exp 6692  df-sqr 6793  df-re 6874  df-im 6875  df-cj 6876  df-abs 6877  df-fac 7055  df-clim 7098  df-sum 7103

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