Theorem efseq1ex 7256

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 3959	. . . . . . . . . . . . 13 |- (A = if((A e. CC /\ A =/= 0), A, 1) -> (A^y) = (if((A e. CC /\ A =/= 0), A, 1)^y))
2	1	opreq1d 3966	. . . . . . . . . . . 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 1483	. . . . . . . . . . 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 615	. . . . . . . . . 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 2665	. . . . . . . . 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 3967	. . . . . . . 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 2624	. . . . . . 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 1277	. . . . . 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 1473	. . . . . . 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 1531	. . . . . . . . . 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 1587	. . . . . . . . . 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 627	. . . . . . . . 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 1531	. . . . . . . . . 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 1587	. . . . . . . . . 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 627	. . . . . . . . 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 5249	. . . . . . . . . 10 |- 1 e. CC
17		ax1ne0 5260	. . . . . . . . . 10 |- 1 =/= 0
18	16, 17	pm3.2i 285	. . . . . . . . 9 |- (1 e. CC /\ 1 =/= 0)
19	12, 15, 18	elimhyp 2386	. . . . . . . 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 7254	. . . . . 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 2379	. . . . 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 3963	. . . . . . 7 |- ( + seq1 F) = ( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))})
26	25	breq1i 2621	. . . . . 6 |- (( + seq1 F) ~~> x <-> ( + seq1 {<.y, z>. | (y e. NN /\ z = ((A^y) / (!` y)))}) ~~> x)
27	26	exbii 1049	. . . . 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 1584	. . 3 |- (A =/= 0 <-> -. A = 0)
31	29, 30	syl5ibr 207	. 2 |- (A e. CC -> (-. A = 0 -> E.x( + seq1 F) ~~> x))
32	24	efcltlem2 7255	. . 3 |- (A = 0 -> ( + seq1 F) ~~> 0)
33		0cn 5308	. . . . 5 |- 0 e. CC
34	33	elisseti 1814	. . . 4 |- 0 e. V
35		breq2 2618	. . . 4 |- (x = 0 -> (( + seq1 F) ~~> x <-> ( + seq1 F) ~~> 0))
36	34, 35	cla4ev 1865	. . 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 954   e. wcel 956  E.wex 978   =/= wne 1582  ifcif 2357   class class class wbr 2614  {copab 2661  ` cfv 3177  (class class class)co 3954  CCcc 5212  0cc0 5214  1c1 5215   + caddc 5217   / cdiv 5274  NNcn 5276   seq1 cseq1 6252  ^cexp 6508  !cfa 6876   ~~> cli 6920

This theorem is referenced by:  dfef2 7257  efseq0ex 7261

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861  ax-inf2 4605

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-nel 1585  df-ral 1646  df-rex 1647  df-reu 1648  df-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-int 2529  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-om 3127  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-1st 4069  df-2nd 4070  df-1o 4123  df-oadd 4125  df-omul 4126  df-er 4251  df-ec 4253  df-qs 4256  df-en 4357  df-dom 4358  df-sdom 4359  df-sup 4554  df-ni 4980  df-pli 4981  df-mi 4982  df-lti 4983  df-plpq 5015  df-mpq 5016  df-enq 5017  df-nq 5018  df-plq 5019  df-mq 5020  df-rq 5021  df-ltq 5022  df-1q 5023  df-np 5066  df-1p 5067  df-plp 5068  df-mp 5069  df-ltp 5070  df-plpr 5144  df-mpr 5145  df-enr 5146  df-nr 5147  df-plr 5148  df-mr 5149  df-ltr 5150  df-0r 5151  df-1r 5152  df-m1r 5153  df-c 5220  df-0 5221  df-1 5222  df-i 5223  df-r 5224  df-plus 5225  df-mul 5226  df-lt 5227  df-sub 5336  df-neg 5338  df-pnf 5467  df-mnf 5468  df-xr 5469  df-ltxr 5470  df-le 5471  df-div 5680  df-n 5881  df-2 5925  df-n0 6055  df-z 6091  df-fl 6180  df-seq1 6253  df-shft 6286  df-uz 6358  df-fz 6408  df-seqz 6473  df-seq0 6474  df-exp 6509  df-sqr 6608  df-re 6690  df-im 6691  df-cj 6692  df-abs 6693  df-fac 6877  df-clim 6921  df-sum 6926

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