Theorem ef0lem 14133

Description: The series defining the exponential function converges in the (trivial) case of a zero argument. (Contributed by Steve Rodriguez, 7-Jun-2006.) (Revised by Mario Carneiro, 28-Apr-2014.)

Hypothesis

Ref	Expression
eftval.1	|- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )

Assertion

Ref	Expression
ef0lem	|- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Distinct variable group:   A, n

Allowed substitution hint:   F( n)

Proof of Theorem ef0lem

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 462	. . . . . 6 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. ( ZZ>= ` 0 ) )
2		nn0uz 11201	. . . . . 6 |- NN0 = ( ZZ>= ` 0 )
3	1, 2	syl6eleqr 2518	. . . . 5 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> k e. NN0 )
4		elnn0 10879	. . . . 5 |- ( k e. NN0 <-> ( k e. NN \/ k = 0 ) )
5	3, 4	sylib 199	. . . 4 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( k e. NN \/ k = 0 ) )
6		nnnn0 10884	. . . . . . . . 9 |- ( k e. NN -> k e. NN0 )
7	6	adantl 467	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> k e. NN0 )
8		eftval.1	. . . . . . . . 9 |- F = ( n e. NN0 |-> ( ( A ^ n ) / ( ! ` n ) ) )
9	8	eftval 14131	. . . . . . . 8 |- ( k e. NN0 -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
10	7, 9	syl 17	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = ( ( A ^ k ) / ( ! ` k ) ) )
11		oveq1 6313	. . . . . . . . 9 |- ( A = 0 -> ( A ^ k ) = ( 0 ^ k ) )
12		0exp 12314	. . . . . . . . 9 |- ( k e. NN -> ( 0 ^ k ) = 0 )
13	11, 12	sylan9eq 2483	. . . . . . . 8 |- ( ( A = 0 /\ k e. NN ) -> ( A ^ k ) = 0 )
14	13	oveq1d 6321	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( ( A ^ k ) / ( ! ` k ) ) = ( 0 / ( ! ` k ) ) )
15		faccl 12476	. . . . . . . 8 |- ( k e. NN0 -> ( ! ` k ) e. NN )
16		nncn 10625	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) e. CC )
17		nnne0 10650	. . . . . . . . 9 |- ( ( ! ` k ) e. NN -> ( ! ` k ) =/= 0 )
18	16, 17	div0d 10390	. . . . . . . 8 |- ( ( ! ` k ) e. NN -> ( 0 / ( ! ` k ) ) = 0 )
19	7, 15, 18	3syl 18	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> ( 0 / ( ! ` k ) ) = 0 )
20	10, 14, 19	3eqtrd 2467	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = 0 )
21		nnne0 10650	. . . . . . . . 9 |- ( k e. NN -> k =/= 0 )
22		elsn 4012	. . . . . . . . . 10 |- ( k e. { 0 } <-> k = 0 )
23	22	necon3bbii 2681	. . . . . . . . 9 |- ( -. k e. { 0 } <-> k =/= 0 )
24	21, 23	sylibr 215	. . . . . . . 8 |- ( k e. NN -> -. k e. { 0 } )
25	24	adantl 467	. . . . . . 7 |- ( ( A = 0 /\ k e. NN ) -> -. k e. { 0 } )
26	25	iffalsed 3922	. . . . . 6 |- ( ( A = 0 /\ k e. NN ) -> if ( k e. { 0 } , 1 , 0 ) = 0 )
27	20, 26	eqtr4d 2466	. . . . 5 |- ( ( A = 0 /\ k e. NN ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
28		fveq2 5882	. . . . . . 7 |- ( k = 0 -> ( F ` k ) = ( F ` 0 ) )
29		oveq1 6313	. . . . . . . . . 10 |- ( A = 0 -> ( A ^ 0 ) = ( 0 ^ 0 ) )
30		0exp0e1 12284	. . . . . . . . . 10 |- ( 0 ^ 0 ) = 1
31	29, 30	syl6eq 2479	. . . . . . . . 9 |- ( A = 0 -> ( A ^ 0 ) = 1 )
32	31	oveq1d 6321	. . . . . . . 8 |- ( A = 0 -> ( ( A ^ 0 ) / ( ! ` 0 ) ) = ( 1 / ( ! ` 0 ) ) )
33		0nn0 10892	. . . . . . . . 9 |- 0 e. NN0
34	8	eftval 14131	. . . . . . . . 9 |- ( 0 e. NN0 -> ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) ) )
35	33, 34	ax-mp 5	. . . . . . . 8 |- ( F ` 0 ) = ( ( A ^ 0 ) / ( ! ` 0 ) )
36		fac0 12469	. . . . . . . . . 10 |- ( ! ` 0 ) = 1
37	36	oveq2i 6317	. . . . . . . . 9 |- ( 1 / ( ! ` 0 ) ) = ( 1 / 1 )
38		1div1e1 10308	. . . . . . . . 9 |- ( 1 / 1 ) = 1
39	37, 38	eqtr2i 2452	. . . . . . . 8 |- 1 = ( 1 / ( ! ` 0 ) )
40	32, 35, 39	3eqtr4g 2488	. . . . . . 7 |- ( A = 0 -> ( F ` 0 ) = 1 )
41	28, 40	sylan9eqr 2485	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = 1 )
42		simpr 462	. . . . . . . 8 |- ( ( A = 0 /\ k = 0 ) -> k = 0 )
43	42, 22	sylibr 215	. . . . . . 7 |- ( ( A = 0 /\ k = 0 ) -> k e. { 0 } )
44	43	iftrued 3919	. . . . . 6 |- ( ( A = 0 /\ k = 0 ) -> if ( k e. { 0 } , 1 , 0 ) = 1 )
45	41, 44	eqtr4d 2466	. . . . 5 |- ( ( A = 0 /\ k = 0 ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
46	27, 45	jaodan 792	. . . 4 |- ( ( A = 0 /\ ( k e. NN \/ k = 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
47	5, 46	syldan 472	. . 3 |- ( ( A = 0 /\ k e. ( ZZ>= ` 0 ) ) -> ( F ` k ) = if ( k e. { 0 } , 1 , 0 ) )
48	33, 2	eleqtri 2505	. . . 4 |- 0 e. ( ZZ>= ` 0 )
49	48	a1i 11	. . 3 |- ( A = 0 -> 0 e. ( ZZ>= ` 0 ) )
50		1cnd 9667	. . 3 |- ( ( A = 0 /\ k e. { 0 } ) -> 1 e. CC )
51		0z 10956	. . . . . 6 |- 0 e. ZZ
52		fzsn 11848	. . . . . 6 |- ( 0 e. ZZ -> ( 0 ... 0 ) = { 0 } )
53	51, 52	ax-mp 5	. . . . 5 |- ( 0 ... 0 ) = { 0 }
54	53	eqimss2i 3519	. . . 4 |- { 0 } C_ ( 0 ... 0 )
55	54	a1i 11	. . 3 |- ( A = 0 -> { 0 } C_ ( 0 ... 0 ) )
56	47, 49, 50, 55	fsumcvg2 13793	. 2 |- ( A = 0 -> seq 0 ( + , F ) ~~> ( seq 0 ( + , F ) ` 0 ) )
57	51, 40	seq1i 12234	. 2 |- ( A = 0 -> ( seq 0 ( + , F ) ` 0 ) = 1 )
58	56, 57	breqtrd 4448	1 |- ( A = 0 -> seq 0 ( + , F ) ~~> 1 )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 369   /\ wa 370   = wceq 1437   e. wcel 1872   =/= wne 2614   C_ wss 3436   ifcif 3911   {csn 3998   class class class wbr 4423   |-> cmpt 4482   ` cfv 5601  (class class class)co 6306   0cc0 9547   1c1 9548   + caddc 9550   / cdiv 10277   NNcn 10617   NN0cn0 10877   ZZcz 10945   ZZ>=cuz 11167   ...cfz 11792   seqcseq 12220   ^cexp 12279   !cfa 12466   ~~> cli 13548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-inf2 8156  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-1st 6808  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-div 10278  df-nn 10618  df-2 10676  df-n0 10878  df-z 10946  df-uz 11168  df-rp 11311  df-fz 11793  df-seq 12221  df-exp 12280  df-fac 12467  df-cj 13163  df-re 13164  df-im 13165  df-sqrt 13299  df-abs 13300  df-clim 13552

This theorem is referenced by:  ef0  14145