Theorem etransc 38187

Description: _e is transcendental. Section *5 of [Juillerat] p. 11 can be used as a reference for this proof. (Contributed by Glauco Siliprandi, 5-Apr-2020.) (Proof shortened by AV, 28-Sep-2020.)

Assertion

Ref	Expression
etransc	|- _e e. ( CC \ AA )

Proof of Theorem etransc

Dummy variables h i l n q k j m are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1red 9684	. . . . 5 |- ( ( k e. ZZ /\ k =/= 0 ) -> 1 e. RR )
2		nn0abscl 13424	. . . . . . 7 |- ( k e. ZZ -> ( abs ` k ) e. NN0 )
3	2	nn0red 10955	. . . . . 6 |- ( k e. ZZ -> ( abs ` k ) e. RR )
4	3	adantr 471	. . . . 5 |- ( ( k e. ZZ /\ k =/= 0 ) -> ( abs ` k ) e. RR )
5		nnabscl 13437	. . . . . 6 |- ( ( k e. ZZ /\ k =/= 0 ) -> ( abs ` k ) e. NN )
6	5	nnge1d 10680	. . . . 5 |- ( ( k e. ZZ /\ k =/= 0 ) -> 1 <_ ( abs ` k ) )
7	1, 4, 6	lensymd 9812	. . . 4 |- ( ( k e. ZZ /\ k =/= 0 ) -> -. ( abs ` k ) < 1 )
8		nan 588	. . . 4 |- ( ( k e. ZZ -> -. ( k =/= 0 /\ ( abs ` k ) < 1 ) ) <-> ( ( k e. ZZ /\ k =/= 0 ) -> -. ( abs ` k ) < 1 ) )
9	7, 8	mpbir 214	. . 3 |- ( k e. ZZ -> -. ( k =/= 0 /\ ( abs ` k ) < 1 ) )
10	9	nrex 2854	. 2 |- -. E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 )
11		ere 14192	. . . . . . . 8 |- _e e. RR
12	11	recni 9681	. . . . . . 7 |- _e e. CC
13		neldif 3570	. . . . . . 7 |- ( ( _e e. CC /\ -. _e e. ( CC \ AA ) ) -> _e e. AA )
14	12, 13	mpan 681	. . . . . 6 |- ( -. _e e. ( CC \ AA ) -> _e e. AA )
15		ene0 14310	. . . . . . . 8 |- _e =/= 0
16		elsncg 4003	. . . . . . . . 9 |- ( _e e. CC -> ( _e e. { 0 } <-> _e = 0 ) )
17	12, 16	ax-mp 5	. . . . . . . 8 |- ( _e e. { 0 } <-> _e = 0 )
18	15, 17	nemtbir 2731	. . . . . . 7 |- -. _e e. { 0 }
19	18	a1i 11	. . . . . 6 |- ( -. _e e. ( CC \ AA ) -> -. _e e. { 0 } )
20	14, 19	eldifd 3427	. . . . 5 |- ( -. _e e. ( CC \ AA ) -> _e e. ( AA \ { 0 } ) )
21		elaa2 38137	. . . . 5 |- ( _e e. ( AA \ { 0 } ) <-> ( _e e. CC /\ E. q e. (Poly ` ZZ ) ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) )
22	20, 21	sylib 201	. . . 4 |- ( -. _e e. ( CC \ AA ) -> ( _e e. CC /\ E. q e. (Poly ` ZZ ) ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) )
23	22	simprd 469	. . 3 |- ( -. _e e. ( CC \ AA ) -> E. q e. (Poly ` ZZ ) ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) )
24		simpl 463	. . . . . . 7 |- ( ( q e. (Poly ` ZZ ) /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> q e. (Poly ` ZZ ) )
25		0nn0 10913	. . . . . . . . 9 |- 0 e. NN0
26		n0p 37414	. . . . . . . . 9 |- ( ( q e. (Poly ` ZZ ) /\ 0 e. NN0 /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> q =/= 0p )
27	25, 26	mp3an2 1361	. . . . . . . 8 |- ( ( q e. (Poly ` ZZ ) /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> q =/= 0p )
28		nelsn 4012	. . . . . . . 8 |- ( q =/= 0p -> -. q e. { 0p } )
29	27, 28	syl 17	. . . . . . 7 |- ( ( q e. (Poly ` ZZ ) /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> -. q e. { 0p } )
30	24, 29	eldifd 3427	. . . . . 6 |- ( ( q e. (Poly ` ZZ ) /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> q e. ( (Poly ` ZZ ) \ { 0p } ) )
31	30	adantrr 728	. . . . 5 |- ( ( q e. (Poly ` ZZ ) /\ ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) -> q e. ( (Poly ` ZZ ) \ { 0p } ) )
32		simprr 771	. . . . 5 |- ( ( q e. (Poly ` ZZ ) /\ ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) -> ( q ` _e ) = 0 )
33		eqid 2462	. . . . 5 |- (coeff ` q ) = (coeff ` q )
34		simprl 769	. . . . 5 |- ( ( q e. (Poly ` ZZ ) /\ ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) -> ( (coeff ` q ) ` 0 ) =/= 0 )
35		eqid 2462	. . . . 5 |- (deg ` q ) = (deg ` q )
36		eqid 2462	. . . . 5 |- sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) = sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) )
37		eqid 2462	. . . . 5 |- ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) = ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) )
38		fveq2 5888	. . . . . . . . . . . . . . . . . . . 20 |- ( h = l -> ( (coeff ` q ) ` h ) = ( (coeff ` q ) ` l ) )
39		oveq2 6323	. . . . . . . . . . . . . . . . . . . 20 |- ( h = l -> ( _e ^c h ) = ( _e ^c l ) )
40	38, 39	oveq12d 6333	. . . . . . . . . . . . . . . . . . 19 |- ( h = l -> ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) = ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) )
41	40	fveq2d 5892	. . . . . . . . . . . . . . . . . 18 |- ( h = l -> ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) = ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) )
42	41	oveq1d 6330	. . . . . . . . . . . . . . . . 17 |- ( h = l -> ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) = ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) )
43	42	cbvsumv 13811	. . . . . . . . . . . . . . . 16 |- sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) = sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) )
44	43	a1i 11	. . . . . . . . . . . . . . 15 |- ( m = n -> sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) = sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) )
45		oveq2 6323	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) = ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) )
46		fveq2 5888	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( ! ` m ) = ( ! ` n ) )
47	45, 46	oveq12d 6333	. . . . . . . . . . . . . . 15 |- ( m = n -> ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) = ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) )
48	44, 47	oveq12d 6333	. . . . . . . . . . . . . 14 |- ( m = n -> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) = ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) )
49	48	cbvmptv 4509	. . . . . . . . . . . . 13 |- ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) = ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) )
50	49	a1i 11	. . . . . . . . . . . 12 |- ( m = n -> ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) = ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) )
51		id 22	. . . . . . . . . . . 12 |- ( m = n -> m = n )
52	50, 51	fveq12d 5894	. . . . . . . . . . 11 |- ( m = n -> ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) = ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) )
53	52	fveq2d 5892	. . . . . . . . . 10 |- ( m = n -> ( abs ` ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) ) = ( abs ` ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) ) )
54	53	breq1d 4426	. . . . . . . . 9 |- ( m = n -> ( ( abs ` ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) ) < 1 <-> ( abs ` ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) ) < 1 ) )
55	54	cbvralv 3031	. . . . . . . 8 |- ( A. m e. ( ZZ>= ` j ) ( abs ` ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) ) < 1 <-> A. n e. ( ZZ>= ` j ) ( abs ` ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) ) < 1 )
56		fveq2 5888	. . . . . . . . 9 |- ( j = i -> ( ZZ>= ` j ) = ( ZZ>= ` i ) )
57	56	raleqdv 3005	. . . . . . . 8 |- ( j = i -> ( A. n e. ( ZZ>= ` j ) ( abs ` ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) ) < 1 <-> A. n e. ( ZZ>= ` i ) ( abs ` ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) ) < 1 ) )
58	55, 57	syl5bb 265	. . . . . . 7 |- ( j = i -> ( A. m e. ( ZZ>= ` j ) ( abs ` ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) ) < 1 <-> A. n e. ( ZZ>= ` i ) ( abs ` ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) ) < 1 ) )
59	58	cbvrabv 3056	. . . . . 6 |- { j e. NN0 | A. m e. ( ZZ>= ` j ) ( abs ` ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) ) < 1 } = { i e. NN0 | A. n e. ( ZZ>= ` i ) ( abs ` ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) ) < 1 }
60	59	infeq1i 8020	. . . . 5 |- inf ( { j e. NN0 | A. m e. ( ZZ>= ` j ) ( abs ` ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) ) < 1 } , RR , < ) = inf ( { i e. NN0 | A. n e. ( ZZ>= ` i ) ( abs ` ( ( n e. NN0 |-> ( sum_ l e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) ) ) ` n ) ) < 1 } , RR , < )
61		eqid 2462	. . . . 5 |- sup ( { ( abs ` ( (coeff ` q ) ` 0 ) ) , ( ! ` (deg ` q ) ) , inf ( { j e. NN0 | A. m e. ( ZZ>= ` j ) ( abs ` ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) ) < 1 } , RR , < ) } , RR* , < ) = sup ( { ( abs ` ( (coeff ` q ) ` 0 ) ) , ( ! ` (deg ` q ) ) , inf ( { j e. NN0 | A. m e. ( ZZ>= ` j ) ( abs ` ( ( m e. NN0 |-> ( sum_ h e. ( 0 ... (deg ` q ) ) ( ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) x. ( (deg ` q ) x. ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ) ) x. ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) ) ) ` m ) ) < 1 } , RR , < ) } , RR* , < )
62	31, 32, 33, 34, 35, 36, 37, 60, 61	etransclem48 38186	. . . 4 |- ( ( q e. (Poly ` ZZ ) /\ ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) -> E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 ) )
63	62	rexlimiva 2887	. . 3 |- ( E. q e. (Poly ` ZZ ) ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) -> E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 ) )
64	23, 63	syl 17	. 2 |- ( -. _e e. ( CC \ AA ) -> E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 ) )
65	10, 64	mt3 185	1 |- _e e. ( CC \ AA )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1455   e. wcel 1898   =/= wne 2633   A.wral 2749   E.wrex 2750   {crab 2753   \ cdif 3413   {csn 3980   {ctp 3984   class class class wbr 4416   |-> cmpt 4475   ` cfv 5601  (class class class)co 6315   supcsup 7980  infcinf 7981   CCcc 9563   RRcr 9564   0cc0 9565   1c1 9566   + caddc 9568   x. cmul 9570   RR*cxr 9700   < clt 9701   / cdiv 10297   NN0cn0 10898   ZZcz 10966   ZZ>=cuz 11188   ...cfz 11813   ^cexp 12304   !cfa 12491   abscabs 13346   sum_csu 13801   _eceu 14164   0pc0p 22676  Polycply 23187  coeffccoe 23189  degcdgr 23190   AAcaa 23316   ^c ccxp 23554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-8 1900  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-rep 4529  ax-sep 4539  ax-nul 4548  ax-pow 4595  ax-pr 4653  ax-un 6610  ax-inf2 8172  ax-cc 8891  ax-cnex 9621  ax-resscn 9622  ax-1cn 9623  ax-icn 9624  ax-addcl 9625  ax-addrcl 9626  ax-mulcl 9627  ax-mulrcl 9628  ax-mulcom 9629  ax-addass 9630  ax-mulass 9631  ax-distr 9632  ax-i2m1 9633  ax-1ne0 9634  ax-1rid 9635  ax-rnegex 9636  ax-rrecex 9637  ax-cnre 9638  ax-pre-lttri 9639  ax-pre-lttrn 9640  ax-pre-ltadd 9641  ax-pre-mulgt0 9642  ax-pre-sup 9643  ax-addf 9644  ax-mulf 9645

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1458  df-fal 1461  df-ex 1675  df-nf 1679  df-sb 1809  df-eu 2314  df-mo 2315  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-nel 2636  df-ral 2754  df-rex 2755  df-reu 2756  df-rmo 2757  df-rab 2758  df-v 3059  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-pss 3432  df-nul 3744  df-if 3894  df-pw 3965  df-sn 3981  df-pr 3983  df-tp 3985  df-op 3987  df-uni 4213  df-int 4249  df-iun 4294  df-iin 4295  df-disj 4388  df-br 4417  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-se 4813  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-isom 5610  df-riota 6277  df-ov 6318  df-oprab 6319  df-mpt2 6320  df-of 6558  df-ofr 6559  df-om 6720  df-1st 6820  df-2nd 6821  df-supp 6942  df-wrecs 7054  df-recs 7116  df-rdg 7154  df-1o 7208  df-2o 7209  df-oadd 7212  df-omul 7213  df-er 7389  df-map 7500  df-pm 7501  df-ixp 7549  df-en 7596  df-dom 7597  df-sdom 7598  df-fin 7599  df-fsupp 7910  df-fi 7951  df-sup 7982  df-inf 7983  df-oi 8051  df-card 8399  df-acn 8402  df-cda 8624  df-pnf 9703  df-mnf 9704  df-xr 9705  df-ltxr 9706  df-le 9707  df-sub 9888  df-neg 9889  df-div 10298  df-nn 10638  df-2 10696  df-3 10697  df-4 10698  df-5 10699  df-6 10700  df-7 10701  df-8 10702  df-9 10703  df-10 10704  df-n0 10899  df-z 10967  df-dec 11081  df-uz 11189  df-q 11294  df-rp 11332  df-xneg 11438  df-xadd 11439  df-xmul 11440  df-ioo 11668  df-ioc 11669  df-ico 11670  df-icc 11671  df-fz 11814  df-fzo 11947  df-fl 12060  df-mod 12129  df-seq 12246  df-exp 12305  df-fac 12492  df-bc 12520  df-hash 12548  df-shft 13179  df-cj 13211  df-re 13212  df-im 13213  df-sqrt 13347  df-abs 13348  df-limsup 13575  df-clim 13601  df-rlim 13602  df-sum 13802  df-prod 14009  df-ef 14170  df-e 14171  df-sin 14172  df-cos 14173  df-tan 14174  df-pi 14175  df-dvds 14355  df-gcd 14518  df-prm 14672  df-struct 15172  df-ndx 15173  df-slot 15174  df-base 15175  df-sets 15176  df-ress 15177  df-plusg 15252  df-mulr 15253  df-starv 15254  df-sca 15255  df-vsca 15256  df-ip 15257  df-tset 15258  df-ple 15259  df-ds 15261  df-unif 15262  df-hom 15263  df-cco 15264  df-rest 15370  df-topn 15371  df-0g 15389  df-gsum 15390  df-topgen 15391  df-pt 15392  df-prds 15395  df-xrs 15449  df-qtop 15455  df-imas 15456  df-xps 15459  df-mre 15541  df-mrc 15542  df-acs 15544  df-mgm 16537  df-sgrp 16576  df-mnd 16586  df-submnd 16632  df-mulg 16725  df-cntz 17020  df-cmn 17481  df-psmet 19011  df-xmet 19012  df-met 19013  df-bl 19014  df-mopn 19015  df-fbas 19016  df-fg 19017  df-cnfld 19020  df-top 19970  df-bases 19971  df-topon 19972  df-topsp 19973  df-cld 20083  df-ntr 20084  df-cls 20085  df-nei 20163  df-lp 20201  df-perf 20202  df-cn 20292  df-cnp 20293  df-haus 20380  df-cmp 20451  df-tx 20626  df-hmeo 20819  df-fil 20910  df-fm 21002  df-flim 21003  df-flf 21004  df-xms 21384  df-ms 21385  df-tms 21386  df-cncf 21959  df-ovol 22465  df-vol 22467  df-mbf 22626  df-itg1 22627  df-itg2 22628  df-ibl 22629  df-itg 22630  df-0p 22677  df-limc 22870  df-dv 22871  df-dvn 22872  df-ply 23191  df-coe 23193  df-dgr 23194  df-aa 23317  df-log 23555  df-cxp 23556

This theorem is referenced by: (None)