Theorem etransc 37968

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 9658	. . . . 5 |- ( ( k e. ZZ /\ k =/= 0 ) -> 1 e. RR )
2		nn0abscl 13363	. . . . . . 7 |- ( k e. ZZ -> ( abs ` k ) e. NN0 )
3	2	nn0red 10926	. . . . . 6 |- ( k e. ZZ -> ( abs ` k ) e. RR )
4	3	adantr 466	. . . . 5 |- ( ( k e. ZZ /\ k =/= 0 ) -> ( abs ` k ) e. RR )
5		nnabscl 13376	. . . . . 6 |- ( ( k e. ZZ /\ k =/= 0 ) -> ( abs ` k ) e. NN )
6	5	nnge1d 10652	. . . . 5 |- ( ( k e. ZZ /\ k =/= 0 ) -> 1 <_ ( abs ` k ) )
7	1, 4, 6	lensymd 9786	. . . 4 |- ( ( k e. ZZ /\ k =/= 0 ) -> -. ( abs ` k ) < 1 )
8		nan 582	. . . 4 |- ( ( k e. ZZ -> -. ( k =/= 0 /\ ( abs ` k ) < 1 ) ) <-> ( ( k e. ZZ /\ k =/= 0 ) -> -. ( abs ` k ) < 1 ) )
9	7, 8	mpbir 212	. . 3 |- ( k e. ZZ -> -. ( k =/= 0 /\ ( abs ` k ) < 1 ) )
10	9	nrex 2880	. 2 |- -. E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 )
11		ere 14130	. . . . . . . 8 |- _e e. RR
12	11	recni 9655	. . . . . . 7 |- _e e. CC
13		neldif 3590	. . . . . . 7 |- ( ( _e e. CC /\ -. _e e. ( CC \ AA ) ) -> _e e. AA )
14	12, 13	mpan 674	. . . . . 6 |- ( -. _e e. ( CC \ AA ) -> _e e. AA )
15		ene0 14248	. . . . . . . 8 |- _e =/= 0
16		elsncg 4019	. . . . . . . . 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 2752	. . . . . . 7 |- -. _e e. { 0 }
19	18	a1i 11	. . . . . 6 |- ( -. _e e. ( CC \ AA ) -> -. _e e. { 0 } )
20	14, 19	eldifd 3447	. . . . 5 |- ( -. _e e. ( CC \ AA ) -> _e e. ( AA \ { 0 } ) )
21		elaa2 37918	. . . . 5 |- ( _e e. ( AA \ { 0 } ) <-> ( _e e. CC /\ E. q e. (Poly ` ZZ ) ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) )
22	20, 21	sylib 199	. . . 4 |- ( -. _e e. ( CC \ AA ) -> ( _e e. CC /\ E. q e. (Poly ` ZZ ) ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) )
23	22	simprd 464	. . 3 |- ( -. _e e. ( CC \ AA ) -> E. q e. (Poly ` ZZ ) ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) )
24		simpl 458	. . . . . . 7 |- ( ( q e. (Poly ` ZZ ) /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> q e. (Poly ` ZZ ) )
25		0nn0 10884	. . . . . . . . 9 |- 0 e. NN0
26		n0p 37236	. . . . . . . . 9 |- ( ( q e. (Poly ` ZZ ) /\ 0 e. NN0 /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> q =/= 0p )
27	25, 26	mp3an2 1348	. . . . . . . 8 |- ( ( q e. (Poly ` ZZ ) /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> q =/= 0p )
28		nelsn 37262	. . . . . . . 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 3447	. . . . . 6 |- ( ( q e. (Poly ` ZZ ) /\ ( (coeff ` q ) ` 0 ) =/= 0 ) -> q e. ( (Poly ` ZZ ) \ { 0p } ) )
31	30	adantrr 721	. . . . 5 |- ( ( q e. (Poly ` ZZ ) /\ ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) -> q e. ( (Poly ` ZZ ) \ { 0p } ) )
32		simprr 764	. . . . 5 |- ( ( q e. (Poly ` ZZ ) /\ ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) -> ( q ` _e ) = 0 )
33		eqid 2422	. . . . 5 |- (coeff ` q ) = (coeff ` q )
34		simprl 762	. . . . 5 |- ( ( q e. (Poly ` ZZ ) /\ ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) -> ( (coeff ` q ) ` 0 ) =/= 0 )
35		eqid 2422	. . . . 5 |- (deg ` q ) = (deg ` q )
36		eqid 2422	. . . . 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 2422	. . . . 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 5877	. . . . . . . . . . . . . . . . . . . 20 |- ( h = l -> ( (coeff ` q ) ` h ) = ( (coeff ` q ) ` l ) )
39		oveq2 6309	. . . . . . . . . . . . . . . . . . . 20 |- ( h = l -> ( _e ^c h ) = ( _e ^c l ) )
40	38, 39	oveq12d 6319	. . . . . . . . . . . . . . . . . . 19 |- ( h = l -> ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) = ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) )
41	40	fveq2d 5881	. . . . . . . . . . . . . . . . . 18 |- ( h = l -> ( abs ` ( ( (coeff ` q ) ` h ) x. ( _e ^c h ) ) ) = ( abs ` ( ( (coeff ` q ) ` l ) x. ( _e ^c l ) ) ) )
42	41	oveq1d 6316	. . . . . . . . . . . . . . . . 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 13749	. . . . . . . . . . . . . . . 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 6309	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) = ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) )
46		fveq2 5877	. . . . . . . . . . . . . . . 16 |- ( m = n -> ( ! ` m ) = ( ! ` n ) )
47	45, 46	oveq12d 6319	. . . . . . . . . . . . . . 15 |- ( m = n -> ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ m ) / ( ! ` m ) ) = ( ( ( (deg ` q ) ^ ( (deg ` q ) + 1 ) ) ^ n ) / ( ! ` n ) ) )
48	44, 47	oveq12d 6319	. . . . . . . . . . . . . 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 4513	. . . . . . . . . . . . 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 23	. . . . . . . . . . . 12 |- ( m = n -> m = n )
52	50, 51	fveq12d 5883	. . . . . . . . . . 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 5881	. . . . . . . . . 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 4430	. . . . . . . . 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 3055	. . . . . . . 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 5877	. . . . . . . . 9 |- ( j = i -> ( ZZ>= ` j ) = ( ZZ>= ` i ) )
57	56	raleqdv 3031	. . . . . . . 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 260	. . . . . . 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 3080	. . . . . 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 7996	. . . . 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 2422	. . . . 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 37967	. . . 4 |- ( ( q e. (Poly ` ZZ ) /\ ( ( (coeff ` q ) ` 0 ) =/= 0 /\ ( q ` _e ) = 0 ) ) -> E. k e. ZZ ( k =/= 0 /\ ( abs ` k ) < 1 ) )
63	62	rexlimiva 2913	. . 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 183	1 |- _e e. ( CC \ AA )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   e. wcel 1868   =/= wne 2618   A.wral 2775   E.wrex 2776   {crab 2779   \ cdif 3433   {csn 3996   {ctp 4000   class class class wbr 4420   |-> cmpt 4479   ` cfv 5597  (class class class)co 6301   supcsup 7956  infcinf 7957   CCcc 9537   RRcr 9538   0cc0 9539   1c1 9540   + caddc 9542   x. cmul 9544   RR*cxr 9674   < clt 9675   / cdiv 10269   NN0cn0 10869   ZZcz 10937   ZZ>=cuz 11159   ...cfz 11784   ^cexp 12271   !cfa 12458   abscabs 13285   sum_csu 13739   _eceu 14102   0pc0p 22613  Polycply 23124  coeffccoe 23126  degcdgr 23127   AAcaa 23253   ^c ccxp 23491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593  ax-inf2 8148  ax-cc 8865  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617  ax-addf 9618  ax-mulf 9619

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  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 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-iin 4299  df-disj 4392  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-se 4809  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-isom 5606  df-riota 6263  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-of 6541  df-ofr 6542  df-om 6703  df-1st 6803  df-2nd 6804  df-supp 6922  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-2o 7187  df-oadd 7190  df-omul 7191  df-er 7367  df-map 7478  df-pm 7479  df-ixp 7527  df-en 7574  df-dom 7575  df-sdom 7576  df-fin 7577  df-fsupp 7886  df-fi 7927  df-sup 7958  df-inf 7959  df-oi 8027  df-card 8374  df-acn 8377  df-cda 8598  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-ioc 11640  df-ico 11641  df-icc 11642  df-fz 11785  df-fzo 11916  df-fl 12027  df-mod 12096  df-seq 12213  df-exp 12272  df-fac 12459  df-bc 12487  df-hash 12515  df-shft 13118  df-cj 13150  df-re 13151  df-im 13152  df-sqrt 13286  df-abs 13287  df-limsup 13513  df-clim 13539  df-rlim 13540  df-sum 13740  df-prod 13947  df-ef 14108  df-e 14109  df-sin 14110  df-cos 14111  df-tan 14112  df-pi 14113  df-dvds 14293  df-gcd 14456  df-prm 14610  df-struct 15110  df-ndx 15111  df-slot 15112  df-base 15113  df-sets 15114  df-ress 15115  df-plusg 15190  df-mulr 15191  df-starv 15192  df-sca 15193  df-vsca 15194  df-ip 15195  df-tset 15196  df-ple 15197  df-ds 15199  df-unif 15200  df-hom 15201  df-cco 15202  df-rest 15308  df-topn 15309  df-0g 15327  df-gsum 15328  df-topgen 15329  df-pt 15330  df-prds 15333  df-xrs 15387  df-qtop 15393  df-imas 15394  df-xps 15397  df-mre 15479  df-mrc 15480  df-acs 15482  df-mgm 16475  df-sgrp 16514  df-mnd 16524  df-submnd 16570  df-mulg 16663  df-cntz 16958  df-cmn 17419  df-psmet 18949  df-xmet 18950  df-met 18951  df-bl 18952  df-mopn 18953  df-fbas 18954  df-fg 18955  df-cnfld 18958  df-top 19907  df-bases 19908  df-topon 19909  df-topsp 19910  df-cld 20020  df-ntr 20021  df-cls 20022  df-nei 20100  df-lp 20138  df-perf 20139  df-cn 20229  df-cnp 20230  df-haus 20317  df-cmp 20388  df-tx 20563  df-hmeo 20756  df-fil 20847  df-fm 20939  df-flim 20940  df-flf 20941  df-xms 21321  df-ms 21322  df-tms 21323  df-cncf 21896  df-ovol 22402  df-vol 22404  df-mbf 22563  df-itg1 22564  df-itg2 22565  df-ibl 22566  df-itg 22567  df-0p 22614  df-limc 22807  df-dv 22808  df-dvn 22809  df-ply 23128  df-coe 23130  df-dgr 23131  df-aa 23254  df-log 23492  df-cxp 23493

This theorem is referenced by: (None)