Step |
Hyp |
Ref |
Expression |
1 |
|
expcllem.1 |
|- F C_ CC |
2 |
|
expcllem.2 |
|- ( ( x e. F /\ y e. F ) -> ( x x. y ) e. F ) |
3 |
|
expcllem.3 |
|- 1 e. F |
4 |
|
expcl2lem.4 |
|- ( ( x e. F /\ x =/= 0 ) -> ( 1 / x ) e. F ) |
5 |
|
elznn0nn |
|- ( B e. ZZ <-> ( B e. NN0 \/ ( B e. RR /\ -u B e. NN ) ) ) |
6 |
1 2 3
|
expcllem |
|- ( ( A e. F /\ B e. NN0 ) -> ( A ^ B ) e. F ) |
7 |
6
|
ex |
|- ( A e. F -> ( B e. NN0 -> ( A ^ B ) e. F ) ) |
8 |
7
|
adantr |
|- ( ( A e. F /\ A =/= 0 ) -> ( B e. NN0 -> ( A ^ B ) e. F ) ) |
9 |
|
simpll |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. F ) |
10 |
1 9
|
sselid |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. CC ) |
11 |
|
simprl |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. RR ) |
12 |
11
|
recnd |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> B e. CC ) |
13 |
|
nnnn0 |
|- ( -u B e. NN -> -u B e. NN0 ) |
14 |
13
|
ad2antll |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> -u B e. NN0 ) |
15 |
|
expneg2 |
|- ( ( A e. CC /\ B e. CC /\ -u B e. NN0 ) -> ( A ^ B ) = ( 1 / ( A ^ -u B ) ) ) |
16 |
10 12 14 15
|
syl3anc |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) = ( 1 / ( A ^ -u B ) ) ) |
17 |
|
difss |
|- ( F \ { 0 } ) C_ F |
18 |
|
simpl |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A e. F /\ A =/= 0 ) ) |
19 |
|
eldifsn |
|- ( A e. ( F \ { 0 } ) <-> ( A e. F /\ A =/= 0 ) ) |
20 |
18 19
|
sylibr |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> A e. ( F \ { 0 } ) ) |
21 |
17 1
|
sstri |
|- ( F \ { 0 } ) C_ CC |
22 |
17
|
sseli |
|- ( x e. ( F \ { 0 } ) -> x e. F ) |
23 |
17
|
sseli |
|- ( y e. ( F \ { 0 } ) -> y e. F ) |
24 |
22 23 2
|
syl2an |
|- ( ( x e. ( F \ { 0 } ) /\ y e. ( F \ { 0 } ) ) -> ( x x. y ) e. F ) |
25 |
|
eldifsn |
|- ( x e. ( F \ { 0 } ) <-> ( x e. F /\ x =/= 0 ) ) |
26 |
1
|
sseli |
|- ( x e. F -> x e. CC ) |
27 |
26
|
anim1i |
|- ( ( x e. F /\ x =/= 0 ) -> ( x e. CC /\ x =/= 0 ) ) |
28 |
25 27
|
sylbi |
|- ( x e. ( F \ { 0 } ) -> ( x e. CC /\ x =/= 0 ) ) |
29 |
|
eldifsn |
|- ( y e. ( F \ { 0 } ) <-> ( y e. F /\ y =/= 0 ) ) |
30 |
1
|
sseli |
|- ( y e. F -> y e. CC ) |
31 |
30
|
anim1i |
|- ( ( y e. F /\ y =/= 0 ) -> ( y e. CC /\ y =/= 0 ) ) |
32 |
29 31
|
sylbi |
|- ( y e. ( F \ { 0 } ) -> ( y e. CC /\ y =/= 0 ) ) |
33 |
|
mulne0 |
|- ( ( ( x e. CC /\ x =/= 0 ) /\ ( y e. CC /\ y =/= 0 ) ) -> ( x x. y ) =/= 0 ) |
34 |
28 32 33
|
syl2an |
|- ( ( x e. ( F \ { 0 } ) /\ y e. ( F \ { 0 } ) ) -> ( x x. y ) =/= 0 ) |
35 |
|
eldifsn |
|- ( ( x x. y ) e. ( F \ { 0 } ) <-> ( ( x x. y ) e. F /\ ( x x. y ) =/= 0 ) ) |
36 |
24 34 35
|
sylanbrc |
|- ( ( x e. ( F \ { 0 } ) /\ y e. ( F \ { 0 } ) ) -> ( x x. y ) e. ( F \ { 0 } ) ) |
37 |
|
ax-1ne0 |
|- 1 =/= 0 |
38 |
|
eldifsn |
|- ( 1 e. ( F \ { 0 } ) <-> ( 1 e. F /\ 1 =/= 0 ) ) |
39 |
3 37 38
|
mpbir2an |
|- 1 e. ( F \ { 0 } ) |
40 |
21 36 39
|
expcllem |
|- ( ( A e. ( F \ { 0 } ) /\ -u B e. NN0 ) -> ( A ^ -u B ) e. ( F \ { 0 } ) ) |
41 |
20 14 40
|
syl2anc |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) e. ( F \ { 0 } ) ) |
42 |
17 41
|
sselid |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) e. F ) |
43 |
|
eldifsn |
|- ( ( A ^ -u B ) e. ( F \ { 0 } ) <-> ( ( A ^ -u B ) e. F /\ ( A ^ -u B ) =/= 0 ) ) |
44 |
41 43
|
sylib |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( ( A ^ -u B ) e. F /\ ( A ^ -u B ) =/= 0 ) ) |
45 |
44
|
simprd |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ -u B ) =/= 0 ) |
46 |
|
neeq1 |
|- ( x = ( A ^ -u B ) -> ( x =/= 0 <-> ( A ^ -u B ) =/= 0 ) ) |
47 |
|
oveq2 |
|- ( x = ( A ^ -u B ) -> ( 1 / x ) = ( 1 / ( A ^ -u B ) ) ) |
48 |
47
|
eleq1d |
|- ( x = ( A ^ -u B ) -> ( ( 1 / x ) e. F <-> ( 1 / ( A ^ -u B ) ) e. F ) ) |
49 |
46 48
|
imbi12d |
|- ( x = ( A ^ -u B ) -> ( ( x =/= 0 -> ( 1 / x ) e. F ) <-> ( ( A ^ -u B ) =/= 0 -> ( 1 / ( A ^ -u B ) ) e. F ) ) ) |
50 |
4
|
ex |
|- ( x e. F -> ( x =/= 0 -> ( 1 / x ) e. F ) ) |
51 |
49 50
|
vtoclga |
|- ( ( A ^ -u B ) e. F -> ( ( A ^ -u B ) =/= 0 -> ( 1 / ( A ^ -u B ) ) e. F ) ) |
52 |
42 45 51
|
sylc |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( 1 / ( A ^ -u B ) ) e. F ) |
53 |
16 52
|
eqeltrd |
|- ( ( ( A e. F /\ A =/= 0 ) /\ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) e. F ) |
54 |
53
|
ex |
|- ( ( A e. F /\ A =/= 0 ) -> ( ( B e. RR /\ -u B e. NN ) -> ( A ^ B ) e. F ) ) |
55 |
8 54
|
jaod |
|- ( ( A e. F /\ A =/= 0 ) -> ( ( B e. NN0 \/ ( B e. RR /\ -u B e. NN ) ) -> ( A ^ B ) e. F ) ) |
56 |
5 55
|
syl5bi |
|- ( ( A e. F /\ A =/= 0 ) -> ( B e. ZZ -> ( A ^ B ) e. F ) ) |
57 |
56
|
3impia |
|- ( ( A e. F /\ A =/= 0 /\ B e. ZZ ) -> ( A ^ B ) e. F ) |