Step |
Hyp |
Ref |
Expression |
1 |
|
elnn0 |
|- ( J e. NN0 <-> ( J e. NN \/ J = 0 ) ) |
2 |
|
elnn0 |
|- ( K e. NN0 <-> ( K e. NN \/ K = 0 ) ) |
3 |
|
relexpmulnn |
|- ( ( ( R e. V /\ I = ( J x. K ) ) /\ ( J e. NN /\ K e. NN ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) |
4 |
3
|
3adantl3 |
|- ( ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) /\ ( J e. NN /\ K e. NN ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) |
5 |
4
|
expcom |
|- ( ( J e. NN /\ K e. NN ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) |
6 |
5
|
expcom |
|- ( K e. NN -> ( J e. NN -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) |
7 |
|
simprr |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> I = ( J x. K ) ) |
8 |
|
simpll |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> K = 0 ) |
9 |
8
|
oveq2d |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> ( J x. K ) = ( J x. 0 ) ) |
10 |
|
simplr |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> J e. NN ) |
11 |
10
|
nncnd |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> J e. CC ) |
12 |
11
|
mul01d |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> ( J x. 0 ) = 0 ) |
13 |
7 9 12
|
3eqtrd |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> I = 0 ) |
14 |
|
simpl |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> ( K = 0 /\ J e. NN ) ) |
15 |
|
nnnle0 |
|- ( J e. NN -> -. J <_ 0 ) |
16 |
15
|
adantl |
|- ( ( K = 0 /\ J e. NN ) -> -. J <_ 0 ) |
17 |
|
simpl |
|- ( ( K = 0 /\ J e. NN ) -> K = 0 ) |
18 |
17
|
breq2d |
|- ( ( K = 0 /\ J e. NN ) -> ( J <_ K <-> J <_ 0 ) ) |
19 |
16 18
|
mtbird |
|- ( ( K = 0 /\ J e. NN ) -> -. J <_ K ) |
20 |
14 19
|
syl |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> -. J <_ K ) |
21 |
13 20
|
jcnd |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> -. ( I = 0 -> J <_ K ) ) |
22 |
21
|
pm2.21d |
|- ( ( ( K = 0 /\ J e. NN ) /\ ( R e. V /\ I = ( J x. K ) ) ) -> ( ( I = 0 -> J <_ K ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) |
23 |
22
|
exp32 |
|- ( ( K = 0 /\ J e. NN ) -> ( R e. V -> ( I = ( J x. K ) -> ( ( I = 0 -> J <_ K ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) ) |
24 |
23
|
3impd |
|- ( ( K = 0 /\ J e. NN ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) |
25 |
24
|
ex |
|- ( K = 0 -> ( J e. NN -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) |
26 |
6 25
|
jaoi |
|- ( ( K e. NN \/ K = 0 ) -> ( J e. NN -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) |
27 |
2 26
|
sylbi |
|- ( K e. NN0 -> ( J e. NN -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) |
28 |
|
simplr |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> J = 0 ) |
29 |
28
|
oveq2d |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( R ^r J ) = ( R ^r 0 ) ) |
30 |
|
simpr1 |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> R e. V ) |
31 |
|
relexp0g |
|- ( R e. V -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) ) |
32 |
30 31
|
syl |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( R ^r 0 ) = ( _I |` ( dom R u. ran R ) ) ) |
33 |
29 32
|
eqtrd |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( R ^r J ) = ( _I |` ( dom R u. ran R ) ) ) |
34 |
33
|
oveq1d |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( ( R ^r J ) ^r K ) = ( ( _I |` ( dom R u. ran R ) ) ^r K ) ) |
35 |
|
dmexg |
|- ( R e. V -> dom R e. _V ) |
36 |
|
rnexg |
|- ( R e. V -> ran R e. _V ) |
37 |
|
unexg |
|- ( ( dom R e. _V /\ ran R e. _V ) -> ( dom R u. ran R ) e. _V ) |
38 |
35 36 37
|
syl2anc |
|- ( R e. V -> ( dom R u. ran R ) e. _V ) |
39 |
30 38
|
syl |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( dom R u. ran R ) e. _V ) |
40 |
|
simpll |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> K e. NN0 ) |
41 |
|
relexpiidm |
|- ( ( ( dom R u. ran R ) e. _V /\ K e. NN0 ) -> ( ( _I |` ( dom R u. ran R ) ) ^r K ) = ( _I |` ( dom R u. ran R ) ) ) |
42 |
39 40 41
|
syl2anc |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( ( _I |` ( dom R u. ran R ) ) ^r K ) = ( _I |` ( dom R u. ran R ) ) ) |
43 |
|
simpr2 |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> I = ( J x. K ) ) |
44 |
28
|
oveq1d |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( J x. K ) = ( 0 x. K ) ) |
45 |
40
|
nn0cnd |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> K e. CC ) |
46 |
45
|
mul02d |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( 0 x. K ) = 0 ) |
47 |
43 44 46
|
3eqtrd |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> I = 0 ) |
48 |
47
|
oveq2d |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( R ^r I ) = ( R ^r 0 ) ) |
49 |
48 32
|
eqtr2d |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( _I |` ( dom R u. ran R ) ) = ( R ^r I ) ) |
50 |
34 42 49
|
3eqtrd |
|- ( ( ( K e. NN0 /\ J = 0 ) /\ ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) |
51 |
50
|
ex |
|- ( ( K e. NN0 /\ J = 0 ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) |
52 |
51
|
ex |
|- ( K e. NN0 -> ( J = 0 -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) |
53 |
27 52
|
jaod |
|- ( K e. NN0 -> ( ( J e. NN \/ J = 0 ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) |
54 |
1 53
|
syl5bi |
|- ( K e. NN0 -> ( J e. NN0 -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) ) |
55 |
54
|
impcom |
|- ( ( J e. NN0 /\ K e. NN0 ) -> ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) ) |
56 |
55
|
impcom |
|- ( ( ( R e. V /\ I = ( J x. K ) /\ ( I = 0 -> J <_ K ) ) /\ ( J e. NN0 /\ K e. NN0 ) ) -> ( ( R ^r J ) ^r K ) = ( R ^r I ) ) |