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Theorem oev2 7173
Description: Alternate value of ordinal exponentiation. Compare oev 7164. (Contributed by NM, 2-Jan-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
oev2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  (
( _V  \  |^| A )  u.  |^| B ) ) )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem oev2
StepHypRef Expression
1 oveq12 6293 . . . . . 6  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( (/)  ^o  (/) ) )
2 oe0m0 7170 . . . . . 6  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2524 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  1o )
4 fveq2 5866 . . . . . . . 8  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) ) )
5 1on 7137 . . . . . . . . . 10  |-  1o  e.  On
65elexi 3123 . . . . . . . . 9  |-  1o  e.  _V
76rdg0 7087 . . . . . . . 8  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
84, 7syl6eq 2524 . . . . . . 7  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  1o )
9 inteq 4285 . . . . . . . 8  |-  ( B  =  (/)  ->  |^| B  =  |^| (/) )
10 int0 4296 . . . . . . . 8  |-  |^| (/)  =  _V
119, 10syl6eq 2524 . . . . . . 7  |-  ( B  =  (/)  ->  |^| B  =  _V )
128, 11ineq12d 3701 . . . . . 6  |-  ( B  =  (/)  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  ( 1o  i^i  _V )
)
13 inv1 3812 . . . . . . 7  |-  ( 1o 
i^i  _V )  =  1o
1413a1i 11 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
i^i  _V )  =  1o )
1512, 14sylan9eqr 2530 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  (
( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 B )  i^i  |^| B )  =  1o )
163, 15eqtr4d 2511 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B ) )
17 oveq1 6291 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
18 oe0m1 7171 . . . . . . . 8  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
1918biimpa 484 . . . . . . 7  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2017, 19sylan9eqr 2530 . . . . . 6  |-  ( ( ( B  e.  On  /\  (/)  e.  B )  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  (/) )
2120an32s 802 . . . . 5  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  (/) )
22 int0el 4313 . . . . . . . 8  |-  ( (/)  e.  B  ->  |^| B  =  (/) )
2322ineq2d 3700 . . . . . . 7  |-  ( (/)  e.  B  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 B )  i^i  (/) ) )
24 in0 3811 . . . . . . 7  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  (/) )  =  (/)
2523, 24syl6eq 2524 . . . . . 6  |-  ( (/)  e.  B  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  (/) )
2625adantl 466 . . . . 5  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B )  =  (/) )
2721, 26eqtr4d 2511 . . . 4  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
2816, 27oe0lem 7163 . . 3  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
29 inteq 4285 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3029, 10syl6eq 2524 . . . . . . . . 9  |-  ( A  =  (/)  ->  |^| A  =  _V )
3130difeq2d 3622 . . . . . . . 8  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  ( _V  \  _V ) )
32 difid 3895 . . . . . . . 8  |-  ( _V 
\  _V )  =  (/)
3331, 32syl6eq 2524 . . . . . . 7  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  (/) )
3433uneq2d 3658 . . . . . 6  |-  ( A  =  (/)  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  (/) ) )
35 uncom 3648 . . . . . 6  |-  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( ( _V 
\  |^| A )  u. 
|^| B )
36 un0 3810 . . . . . 6  |-  ( |^| B  u.  (/) )  = 
|^| B
3734, 35, 363eqtr3g 2531 . . . . 5  |-  ( A  =  (/)  ->  ( ( _V  \  |^| A
)  u.  |^| B
)  =  |^| B
)
3837adantl 466 . . . 4  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( ( _V  \  |^| A )  u.  |^| B )  =  |^| B )
3938ineq2d 3700 . . 3  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  ( ( _V  \  |^| A )  u.  |^| B ) )  =  ( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B ) )
4028, 39eqtr4d 2511 . 2  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  (
( _V  \  |^| A )  u.  |^| B ) ) )
41 oevn0 7165 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
42 int0el 4313 . . . . . . . . . 10  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
4342difeq2d 3622 . . . . . . . . 9  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  ( _V  \  (/) ) )
44 dif0 3897 . . . . . . . . 9  |-  ( _V 
\  (/) )  =  _V
4543, 44syl6eq 2524 . . . . . . . 8  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  _V )
4645uneq2d 3658 . . . . . . 7  |-  ( (/)  e.  A  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  _V ) )
47 unv 3813 . . . . . . 7  |-  ( |^| B  u.  _V )  =  _V
4846, 35, 473eqtr3g 2531 . . . . . 6  |-  ( (/)  e.  A  ->  ( ( _V  \  |^| A
)  u.  |^| B
)  =  _V )
4948adantl 466 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( ( _V 
\  |^| A )  u. 
|^| B )  =  _V )
5049ineq2d 3700 . . . 4  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  i^i  ( ( _V  \  |^| A )  u.  |^| B ) )  =  ( ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  i^i  _V )
)
51 inv1 3812 . . . 4  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  _V )  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)
5250, 51syl6req 2525 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  =  ( ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  i^i  ( ( _V  \  |^| A )  u.  |^| B ) ) )
5341, 52eqtrd 2508 . 2  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 B )  i^i  ( ( _V  \  |^| A )  u.  |^| B ) ) )
5440, 53oe0lem 7163 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  (
( _V  \  |^| A )  u.  |^| B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475   (/)c0 3785   |^|cint 4282    |-> cmpt 4505   Oncon0 4878   ` cfv 5588  (class class class)co 6284   reccrdg 7075   1oc1o 7123    .o comu 7128    ^o coe 7129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-recs 7042  df-rdg 7076  df-1o 7130  df-oexp 7136
This theorem is referenced by: (None)
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