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Theorem oev2 6538
Description: Alternate value of ordinal exponentiation. Compare oev 6529. (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 5883 . . . . . 6  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( (/)  ^o  (/) ) )
2 oe0m0 6535 . . . . . 6  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2344 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  1o )
4 fveq2 5541 . . . . . . . 8  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) ) )
5 1on 6502 . . . . . . . . . 10  |-  1o  e.  On
65elexi 2810 . . . . . . . . 9  |-  1o  e.  _V
76rdg0 6450 . . . . . . . 8  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
84, 7syl6eq 2344 . . . . . . 7  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  1o )
9 inteq 3881 . . . . . . . 8  |-  ( B  =  (/)  ->  |^| B  =  |^| (/) )
10 int0 3892 . . . . . . . 8  |-  |^| (/)  =  _V
119, 10syl6eq 2344 . . . . . . 7  |-  ( B  =  (/)  ->  |^| B  =  _V )
128, 11ineq12d 3384 . . . . . 6  |-  ( B  =  (/)  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  ( 1o  i^i  _V )
)
13 inv1 3494 . . . . . . 7  |-  ( 1o 
i^i  _V )  =  1o
1413a1i 10 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
i^i  _V )  =  1o )
1512, 14sylan9eqr 2350 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  (
( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 B )  i^i  |^| B )  =  1o )
163, 15eqtr4d 2331 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B ) )
17 oveq1 5881 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
18 oe0m1 6536 . . . . . . . 8  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
1918biimpa 470 . . . . . . 7  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2017, 19sylan9eqr 2350 . . . . . 6  |-  ( ( ( B  e.  On  /\  (/)  e.  B )  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  (/) )
2120an32s 779 . . . . 5  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  (/) )
22 int0el 3909 . . . . . . . 8  |-  ( (/)  e.  B  ->  |^| B  =  (/) )
2322ineq2d 3383 . . . . . . 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 3493 . . . . . . 7  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  (/) )  =  (/)
2523, 24syl6eq 2344 . . . . . 6  |-  ( (/)  e.  B  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  (/) )
2625adantl 452 . . . . 5  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B )  =  (/) )
2721, 26eqtr4d 2331 . . . 4  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
2816, 27oe0lem 6528 . . 3  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
29 inteq 3881 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3029, 10syl6eq 2344 . . . . . . . . 9  |-  ( A  =  (/)  ->  |^| A  =  _V )
3130difeq2d 3307 . . . . . . . 8  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  ( _V  \  _V ) )
32 difid 3535 . . . . . . . 8  |-  ( _V 
\  _V )  =  (/)
3331, 32syl6eq 2344 . . . . . . 7  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  (/) )
3433uneq2d 3342 . . . . . 6  |-  ( A  =  (/)  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  (/) ) )
35 uncom 3332 . . . . . 6  |-  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( ( _V 
\  |^| A )  u. 
|^| B )
36 un0 3492 . . . . . 6  |-  ( |^| B  u.  (/) )  = 
|^| B
3734, 35, 363eqtr3g 2351 . . . . 5  |-  ( A  =  (/)  ->  ( ( _V  \  |^| A
)  u.  |^| B
)  =  |^| B
)
3837adantl 452 . . . 4  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( ( _V  \  |^| A )  u.  |^| B )  =  |^| B )
3938ineq2d 3383 . . 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 2331 . 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 6530 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
42 int0el 3909 . . . . . . . . . 10  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
4342difeq2d 3307 . . . . . . . . 9  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  ( _V  \  (/) ) )
44 dif0 3537 . . . . . . . . 9  |-  ( _V 
\  (/) )  =  _V
4543, 44syl6eq 2344 . . . . . . . 8  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  _V )
4645uneq2d 3342 . . . . . . 7  |-  ( (/)  e.  A  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  _V ) )
47 unv 3495 . . . . . . 7  |-  ( |^| B  u.  _V )  =  _V
4846, 35, 473eqtr3g 2351 . . . . . 6  |-  ( (/)  e.  A  ->  ( ( _V  \  |^| A
)  u.  |^| B
)  =  _V )
4948adantl 452 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( ( _V 
\  |^| A )  u. 
|^| B )  =  _V )
5049ineq2d 3383 . . . 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 3494 . . . 4  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  _V )  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)
5250, 51syl6req 2345 . . 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 2328 . 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 6528 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 set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   |^|cint 3878    e. cmpt 4093   Oncon0 4408   ` cfv 5271  (class class class)co 5874   reccrdg 6438   1oc1o 6488    .o comu 6493    ^o coe 6494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-1o 6495  df-oexp 6501
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