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Theorem oev2 6522
Description: Alternate value of ordinal exponentiation. Compare oev 6513. (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 5867 . . . . . 6  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( (/)  ^o  (/) ) )
2 oe0m0 6519 . . . . . 6  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2331 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  1o )
4 fveq2 5525 . . . . . . . 8  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) ) )
5 1on 6486 . . . . . . . . . 10  |-  1o  e.  On
65elexi 2797 . . . . . . . . 9  |-  1o  e.  _V
76rdg0 6434 . . . . . . . 8  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
84, 7syl6eq 2331 . . . . . . 7  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  1o )
9 inteq 3865 . . . . . . . 8  |-  ( B  =  (/)  ->  |^| B  =  |^| (/) )
10 int0 3876 . . . . . . . 8  |-  |^| (/)  =  _V
119, 10syl6eq 2331 . . . . . . 7  |-  ( B  =  (/)  ->  |^| B  =  _V )
128, 11ineq12d 3371 . . . . . 6  |-  ( B  =  (/)  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  ( 1o  i^i  _V )
)
13 inv1 3481 . . . . . . 7  |-  ( 1o 
i^i  _V )  =  1o
1413a1i 10 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
i^i  _V )  =  1o )
1512, 14sylan9eqr 2337 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  (
( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 B )  i^i  |^| B )  =  1o )
163, 15eqtr4d 2318 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B ) )
17 oveq1 5865 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
18 oe0m1 6520 . . . . . . . 8  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
1918biimpa 470 . . . . . . 7  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2017, 19sylan9eqr 2337 . . . . . 6  |-  ( ( ( B  e.  On  /\  (/)  e.  B )  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  (/) )
2120an32s 779 . . . . 5  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  (/) )
22 int0el 3893 . . . . . . . 8  |-  ( (/)  e.  B  ->  |^| B  =  (/) )
2322ineq2d 3370 . . . . . . 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 3480 . . . . . . 7  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  (/) )  =  (/)
2523, 24syl6eq 2331 . . . . . 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 2318 . . . 4  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
2816, 27oe0lem 6512 . . 3  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
29 inteq 3865 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3029, 10syl6eq 2331 . . . . . . . . 9  |-  ( A  =  (/)  ->  |^| A  =  _V )
3130difeq2d 3294 . . . . . . . 8  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  ( _V  \  _V ) )
32 difid 3522 . . . . . . . 8  |-  ( _V 
\  _V )  =  (/)
3331, 32syl6eq 2331 . . . . . . 7  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  (/) )
3433uneq2d 3329 . . . . . 6  |-  ( A  =  (/)  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  (/) ) )
35 uncom 3319 . . . . . 6  |-  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( ( _V 
\  |^| A )  u. 
|^| B )
36 un0 3479 . . . . . 6  |-  ( |^| B  u.  (/) )  = 
|^| B
3734, 35, 363eqtr3g 2338 . . . . 5  |-  ( A  =  (/)  ->  ( ( _V  \  |^| A
)  u.  |^| B
)  =  |^| B
)
3837adantl 452 . . . 4  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( ( _V  \  |^| A )  u.  |^| B )  =  |^| B )
3938ineq2d 3370 . . 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 2318 . 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 6514 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
42 int0el 3893 . . . . . . . . . 10  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
4342difeq2d 3294 . . . . . . . . 9  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  ( _V  \  (/) ) )
44 dif0 3524 . . . . . . . . 9  |-  ( _V 
\  (/) )  =  _V
4543, 44syl6eq 2331 . . . . . . . 8  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  _V )
4645uneq2d 3329 . . . . . . 7  |-  ( (/)  e.  A  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  _V ) )
47 unv 3482 . . . . . . 7  |-  ( |^| B  u.  _V )  =  _V
4846, 35, 473eqtr3g 2338 . . . . . 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 3370 . . . 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 3481 . . . 4  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  _V )  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)
5250, 51syl6req 2332 . . 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 2315 . 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 6512 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 1623    e. wcel 1684   _Vcvv 2788    \ cdif 3149    u. cun 3150    i^i cin 3151   (/)c0 3455   |^|cint 3862    e. cmpt 4077   Oncon0 4392   ` cfv 5255  (class class class)co 5858   reccrdg 6422   1oc1o 6472    .o comu 6477    ^o coe 6478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-1o 6479  df-oexp 6485
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