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Theorem oev2 7191
Description: Alternate value of ordinal exponentiation. Compare oev 7182. (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 6305 . . . . . 6  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( (/)  ^o  (/) ) )
2 oe0m0 7188 . . . . . 6  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2514 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  1o )
4 fveq2 5872 . . . . . . . 8  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) ) )
5 1on 7155 . . . . . . . . . 10  |-  1o  e.  On
65elexi 3119 . . . . . . . . 9  |-  1o  e.  _V
76rdg0 7105 . . . . . . . 8  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
84, 7syl6eq 2514 . . . . . . 7  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  1o )
9 inteq 4291 . . . . . . . 8  |-  ( B  =  (/)  ->  |^| B  =  |^| (/) )
10 int0 4302 . . . . . . . 8  |-  |^| (/)  =  _V
119, 10syl6eq 2514 . . . . . . 7  |-  ( B  =  (/)  ->  |^| B  =  _V )
128, 11ineq12d 3697 . . . . . 6  |-  ( B  =  (/)  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  ( 1o  i^i  _V )
)
13 inv1 3821 . . . . . . 7  |-  ( 1o 
i^i  _V )  =  1o
1413a1i 11 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
i^i  _V )  =  1o )
1512, 14sylan9eqr 2520 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  (
( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 B )  i^i  |^| B )  =  1o )
163, 15eqtr4d 2501 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B ) )
17 oveq1 6303 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
18 oe0m1 7189 . . . . . . . 8  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
1918biimpa 484 . . . . . . 7  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2017, 19sylan9eqr 2520 . . . . . 6  |-  ( ( ( B  e.  On  /\  (/)  e.  B )  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  (/) )
2120an32s 804 . . . . 5  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  (/) )
22 int0el 4320 . . . . . . . 8  |-  ( (/)  e.  B  ->  |^| B  =  (/) )
2322ineq2d 3696 . . . . . . 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 3820 . . . . . . 7  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  (/) )  =  (/)
2523, 24syl6eq 2514 . . . . . 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 2501 . . . 4  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
2816, 27oe0lem 7181 . . 3  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
29 inteq 4291 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3029, 10syl6eq 2514 . . . . . . . . 9  |-  ( A  =  (/)  ->  |^| A  =  _V )
3130difeq2d 3618 . . . . . . . 8  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  ( _V  \  _V ) )
32 difid 3899 . . . . . . . 8  |-  ( _V 
\  _V )  =  (/)
3331, 32syl6eq 2514 . . . . . . 7  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  (/) )
3433uneq2d 3654 . . . . . 6  |-  ( A  =  (/)  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  (/) ) )
35 uncom 3644 . . . . . 6  |-  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( ( _V 
\  |^| A )  u. 
|^| B )
36 un0 3819 . . . . . 6  |-  ( |^| B  u.  (/) )  = 
|^| B
3734, 35, 363eqtr3g 2521 . . . . 5  |-  ( A  =  (/)  ->  ( ( _V  \  |^| A
)  u.  |^| B
)  =  |^| B
)
3837adantl 466 . . . 4  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( ( _V  \  |^| A )  u.  |^| B )  =  |^| B )
3938ineq2d 3696 . . 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 2501 . 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 7183 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
42 int0el 4320 . . . . . . . . . 10  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
4342difeq2d 3618 . . . . . . . . 9  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  ( _V  \  (/) ) )
44 dif0 3901 . . . . . . . . 9  |-  ( _V 
\  (/) )  =  _V
4543, 44syl6eq 2514 . . . . . . . 8  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  _V )
4645uneq2d 3654 . . . . . . 7  |-  ( (/)  e.  A  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  _V ) )
47 unv 3822 . . . . . . 7  |-  ( |^| B  u.  _V )  =  _V
4846, 35, 473eqtr3g 2521 . . . . . 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 3696 . . . 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 3821 . . . 4  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  _V )  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)
5250, 51syl6req 2515 . . 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 2498 . 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 7181 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 1395    e. wcel 1819   _Vcvv 3109    \ cdif 3468    u. cun 3469    i^i cin 3470   (/)c0 3793   |^|cint 4288    |-> cmpt 4515   Oncon0 4887   ` cfv 5594  (class class class)co 6296   reccrdg 7093   1oc1o 7141    .o comu 7146    ^o coe 7147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-oexp 7154
This theorem is referenced by: (None)
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