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Theorem oev2 7229
Description: Alternate value of ordinal exponentiation. Compare oev 7220. (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 6310 . . . . . 6  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( (/)  ^o  (/) ) )
2 oe0m0 7226 . . . . . 6  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2479 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  1o )
4 fveq2 5877 . . . . . . . 8  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) ) )
5 1on 7193 . . . . . . . . . 10  |-  1o  e.  On
65elexi 3091 . . . . . . . . 9  |-  1o  e.  _V
76rdg0 7143 . . . . . . . 8  |-  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  (/) )  =  1o
84, 7syl6eq 2479 . . . . . . 7  |-  ( B  =  (/)  ->  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)  =  1o )
9 inteq 4255 . . . . . . . 8  |-  ( B  =  (/)  ->  |^| B  =  |^| (/) )
10 int0 4266 . . . . . . . 8  |-  |^| (/)  =  _V
119, 10syl6eq 2479 . . . . . . 7  |-  ( B  =  (/)  ->  |^| B  =  _V )
128, 11ineq12d 3665 . . . . . 6  |-  ( B  =  (/)  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  ( 1o  i^i  _V )
)
13 inv1 3789 . . . . . . 7  |-  ( 1o 
i^i  _V )  =  1o
1413a1i 11 . . . . . 6  |-  ( A  =  (/)  ->  ( 1o 
i^i  _V )  =  1o )
1512, 14sylan9eqr 2485 . . . . 5  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  (
( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `
 B )  i^i  |^| B )  =  1o )
163, 15eqtr4d 2466 . . . 4  |-  ( ( A  =  (/)  /\  B  =  (/) )  ->  ( A  ^o  B )  =  ( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B ) )
17 oveq1 6308 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
18 oe0m1 7227 . . . . . . . 8  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
1918biimpa 486 . . . . . . 7  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2017, 19sylan9eqr 2485 . . . . . 6  |-  ( ( ( B  e.  On  /\  (/)  e.  B )  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  (/) )
2120an32s 811 . . . . 5  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  (/) )
22 int0el 4284 . . . . . . . 8  |-  ( (/)  e.  B  ->  |^| B  =  (/) )
2322ineq2d 3664 . . . . . . 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 3788 . . . . . . 7  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  (/) )  =  (/)
2523, 24syl6eq 2479 . . . . . 6  |-  ( (/)  e.  B  ->  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B )  =  (/) )
2625adantl 467 . . . . 5  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( ( rec (
( x  e.  _V  |->  ( x  .o  A
) ) ,  1o ) `  B )  i^i  |^| B )  =  (/) )
2721, 26eqtr4d 2466 . . . 4  |-  ( ( ( B  e.  On  /\  A  =  (/) )  /\  (/) 
e.  B )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
2816, 27oe0lem 7219 . . 3  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( A  ^o  B
)  =  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  |^| B ) )
29 inteq 4255 . . . . . . . . . 10  |-  ( A  =  (/)  ->  |^| A  =  |^| (/) )
3029, 10syl6eq 2479 . . . . . . . . 9  |-  ( A  =  (/)  ->  |^| A  =  _V )
3130difeq2d 3583 . . . . . . . 8  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  ( _V  \  _V ) )
32 difid 3863 . . . . . . . 8  |-  ( _V 
\  _V )  =  (/)
3331, 32syl6eq 2479 . . . . . . 7  |-  ( A  =  (/)  ->  ( _V 
\  |^| A )  =  (/) )
3433uneq2d 3620 . . . . . 6  |-  ( A  =  (/)  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  (/) ) )
35 uncom 3610 . . . . . 6  |-  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( ( _V 
\  |^| A )  u. 
|^| B )
36 un0 3787 . . . . . 6  |-  ( |^| B  u.  (/) )  = 
|^| B
3734, 35, 363eqtr3g 2486 . . . . 5  |-  ( A  =  (/)  ->  ( ( _V  \  |^| A
)  u.  |^| B
)  =  |^| B
)
3837adantl 467 . . . 4  |-  ( ( B  e.  On  /\  A  =  (/) )  -> 
( ( _V  \  |^| A )  u.  |^| B )  =  |^| B )
3938ineq2d 3664 . . 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 2466 . 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 7221 . . 3  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( A  ^o  B )  =  ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B ) )
42 int0el 4284 . . . . . . . . . 10  |-  ( (/)  e.  A  ->  |^| A  =  (/) )
4342difeq2d 3583 . . . . . . . . 9  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  ( _V  \  (/) ) )
44 dif0 3865 . . . . . . . . 9  |-  ( _V 
\  (/) )  =  _V
4543, 44syl6eq 2479 . . . . . . . 8  |-  ( (/)  e.  A  ->  ( _V 
\  |^| A )  =  _V )
4645uneq2d 3620 . . . . . . 7  |-  ( (/)  e.  A  ->  ( |^| B  u.  ( _V  \ 
|^| A ) )  =  ( |^| B  u.  _V ) )
47 unv 3790 . . . . . . 7  |-  ( |^| B  u.  _V )  =  _V
4846, 35, 473eqtr3g 2486 . . . . . 6  |-  ( (/)  e.  A  ->  ( ( _V  \  |^| A
)  u.  |^| B
)  =  _V )
4948adantl 467 . . . . 5  |-  ( ( ( A  e.  On  /\  B  e.  On )  /\  (/)  e.  A )  ->  ( ( _V 
\  |^| A )  u. 
|^| B )  =  _V )
5049ineq2d 3664 . . . 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 3789 . . . 4  |-  ( ( rec ( ( x  e.  _V  |->  ( x  .o  A ) ) ,  1o ) `  B )  i^i  _V )  =  ( rec ( ( x  e. 
_V  |->  ( x  .o  A ) ) ,  1o ) `  B
)
5250, 51syl6req 2480 . . 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 2463 . 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 7219 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 370    = wceq 1437    e. wcel 1868   _Vcvv 3081    \ cdif 3433    u. cun 3434    i^i cin 3435   (/)c0 3761   |^|cint 4252    |-> cmpt 4479   Oncon0 5438   ` cfv 5597  (class class class)co 6301   reccrdg 7131   1oc1o 7179    .o comu 7184    ^o coe 7185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-int 4253  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-pred 5395  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-iota 5561  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-fv 5605  df-ov 6304  df-oprab 6305  df-mpt2 6306  df-om 6703  df-wrecs 7032  df-recs 7094  df-rdg 7132  df-1o 7186  df-oexp 7192
This theorem is referenced by: (None)
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