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Theorem oeoe 6530
Description: Product of exponents law for ordinal exponentiation. Theorem 8S of [Enderton] p. 238. Also Proposition 8.42 of [TakeutiZaring] p. 70. (Contributed by Eric Schmidt, 26-May-2009.)
Assertion
Ref Expression
oeoe  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) ) )

Proof of Theorem oeoe
StepHypRef Expression
1 oveq2 5765 . . . . . . . . . . . 12  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6452 . . . . . . . . . . . 12  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2304 . . . . . . . . . . 11  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  1o )
43oveq1d 5772 . . . . . . . . . 10  |-  ( B  =  (/)  ->  ( (
(/)  ^o  B )  ^o  C )  =  ( 1o  ^o  C ) )
5 oe1m 6476 . . . . . . . . . 10  |-  ( C  e.  On  ->  ( 1o  ^o  C )  =  1o )
64, 5sylan9eqr 2310 . . . . . . . . 9  |-  ( ( C  e.  On  /\  B  =  (/) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  1o )
76adantll 697 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  B  =  (/) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  1o )
8 oveq2 5765 . . . . . . . . . 10  |-  ( C  =  (/)  ->  ( (
(/)  ^o  B )  ^o  C )  =  ( ( (/)  ^o  B )  ^o  (/) ) )
9 0elon 4382 . . . . . . . . . . . 12  |-  (/)  e.  On
10 oecl 6469 . . . . . . . . . . . 12  |-  ( (
(/)  e.  On  /\  B  e.  On )  ->  ( (/) 
^o  B )  e.  On )
119, 10mpan 654 . . . . . . . . . . 11  |-  ( B  e.  On  ->  ( (/) 
^o  B )  e.  On )
12 oe0 6454 . . . . . . . . . . 11  |-  ( (
(/)  ^o  B )  e.  On  ->  ( ( (/) 
^o  B )  ^o  (/) )  =  1o )
1311, 12syl 17 . . . . . . . . . 10  |-  ( B  e.  On  ->  (
( (/)  ^o  B )  ^o  (/) )  =  1o )
148, 13sylan9eqr 2310 . . . . . . . . 9  |-  ( ( B  e.  On  /\  C  =  (/) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  1o )
1514adantlr 698 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  C  =  (/) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  1o )
167, 15jaodan 763 . . . . . . 7  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  (
( (/)  ^o  B )  ^o  C )  =  1o )
17 om00 6506 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( B  .o  C )  =  (/)  <->  ( B  =  (/)  \/  C  =  (/) ) ) )
1817biimpar 473 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( B  .o  C )  =  (/) )
1918oveq2d 5773 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( (/) 
^o  ( B  .o  C ) )  =  ( (/)  ^o  (/) ) )
2019, 2syl6eq 2304 . . . . . . 7  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( (/) 
^o  ( B  .o  C ) )  =  1o )
2116, 20eqtr4d 2291 . . . . . 6  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  (
( (/)  ^o  B )  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
22 on0eln0 4384 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
23 on0eln0 4384 . . . . . . . . . 10  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  C  =/=  (/) ) )
2422, 23bi2anan9 848 . . . . . . . . 9  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <-> 
( B  =/=  (/)  /\  C  =/=  (/) ) ) )
25 neanior 2504 . . . . . . . . 9  |-  ( ( B  =/=  (/)  /\  C  =/=  (/) )  <->  -.  ( B  =  (/)  \/  C  =  (/) ) )
2624, 25syl6bb 254 . . . . . . . 8  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <->  -.  ( B  =  (/)  \/  C  =  (/) ) ) )
27 oe0m1 6453 . . . . . . . . . . . . . 14  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
2827biimpa 472 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2928oveq1d 5772 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  C
) )
30 oe0m1 6453 . . . . . . . . . . . . 13  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  ( (/)  ^o  C
)  =  (/) ) )
3130biimpa 472 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  (/) 
e.  C )  -> 
( (/)  ^o  C )  =  (/) )
3229, 31sylan9eq 2308 . . . . . . . . . . 11  |-  ( ( ( B  e.  On  /\  (/)  e.  B )  /\  ( C  e.  On  /\  (/)  e.  C ) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  (/) )
3332an4s 802 . . . . . . . . . 10  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( ( (/) 
^o  B )  ^o  C )  =  (/) )
34 om00el 6507 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  ( B  .o  C )  <->  ( (/)  e.  B  /\  (/)  e.  C ) ) )
35 omcl 6468 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  .o  C
)  e.  On )
36 oe0m1 6453 . . . . . . . . . . . . 13  |-  ( ( B  .o  C )  e.  On  ->  ( (/) 
e.  ( B  .o  C )  <->  ( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3735, 36syl 17 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  ( B  .o  C )  <->  ( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3834, 37bitr3d 248 . . . . . . . . . . 11  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <-> 
( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3938biimpa 472 . . . . . . . . . 10  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( (/)  ^o  ( B  .o  C ) )  =  (/) )
4033, 39eqtr4d 2291 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( ( (/) 
^o  B )  ^o  C )  =  (
(/)  ^o  ( B  .o  C ) ) )
4140ex 425 . . . . . . . 8  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  ( (/)  ^o  ( B  .o  C
) ) ) )
4226, 41sylbird 228 . . . . . . 7  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( -.  ( B  =  (/)  \/  C  =  (/) )  ->  (
( (/)  ^o  B )  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) ) )
4342imp 420 . . . . . 6  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  -.  ( B  =  (/)  \/  C  =  (/) ) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
4421, 43pm2.61dan 769 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
45 oveq1 5764 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
4645oveq1d 5772 . . . . . 6  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  ^o  C )  =  ( ( (/)  ^o  B
)  ^o  C )
)
47 oveq1 5764 . . . . . 6  |-  ( A  =  (/)  ->  ( A  ^o  ( B  .o  C ) )  =  ( (/)  ^o  ( B  .o  C ) ) )
4846, 47eqeq12d 2270 . . . . 5  |-  ( A  =  (/)  ->  ( ( ( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) )  <->  ( ( (/) 
^o  B )  ^o  C )  =  (
(/)  ^o  ( B  .o  C ) ) ) )
4944, 48syl5ibr 214 . . . 4  |-  ( A  =  (/)  ->  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  ^o  C
)  =  ( A  ^o  ( B  .o  C ) ) ) )
5049impcom 421 . . 3  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  A  =  (/) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
51 oveq1 5764 . . . . . . . . 9  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( A  ^o  B )  =  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  ^o  B ) )
5251oveq1d 5772 . . . . . . . 8  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( ( A  ^o  B )  ^o  C )  =  ( ( if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ^o  B )  ^o  C ) )
53 oveq1 5764 . . . . . . . 8  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( A  ^o  ( B  .o  C
) )  =  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  ^o  ( B  .o  C ) ) )
5452, 53eqeq12d 2270 . . . . . . 7  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) )  <-> 
( ( if ( ( A  e.  On  /\  (/)  e.  A ) ,  A ,  1o )  ^o  B )  ^o  C )  =  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  ^o  ( B  .o  C ) ) ) )
5554imbi2d 309 . . . . . 6  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( ( ( B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  ^o  C
)  =  ( A  ^o  ( B  .o  C ) ) )  <-> 
( ( B  e.  On  /\  C  e.  On )  ->  (
( if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ^o  B )  ^o  C )  =  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  ^o  ( B  .o  C ) ) ) ) )
56 eleq1 2316 . . . . . . . . . 10  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( A  e.  On  <->  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  e.  On ) )
57 eleq2 2317 . . . . . . . . . 10  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( (/)  e.  A  <->  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o ) ) )
5856, 57anbi12d 694 . . . . . . . . 9  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( ( A  e.  On  /\  (/)  e.  A
)  <->  ( if ( ( A  e.  On  /\  (/)  e.  A ) ,  A ,  1o )  e.  On  /\  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o ) ) ) )
59 eleq1 2316 . . . . . . . . . 10  |-  ( 1o  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( 1o  e.  On 
<->  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  e.  On ) )
60 eleq2 2317 . . . . . . . . . 10  |-  ( 1o  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( (/)  e.  1o  <->  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o ) ) )
6159, 60anbi12d 694 . . . . . . . . 9  |-  ( 1o  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( ( 1o  e.  On  /\  (/)  e.  1o ) 
<->  ( if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  e.  On  /\  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o ) ) ) )
62 1on 6419 . . . . . . . . . 10  |-  1o  e.  On
63 0lt1o 6436 . . . . . . . . . 10  |-  (/)  e.  1o
6462, 63pm3.2i 443 . . . . . . . . 9  |-  ( 1o  e.  On  /\  (/)  e.  1o )
6558, 61, 64elimhyp 3554 . . . . . . . 8  |-  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  e.  On  /\  (/)  e.  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o ) )
6665simpli 446 . . . . . . 7  |-  if ( ( A  e.  On  /\  (/)  e.  A ) ,  A ,  1o )  e.  On
6765simpri 450 . . . . . . 7  |-  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )
6866, 67oeoelem 6529 . . . . . 6  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( if ( ( A  e.  On  /\  (/)  e.  A ) ,  A ,  1o )  ^o  B )  ^o  C )  =  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  ^o  ( B  .o  C ) ) )
6955, 68dedth 3547 . . . . 5  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( ( B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) ) ) )
7069imp 420 . . . 4  |-  ( ( ( A  e.  On  /\  (/)  e.  A )  /\  ( B  e.  On  /\  C  e.  On ) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
7170an32s 782 . . 3  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\  C  e.  On ) )  /\  (/)  e.  A
)  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
7250, 71oe0lem 6445 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  C  e.  On ) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
73723impb 1152 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2419   (/)c0 3397   ifcif 3506   Oncon0 4329  (class class class)co 5757   1oc1o 6405    .o comu 6410    ^o coe 6411
This theorem is referenced by:  infxpenc  7578
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-1st 6021  df-2nd 6022  df-recs 6321  df-rdg 6356  df-1o 6412  df-2o 6413  df-oadd 6416  df-omul 6417  df-oexp 6418
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