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Theorem oeoe 6597
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 5866 . . . . . . . . . . . 12  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6519 . . . . . . . . . . . 12  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2331 . . . . . . . . . . 11  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  1o )
43oveq1d 5873 . . . . . . . . . 10  |-  ( B  =  (/)  ->  ( (
(/)  ^o  B )  ^o  C )  =  ( 1o  ^o  C ) )
5 oe1m 6543 . . . . . . . . . 10  |-  ( C  e.  On  ->  ( 1o  ^o  C )  =  1o )
64, 5sylan9eqr 2337 . . . . . . . . 9  |-  ( ( C  e.  On  /\  B  =  (/) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  1o )
76adantll 694 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  B  =  (/) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  1o )
8 oveq2 5866 . . . . . . . . . 10  |-  ( C  =  (/)  ->  ( (
(/)  ^o  B )  ^o  C )  =  ( ( (/)  ^o  B )  ^o  (/) ) )
9 0elon 4445 . . . . . . . . . . . 12  |-  (/)  e.  On
10 oecl 6536 . . . . . . . . . . . 12  |-  ( (
(/)  e.  On  /\  B  e.  On )  ->  ( (/) 
^o  B )  e.  On )
119, 10mpan 651 . . . . . . . . . . 11  |-  ( B  e.  On  ->  ( (/) 
^o  B )  e.  On )
12 oe0 6521 . . . . . . . . . . 11  |-  ( (
(/)  ^o  B )  e.  On  ->  ( ( (/) 
^o  B )  ^o  (/) )  =  1o )
1311, 12syl 15 . . . . . . . . . 10  |-  ( B  e.  On  ->  (
( (/)  ^o  B )  ^o  (/) )  =  1o )
148, 13sylan9eqr 2337 . . . . . . . . 9  |-  ( ( B  e.  On  /\  C  =  (/) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  1o )
1514adantlr 695 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  C  =  (/) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  1o )
167, 15jaodan 760 . . . . . . 7  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  (
( (/)  ^o  B )  ^o  C )  =  1o )
17 om00 6573 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( B  .o  C )  =  (/)  <->  ( B  =  (/)  \/  C  =  (/) ) ) )
1817biimpar 471 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( B  .o  C )  =  (/) )
1918oveq2d 5874 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( (/) 
^o  ( B  .o  C ) )  =  ( (/)  ^o  (/) ) )
2019, 2syl6eq 2331 . . . . . . 7  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( (/) 
^o  ( B  .o  C ) )  =  1o )
2116, 20eqtr4d 2318 . . . . . 6  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  (
( (/)  ^o  B )  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
22 on0eln0 4447 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
23 on0eln0 4447 . . . . . . . . . 10  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  C  =/=  (/) ) )
2422, 23bi2anan9 843 . . . . . . . . 9  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <-> 
( B  =/=  (/)  /\  C  =/=  (/) ) ) )
25 neanior 2531 . . . . . . . . 9  |-  ( ( B  =/=  (/)  /\  C  =/=  (/) )  <->  -.  ( B  =  (/)  \/  C  =  (/) ) )
2624, 25syl6bb 252 . . . . . . . 8  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <->  -.  ( B  =  (/)  \/  C  =  (/) ) ) )
27 oe0m1 6520 . . . . . . . . . . . . . 14  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
2827biimpa 470 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2928oveq1d 5873 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  C
) )
30 oe0m1 6520 . . . . . . . . . . . . 13  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  ( (/)  ^o  C
)  =  (/) ) )
3130biimpa 470 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  (/) 
e.  C )  -> 
( (/)  ^o  C )  =  (/) )
3229, 31sylan9eq 2335 . . . . . . . . . . 11  |-  ( ( ( B  e.  On  /\  (/)  e.  B )  /\  ( C  e.  On  /\  (/)  e.  C ) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  (/) )
3332an4s 799 . . . . . . . . . 10  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( ( (/) 
^o  B )  ^o  C )  =  (/) )
34 om00el 6574 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  ( B  .o  C )  <->  ( (/)  e.  B  /\  (/)  e.  C ) ) )
35 omcl 6535 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  .o  C
)  e.  On )
36 oe0m1 6520 . . . . . . . . . . . . 13  |-  ( ( B  .o  C )  e.  On  ->  ( (/) 
e.  ( B  .o  C )  <->  ( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3735, 36syl 15 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  ( B  .o  C )  <->  ( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3834, 37bitr3d 246 . . . . . . . . . . 11  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <-> 
( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3938biimpa 470 . . . . . . . . . 10  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( (/)  ^o  ( B  .o  C ) )  =  (/) )
4033, 39eqtr4d 2318 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( ( (/) 
^o  B )  ^o  C )  =  (
(/)  ^o  ( B  .o  C ) ) )
4140ex 423 . . . . . . . 8  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  ( (/)  ^o  ( B  .o  C
) ) ) )
4226, 41sylbird 226 . . . . . . 7  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( -.  ( B  =  (/)  \/  C  =  (/) )  ->  (
( (/)  ^o  B )  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) ) )
4342imp 418 . . . . . 6  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  -.  ( B  =  (/)  \/  C  =  (/) ) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
4421, 43pm2.61dan 766 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
45 oveq1 5865 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
4645oveq1d 5873 . . . . . 6  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  ^o  C )  =  ( ( (/)  ^o  B
)  ^o  C )
)
47 oveq1 5865 . . . . . 6  |-  ( A  =  (/)  ->  ( A  ^o  ( B  .o  C ) )  =  ( (/)  ^o  ( B  .o  C ) ) )
4846, 47eqeq12d 2297 . . . . 5  |-  ( A  =  (/)  ->  ( ( ( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) )  <->  ( ( (/) 
^o  B )  ^o  C )  =  (
(/)  ^o  ( B  .o  C ) ) ) )
4944, 48syl5ibr 212 . . . 4  |-  ( A  =  (/)  ->  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  ^o  C
)  =  ( A  ^o  ( B  .o  C ) ) ) )
5049impcom 419 . . 3  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  A  =  (/) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
51 oveq1 5865 . . . . . . . . 9  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( A  ^o  B )  =  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  ^o  B ) )
5251oveq1d 5873 . . . . . . . 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 5865 . . . . . . . 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 2297 . . . . . . 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 307 . . . . . 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 2343 . . . . . . . . . 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 2344 . . . . . . . . . 10  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( (/)  e.  A  <->  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o ) ) )
5856, 57anbi12d 691 . . . . . . . . 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 2343 . . . . . . . . . 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 2344 . . . . . . . . . 10  |-  ( 1o  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( (/)  e.  1o  <->  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o ) ) )
6159, 60anbi12d 691 . . . . . . . . 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 6486 . . . . . . . . . 10  |-  1o  e.  On
63 0lt1o 6503 . . . . . . . . . 10  |-  (/)  e.  1o
6462, 63pm3.2i 441 . . . . . . . . 9  |-  ( 1o  e.  On  /\  (/)  e.  1o )
6558, 61, 64elimhyp 3613 . . . . . . . 8  |-  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  e.  On  /\  (/)  e.  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o ) )
6665simpli 444 . . . . . . 7  |-  if ( ( A  e.  On  /\  (/)  e.  A ) ,  A ,  1o )  e.  On
6765simpri 448 . . . . . . 7  |-  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )
6866, 67oeoelem 6596 . . . . . 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 3606 . . . . 5  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( ( B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) ) ) )
7069imp 418 . . . 4  |-  ( ( ( A  e.  On  /\  (/)  e.  A )  /\  ( B  e.  On  /\  C  e.  On ) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
7170an32s 779 . . 3  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\  C  e.  On ) )  /\  (/)  e.  A
)  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
7250, 71oe0lem 6512 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  C  e.  On ) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
73723impb 1147 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 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   ifcif 3565   Oncon0 4392  (class class class)co 5858   1oc1o 6472    .o comu 6477    ^o coe 6478
This theorem is referenced by:  infxpenc  7645
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-rep 4131  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-rmo 2551  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-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-omul 6484  df-oexp 6485
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