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Theorem oeoe 6483
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 5718 . . . . . . . . . . . 12  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
2 oe0m0 6405 . . . . . . . . . . . 12  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2301 . . . . . . . . . . 11  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  1o )
43oveq1d 5725 . . . . . . . . . 10  |-  ( B  =  (/)  ->  ( (
(/)  ^o  B )  ^o  C )  =  ( 1o  ^o  C ) )
5 oe1m 6429 . . . . . . . . . 10  |-  ( C  e.  On  ->  ( 1o  ^o  C )  =  1o )
64, 5sylan9eqr 2307 . . . . . . . . 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 5718 . . . . . . . . . 10  |-  ( C  =  (/)  ->  ( (
(/)  ^o  B )  ^o  C )  =  ( ( (/)  ^o  B )  ^o  (/) ) )
9 0elon 4338 . . . . . . . . . . . 12  |-  (/)  e.  On
10 oecl 6422 . . . . . . . . . . . 12  |-  ( (
(/)  e.  On  /\  B  e.  On )  ->  ( (/) 
^o  B )  e.  On )
119, 10mpan 654 . . . . . . . . . . 11  |-  ( B  e.  On  ->  ( (/) 
^o  B )  e.  On )
12 oe0 6407 . . . . . . . . . . 11  |-  ( (
(/)  ^o  B )  e.  On  ->  ( ( (/) 
^o  B )  ^o  (/) )  =  1o )
1311, 12syl 17 . . . . . . . . . 10  |-  ( B  e.  On  ->  (
( (/)  ^o  B )  ^o  (/) )  =  1o )
148, 13sylan9eqr 2307 . . . . . . . . 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 6459 . . . . . . . . . 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 5726 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( (/) 
^o  ( B  .o  C ) )  =  ( (/)  ^o  (/) ) )
2019, 2syl6eq 2301 . . . . . . 7  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( (/) 
^o  ( B  .o  C ) )  =  1o )
2116, 20eqtr4d 2288 . . . . . 6  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  (
( (/)  ^o  B )  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
22 on0eln0 4340 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
23 on0eln0 4340 . . . . . . . . . 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 2497 . . . . . . . . 9  |-  ( ( B  =/=  (/)  /\  C  =/=  (/) )  <->  -.  ( B  =  (/)  \/  C  =  (/) ) )
2624, 25syl6bb 254 . . . . . . . 8  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <->  -.  ( B  =  (/)  \/  C  =  (/) ) ) )
27 oe0m1 6406 . . . . . . . . . . . . . 14  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
2827biimpa 472 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2928oveq1d 5725 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  C
) )
30 oe0m1 6406 . . . . . . . . . . . . 13  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  ( (/)  ^o  C
)  =  (/) ) )
3130biimpa 472 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  (/) 
e.  C )  -> 
( (/)  ^o  C )  =  (/) )
3229, 31sylan9eq 2305 . . . . . . . . . . 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 6460 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  ( B  .o  C )  <->  ( (/)  e.  B  /\  (/)  e.  C ) ) )
35 omcl 6421 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  .o  C
)  e.  On )
36 oe0m1 6406 . . . . . . . . . . . . 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 2288 . . . . . . . . 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 5717 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
4645oveq1d 5725 . . . . . 6  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  ^o  C )  =  ( ( (/)  ^o  B
)  ^o  C )
)
47 oveq1 5717 . . . . . 6  |-  ( A  =  (/)  ->  ( A  ^o  ( B  .o  C ) )  =  ( (/)  ^o  ( B  .o  C ) ) )
4846, 47eqeq12d 2267 . . . . 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 5717 . . . . . . . . 9  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( A  ^o  B )  =  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  ^o  B ) )
5251oveq1d 5725 . . . . . . . 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 5717 . . . . . . . 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 2267 . . . . . . 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 2313 . . . . . . . . . 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 2314 . . . . . . . . . 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 2313 . . . . . . . . . 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 2314 . . . . . . . . . 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 6372 . . . . . . . . . 10  |-  1o  e.  On
63 0lt1o 6389 . . . . . . . . . 10  |-  (/)  e.  1o
6462, 63pm3.2i 443 . . . . . . . . 9  |-  ( 1o  e.  On  /\  (/)  e.  1o )
6558, 61, 64elimhyp 3518 . . . . . . . 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 6482 . . . . . 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 3511 . . . . 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 6398 . 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 2412   (/)c0 3362   ifcif 3470   Oncon0 4285  (class class class)co 5710   1oc1o 6358    .o comu 6363    ^o coe 6364
This theorem is referenced by:  infxpenc  7529
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-omul 6370  df-oexp 6371
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