MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oeoe Structured version   Unicode version

Theorem oeoe 7266
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 6304 . . . . . . . . . . . 12  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  (
(/)  ^o  (/) ) )
2 oe0m0 7188 . . . . . . . . . . . 12  |-  ( (/)  ^o  (/) )  =  1o
31, 2syl6eq 2514 . . . . . . . . . . 11  |-  ( B  =  (/)  ->  ( (/)  ^o  B )  =  1o )
43oveq1d 6311 . . . . . . . . . 10  |-  ( B  =  (/)  ->  ( (
(/)  ^o  B )  ^o  C )  =  ( 1o  ^o  C ) )
5 oe1m 7212 . . . . . . . . . 10  |-  ( C  e.  On  ->  ( 1o  ^o  C )  =  1o )
64, 5sylan9eqr 2520 . . . . . . . . 9  |-  ( ( C  e.  On  /\  B  =  (/) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  1o )
76adantll 713 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  B  =  (/) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  1o )
8 oveq2 6304 . . . . . . . . . 10  |-  ( C  =  (/)  ->  ( (
(/)  ^o  B )  ^o  C )  =  ( ( (/)  ^o  B )  ^o  (/) ) )
9 0elon 4940 . . . . . . . . . . . 12  |-  (/)  e.  On
10 oecl 7205 . . . . . . . . . . . 12  |-  ( (
(/)  e.  On  /\  B  e.  On )  ->  ( (/) 
^o  B )  e.  On )
119, 10mpan 670 . . . . . . . . . . 11  |-  ( B  e.  On  ->  ( (/) 
^o  B )  e.  On )
12 oe0 7190 . . . . . . . . . . 11  |-  ( (
(/)  ^o  B )  e.  On  ->  ( ( (/) 
^o  B )  ^o  (/) )  =  1o )
1311, 12syl 16 . . . . . . . . . 10  |-  ( B  e.  On  ->  (
( (/)  ^o  B )  ^o  (/) )  =  1o )
148, 13sylan9eqr 2520 . . . . . . . . 9  |-  ( ( B  e.  On  /\  C  =  (/) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  1o )
1514adantlr 714 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  C  =  (/) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  1o )
167, 15jaodan 785 . . . . . . 7  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  (
( (/)  ^o  B )  ^o  C )  =  1o )
17 om00 7242 . . . . . . . . . 10  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( B  .o  C )  =  (/)  <->  ( B  =  (/)  \/  C  =  (/) ) ) )
1817biimpar 485 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( B  .o  C )  =  (/) )
1918oveq2d 6312 . . . . . . . 8  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( (/) 
^o  ( B  .o  C ) )  =  ( (/)  ^o  (/) ) )
2019, 2syl6eq 2514 . . . . . . 7  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  ( (/) 
^o  ( B  .o  C ) )  =  1o )
2116, 20eqtr4d 2501 . . . . . 6  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( B  =  (/)  \/  C  =  (/) ) )  ->  (
( (/)  ^o  B )  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
22 on0eln0 4942 . . . . . . . . . 10  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  B  =/=  (/) ) )
23 on0eln0 4942 . . . . . . . . . 10  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  C  =/=  (/) ) )
2422, 23bi2anan9 873 . . . . . . . . 9  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <-> 
( B  =/=  (/)  /\  C  =/=  (/) ) ) )
25 neanior 2782 . . . . . . . . 9  |-  ( ( B  =/=  (/)  /\  C  =/=  (/) )  <->  -.  ( B  =  (/)  \/  C  =  (/) ) )
2624, 25syl6bb 261 . . . . . . . 8  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <->  -.  ( B  =  (/)  \/  C  =  (/) ) ) )
27 oe0m1 7189 . . . . . . . . . . . . . 14  |-  ( B  e.  On  ->  ( (/) 
e.  B  <->  ( (/)  ^o  B
)  =  (/) ) )
2827biimpa 484 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( (/)  ^o  B )  =  (/) )
2928oveq1d 6311 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  (/) 
e.  B )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  C
) )
30 oe0m1 7189 . . . . . . . . . . . . 13  |-  ( C  e.  On  ->  ( (/) 
e.  C  <->  ( (/)  ^o  C
)  =  (/) ) )
3130biimpa 484 . . . . . . . . . . . 12  |-  ( ( C  e.  On  /\  (/) 
e.  C )  -> 
( (/)  ^o  C )  =  (/) )
3229, 31sylan9eq 2518 . . . . . . . . . . 11  |-  ( ( ( B  e.  On  /\  (/)  e.  B )  /\  ( C  e.  On  /\  (/)  e.  C ) )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  (/) )
3332an4s 826 . . . . . . . . . 10  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( ( (/) 
^o  B )  ^o  C )  =  (/) )
34 om00el 7243 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  ( B  .o  C )  <->  ( (/)  e.  B  /\  (/)  e.  C ) ) )
35 omcl 7204 . . . . . . . . . . . . 13  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( B  .o  C
)  e.  On )
36 oe0m1 7189 . . . . . . . . . . . . 13  |-  ( ( B  .o  C )  e.  On  ->  ( (/) 
e.  ( B  .o  C )  <->  ( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3735, 36syl 16 . . . . . . . . . . . 12  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( (/)  e.  ( B  .o  C )  <->  ( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3834, 37bitr3d 255 . . . . . . . . . . 11  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  <-> 
( (/)  ^o  ( B  .o  C ) )  =  (/) ) )
3938biimpa 484 . . . . . . . . . 10  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( (/)  ^o  ( B  .o  C ) )  =  (/) )
4033, 39eqtr4d 2501 . . . . . . . . 9  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  ( (/)  e.  B  /\  (/)  e.  C ) )  ->  ( ( (/) 
^o  B )  ^o  C )  =  (
(/)  ^o  ( B  .o  C ) ) )
4140ex 434 . . . . . . . 8  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  e.  B  /\  (/)  e.  C )  ->  ( ( (/)  ^o  B )  ^o  C
)  =  ( (/)  ^o  ( B  .o  C
) ) ) )
4226, 41sylbird 235 . . . . . . 7  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( -.  ( B  =  (/)  \/  C  =  (/) )  ->  (
( (/)  ^o  B )  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) ) )
4342imp 429 . . . . . 6  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  -.  ( B  =  (/)  \/  C  =  (/) ) )  -> 
( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
4421, 43pm2.61dan 791 . . . . 5  |-  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( (/)  ^o  B
)  ^o  C )  =  ( (/)  ^o  ( B  .o  C ) ) )
45 oveq1 6303 . . . . . . 7  |-  ( A  =  (/)  ->  ( A  ^o  B )  =  ( (/)  ^o  B ) )
4645oveq1d 6311 . . . . . 6  |-  ( A  =  (/)  ->  ( ( A  ^o  B )  ^o  C )  =  ( ( (/)  ^o  B
)  ^o  C )
)
47 oveq1 6303 . . . . . 6  |-  ( A  =  (/)  ->  ( A  ^o  ( B  .o  C ) )  =  ( (/)  ^o  ( B  .o  C ) ) )
4846, 47eqeq12d 2479 . . . . 5  |-  ( A  =  (/)  ->  ( ( ( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) )  <->  ( ( (/) 
^o  B )  ^o  C )  =  (
(/)  ^o  ( B  .o  C ) ) ) )
4944, 48syl5ibr 221 . . . 4  |-  ( A  =  (/)  ->  ( ( B  e.  On  /\  C  e.  On )  ->  ( ( A  ^o  B )  ^o  C
)  =  ( A  ^o  ( B  .o  C ) ) ) )
5049impcom 430 . . 3  |-  ( ( ( B  e.  On  /\  C  e.  On )  /\  A  =  (/) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
51 oveq1 6303 . . . . . . . . 9  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( A  ^o  B )  =  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  ^o  B ) )
5251oveq1d 6311 . . . . . . . 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 6303 . . . . . . . 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 2479 . . . . . . 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 316 . . . . . 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 2529 . . . . . . . . . 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 2530 . . . . . . . . . 10  |-  ( A  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( (/)  e.  A  <->  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o ) ) )
5856, 57anbi12d 710 . . . . . . . . 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 2529 . . . . . . . . . 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 2530 . . . . . . . . . 10  |-  ( 1o  =  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o )  ->  ( (/)  e.  1o  <->  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o ) ) )
6159, 60anbi12d 710 . . . . . . . . 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 7155 . . . . . . . . . 10  |-  1o  e.  On
63 0lt1o 7172 . . . . . . . . . 10  |-  (/)  e.  1o
6462, 63pm3.2i 455 . . . . . . . . 9  |-  ( 1o  e.  On  /\  (/)  e.  1o )
6558, 61, 64elimhyp 4003 . . . . . . . 8  |-  ( if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )  e.  On  /\  (/)  e.  if ( ( A  e.  On  /\  (/) 
e.  A ) ,  A ,  1o ) )
6665simpli 458 . . . . . . 7  |-  if ( ( A  e.  On  /\  (/)  e.  A ) ,  A ,  1o )  e.  On
6765simpri 462 . . . . . . 7  |-  (/)  e.  if ( ( A  e.  On  /\  (/)  e.  A
) ,  A ,  1o )
6866, 67oeoelem 7265 . . . . . 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 3996 . . . . 5  |-  ( ( A  e.  On  /\  (/) 
e.  A )  -> 
( ( B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) ) ) )
7069imp 429 . . . 4  |-  ( ( ( A  e.  On  /\  (/)  e.  A )  /\  ( B  e.  On  /\  C  e.  On ) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
7170an32s 804 . . 3  |-  ( ( ( A  e.  On  /\  ( B  e.  On  /\  C  e.  On ) )  /\  (/)  e.  A
)  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
7250, 71oe0lem 7181 . 2  |-  ( ( A  e.  On  /\  ( B  e.  On  /\  C  e.  On ) )  ->  ( ( A  ^o  B )  ^o  C )  =  ( A  ^o  ( B  .o  C ) ) )
73723impb 1192 1  |-  ( ( A  e.  On  /\  B  e.  On  /\  C  e.  On )  ->  (
( A  ^o  B
)  ^o  C )  =  ( A  ^o  ( B  .o  C
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    =/= wne 2652   (/)c0 3793   ifcif 3944   Oncon0 4887  (class class class)co 6296   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-rep 4568  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-rmo 2815  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-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-2o 7149  df-oadd 7152  df-omul 7153  df-oexp 7154
This theorem is referenced by:  infxpenc  8412  infxpencOLD  8417
  Copyright terms: Public domain W3C validator