MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oe1m Structured version   Unicode version

Theorem oe1m 6780
Description: Ordinal exponentiation with a mantissa of 1. Proposition 8.31(3) of [TakeutiZaring] p. 67. (Contributed by NM, 2-Jan-2005.)
Assertion
Ref Expression
oe1m  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )

Proof of Theorem oe1m
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6081 . . 3  |-  ( x  =  (/)  ->  ( 1o 
^o  x )  =  ( 1o  ^o  (/) ) )
21eqeq1d 2443 . 2  |-  ( x  =  (/)  ->  ( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  (/) )  =  1o ) )
3 oveq2 6081 . . 3  |-  ( x  =  y  ->  ( 1o  ^o  x )  =  ( 1o  ^o  y
) )
43eqeq1d 2443 . 2  |-  ( x  =  y  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  y )  =  1o ) )
5 oveq2 6081 . . 3  |-  ( x  =  suc  y  -> 
( 1o  ^o  x
)  =  ( 1o 
^o  suc  y )
)
65eqeq1d 2443 . 2  |-  ( x  =  suc  y  -> 
( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  suc  y
)  =  1o ) )
7 oveq2 6081 . . 3  |-  ( x  =  A  ->  ( 1o  ^o  x )  =  ( 1o  ^o  A
) )
87eqeq1d 2443 . 2  |-  ( x  =  A  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  A )  =  1o ) )
9 1on 6723 . . 3  |-  1o  e.  On
10 oe0 6758 . . 3  |-  ( 1o  e.  On  ->  ( 1o  ^o  (/) )  =  1o )
119, 10ax-mp 8 . 2  |-  ( 1o 
^o  (/) )  =  1o
12 oesuc 6763 . . . . 5  |-  ( ( 1o  e.  On  /\  y  e.  On )  ->  ( 1o  ^o  suc  y )  =  ( ( 1o  ^o  y
)  .o  1o ) )
139, 12mpan 652 . . . 4  |-  ( y  e.  On  ->  ( 1o  ^o  suc  y )  =  ( ( 1o 
^o  y )  .o  1o ) )
14 oveq1 6080 . . . . 5  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  ( 1o  .o  1o ) )
15 om1 6777 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  .o  1o )  =  1o )
169, 15ax-mp 8 . . . . 5  |-  ( 1o 
.o  1o )  =  1o
1714, 16syl6eq 2483 . . . 4  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  1o )
1813, 17sylan9eq 2487 . . 3  |-  ( ( y  e.  On  /\  ( 1o  ^o  y
)  =  1o )  ->  ( 1o  ^o  suc  y )  =  1o )
1918ex 424 . 2  |-  ( y  e.  On  ->  (
( 1o  ^o  y
)  =  1o  ->  ( 1o  ^o  suc  y
)  =  1o ) )
20 iuneq2 4101 . . 3  |-  ( A. y  e.  x  ( 1o  ^o  y )  =  1o  ->  U_ y  e.  x  ( 1o  ^o  y )  =  U_ y  e.  x  1o )
21 vex 2951 . . . . . 6  |-  x  e. 
_V
22 0lt1o 6740 . . . . . . . 8  |-  (/)  e.  1o
23 oelim 6770 . . . . . . . 8  |-  ( ( ( 1o  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  /\  (/)  e.  1o )  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
2422, 23mpan2 653 . . . . . . 7  |-  ( ( 1o  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
259, 24mpan 652 . . . . . 6  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( 1o  ^o  x )  = 
U_ y  e.  x  ( 1o  ^o  y
) )
2621, 25mpan 652 . . . . 5  |-  ( Lim  x  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
2726eqeq1d 2443 . . . 4  |-  ( Lim  x  ->  ( ( 1o  ^o  x )  =  1o  <->  U_ y  e.  x  ( 1o  ^o  y
)  =  1o ) )
28 0ellim 4635 . . . . . 6  |-  ( Lim  x  ->  (/)  e.  x
)
29 ne0i 3626 . . . . . 6  |-  ( (/)  e.  x  ->  x  =/=  (/) )
30 iunconst 4093 . . . . . 6  |-  ( x  =/=  (/)  ->  U_ y  e.  x  1o  =  1o )
3128, 29, 303syl 19 . . . . 5  |-  ( Lim  x  ->  U_ y  e.  x  1o  =  1o )
3231eqeq2d 2446 . . . 4  |-  ( Lim  x  ->  ( U_ y  e.  x  ( 1o  ^o  y )  = 
U_ y  e.  x  1o 
<-> 
U_ y  e.  x  ( 1o  ^o  y
)  =  1o ) )
3327, 32bitr4d 248 . . 3  |-  ( Lim  x  ->  ( ( 1o  ^o  x )  =  1o  <->  U_ y  e.  x  ( 1o  ^o  y
)  =  U_ y  e.  x  1o )
)
3420, 33syl5ibr 213 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( 1o  ^o  y )  =  1o  ->  ( 1o  ^o  x )  =  1o ) )
352, 4, 6, 8, 11, 19, 34tfinds 4831 1  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   _Vcvv 2948   (/)c0 3620   U_ciun 4085   Oncon0 4573   Lim wlim 4574   suc csuc 4575  (class class class)co 6073   1oc1o 6709    .o comu 6714    ^o coe 6715
This theorem is referenced by:  oewordi  6826  oeoe  6834  cantnflem2  7638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-omul 6721  df-oexp 6722
  Copyright terms: Public domain W3C validator