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Theorem oe1m 6545
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 5868 . . 3  |-  ( x  =  (/)  ->  ( 1o 
^o  x )  =  ( 1o  ^o  (/) ) )
21eqeq1d 2293 . 2  |-  ( x  =  (/)  ->  ( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  (/) )  =  1o ) )
3 oveq2 5868 . . 3  |-  ( x  =  y  ->  ( 1o  ^o  x )  =  ( 1o  ^o  y
) )
43eqeq1d 2293 . 2  |-  ( x  =  y  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  y )  =  1o ) )
5 oveq2 5868 . . 3  |-  ( x  =  suc  y  -> 
( 1o  ^o  x
)  =  ( 1o 
^o  suc  y )
)
65eqeq1d 2293 . 2  |-  ( x  =  suc  y  -> 
( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  suc  y
)  =  1o ) )
7 oveq2 5868 . . 3  |-  ( x  =  A  ->  ( 1o  ^o  x )  =  ( 1o  ^o  A
) )
87eqeq1d 2293 . 2  |-  ( x  =  A  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  A )  =  1o ) )
9 1on 6488 . . 3  |-  1o  e.  On
10 oe0 6523 . . 3  |-  ( 1o  e.  On  ->  ( 1o  ^o  (/) )  =  1o )
119, 10ax-mp 8 . 2  |-  ( 1o 
^o  (/) )  =  1o
12 oesuc 6528 . . . . 5  |-  ( ( 1o  e.  On  /\  y  e.  On )  ->  ( 1o  ^o  suc  y )  =  ( ( 1o  ^o  y
)  .o  1o ) )
139, 12mpan 651 . . . 4  |-  ( y  e.  On  ->  ( 1o  ^o  suc  y )  =  ( ( 1o 
^o  y )  .o  1o ) )
14 oveq1 5867 . . . . 5  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  ( 1o  .o  1o ) )
15 om1 6542 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  .o  1o )  =  1o )
169, 15ax-mp 8 . . . . 5  |-  ( 1o 
.o  1o )  =  1o
1714, 16syl6eq 2333 . . . 4  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  1o )
1813, 17sylan9eq 2337 . . 3  |-  ( ( y  e.  On  /\  ( 1o  ^o  y
)  =  1o )  ->  ( 1o  ^o  suc  y )  =  1o )
1918ex 423 . 2  |-  ( y  e.  On  ->  (
( 1o  ^o  y
)  =  1o  ->  ( 1o  ^o  suc  y
)  =  1o ) )
20 iuneq2 3923 . . 3  |-  ( A. y  e.  x  ( 1o  ^o  y )  =  1o  ->  U_ y  e.  x  ( 1o  ^o  y )  =  U_ y  e.  x  1o )
21 vex 2793 . . . . . 6  |-  x  e. 
_V
22 0lt1o 6505 . . . . . . . 8  |-  (/)  e.  1o
23 oelim 6535 . . . . . . . 8  |-  ( ( ( 1o  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  /\  (/)  e.  1o )  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
2422, 23mpan2 652 . . . . . . 7  |-  ( ( 1o  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
259, 24mpan 651 . . . . . 6  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( 1o  ^o  x )  = 
U_ y  e.  x  ( 1o  ^o  y
) )
2621, 25mpan 651 . . . . 5  |-  ( Lim  x  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
2726eqeq1d 2293 . . . 4  |-  ( Lim  x  ->  ( ( 1o  ^o  x )  =  1o  <->  U_ y  e.  x  ( 1o  ^o  y
)  =  1o ) )
28 0ellim 4456 . . . . . 6  |-  ( Lim  x  ->  (/)  e.  x
)
29 ne0i 3463 . . . . . 6  |-  ( (/)  e.  x  ->  x  =/=  (/) )
30 iunconst 3915 . . . . . 6  |-  ( x  =/=  (/)  ->  U_ y  e.  x  1o  =  1o )
3128, 29, 303syl 18 . . . . 5  |-  ( Lim  x  ->  U_ y  e.  x  1o  =  1o )
3231eqeq2d 2296 . . . 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 247 . . 3  |-  ( Lim  x  ->  ( ( 1o  ^o  x )  =  1o  <->  U_ y  e.  x  ( 1o  ^o  y
)  =  U_ y  e.  x  1o )
)
3420, 33syl5ibr 212 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( 1o  ^o  y )  =  1o  ->  ( 1o  ^o  x )  =  1o ) )
352, 4, 6, 8, 11, 19, 34tfinds 4652 1  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1625    e. wcel 1686    =/= wne 2448   A.wral 2545   _Vcvv 2790   (/)c0 3457   U_ciun 3907   Oncon0 4394   Lim wlim 4395   suc csuc 4396  (class class class)co 5860   1oc1o 6474    .o comu 6479    ^o coe 6480
This theorem is referenced by:  oewordi  6591  oeoe  6599  cantnflem2  7394
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-recs 6390  df-rdg 6425  df-1o 6481  df-oadd 6485  df-omul 6486  df-oexp 6487
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