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Theorem oe1m 6755
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 6056 . . 3  |-  ( x  =  (/)  ->  ( 1o 
^o  x )  =  ( 1o  ^o  (/) ) )
21eqeq1d 2420 . 2  |-  ( x  =  (/)  ->  ( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  (/) )  =  1o ) )
3 oveq2 6056 . . 3  |-  ( x  =  y  ->  ( 1o  ^o  x )  =  ( 1o  ^o  y
) )
43eqeq1d 2420 . 2  |-  ( x  =  y  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  y )  =  1o ) )
5 oveq2 6056 . . 3  |-  ( x  =  suc  y  -> 
( 1o  ^o  x
)  =  ( 1o 
^o  suc  y )
)
65eqeq1d 2420 . 2  |-  ( x  =  suc  y  -> 
( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  suc  y
)  =  1o ) )
7 oveq2 6056 . . 3  |-  ( x  =  A  ->  ( 1o  ^o  x )  =  ( 1o  ^o  A
) )
87eqeq1d 2420 . 2  |-  ( x  =  A  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  A )  =  1o ) )
9 1on 6698 . . 3  |-  1o  e.  On
10 oe0 6733 . . 3  |-  ( 1o  e.  On  ->  ( 1o  ^o  (/) )  =  1o )
119, 10ax-mp 8 . 2  |-  ( 1o 
^o  (/) )  =  1o
12 oesuc 6738 . . . . 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 6055 . . . . 5  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  ( 1o  .o  1o ) )
15 om1 6752 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  .o  1o )  =  1o )
169, 15ax-mp 8 . . . . 5  |-  ( 1o 
.o  1o )  =  1o
1714, 16syl6eq 2460 . . . 4  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  1o )
1813, 17sylan9eq 2464 . . 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 4077 . . 3  |-  ( A. y  e.  x  ( 1o  ^o  y )  =  1o  ->  U_ y  e.  x  ( 1o  ^o  y )  =  U_ y  e.  x  1o )
21 vex 2927 . . . . . 6  |-  x  e. 
_V
22 0lt1o 6715 . . . . . . . 8  |-  (/)  e.  1o
23 oelim 6745 . . . . . . . 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 2420 . . . 4  |-  ( Lim  x  ->  ( ( 1o  ^o  x )  =  1o  <->  U_ y  e.  x  ( 1o  ^o  y
)  =  1o ) )
28 0ellim 4611 . . . . . 6  |-  ( Lim  x  ->  (/)  e.  x
)
29 ne0i 3602 . . . . . 6  |-  ( (/)  e.  x  ->  x  =/=  (/) )
30 iunconst 4069 . . . . . 6  |-  ( x  =/=  (/)  ->  U_ y  e.  x  1o  =  1o )
3128, 29, 303syl 19 . . . . 5  |-  ( Lim  x  ->  U_ y  e.  x  1o  =  1o )
3231eqeq2d 2423 . . . 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 4806 1  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    =/= wne 2575   A.wral 2674   _Vcvv 2924   (/)c0 3596   U_ciun 4061   Oncon0 4549   Lim wlim 4550   suc csuc 4551  (class class class)co 6048   1oc1o 6684    .o comu 6689    ^o coe 6690
This theorem is referenced by:  oewordi  6801  oeoe  6809  cantnflem2  7610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-omul 6696  df-oexp 6697
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