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Theorem oe1m 6685
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 5989 . . 3  |-  ( x  =  (/)  ->  ( 1o 
^o  x )  =  ( 1o  ^o  (/) ) )
21eqeq1d 2374 . 2  |-  ( x  =  (/)  ->  ( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  (/) )  =  1o ) )
3 oveq2 5989 . . 3  |-  ( x  =  y  ->  ( 1o  ^o  x )  =  ( 1o  ^o  y
) )
43eqeq1d 2374 . 2  |-  ( x  =  y  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  y )  =  1o ) )
5 oveq2 5989 . . 3  |-  ( x  =  suc  y  -> 
( 1o  ^o  x
)  =  ( 1o 
^o  suc  y )
)
65eqeq1d 2374 . 2  |-  ( x  =  suc  y  -> 
( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  suc  y
)  =  1o ) )
7 oveq2 5989 . . 3  |-  ( x  =  A  ->  ( 1o  ^o  x )  =  ( 1o  ^o  A
) )
87eqeq1d 2374 . 2  |-  ( x  =  A  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  A )  =  1o ) )
9 1on 6628 . . 3  |-  1o  e.  On
10 oe0 6663 . . 3  |-  ( 1o  e.  On  ->  ( 1o  ^o  (/) )  =  1o )
119, 10ax-mp 8 . 2  |-  ( 1o 
^o  (/) )  =  1o
12 oesuc 6668 . . . . 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 5988 . . . . 5  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  ( 1o  .o  1o ) )
15 om1 6682 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  .o  1o )  =  1o )
169, 15ax-mp 8 . . . . 5  |-  ( 1o 
.o  1o )  =  1o
1714, 16syl6eq 2414 . . . 4  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  1o )
1813, 17sylan9eq 2418 . . 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 4023 . . 3  |-  ( A. y  e.  x  ( 1o  ^o  y )  =  1o  ->  U_ y  e.  x  ( 1o  ^o  y )  =  U_ y  e.  x  1o )
21 vex 2876 . . . . . 6  |-  x  e. 
_V
22 0lt1o 6645 . . . . . . . 8  |-  (/)  e.  1o
23 oelim 6675 . . . . . . . 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 2374 . . . 4  |-  ( Lim  x  ->  ( ( 1o  ^o  x )  =  1o  <->  U_ y  e.  x  ( 1o  ^o  y
)  =  1o ) )
28 0ellim 4557 . . . . . 6  |-  ( Lim  x  ->  (/)  e.  x
)
29 ne0i 3549 . . . . . 6  |-  ( (/)  e.  x  ->  x  =/=  (/) )
30 iunconst 4015 . . . . . 6  |-  ( x  =/=  (/)  ->  U_ y  e.  x  1o  =  1o )
3128, 29, 303syl 18 . . . . 5  |-  ( Lim  x  ->  U_ y  e.  x  1o  =  1o )
3231eqeq2d 2377 . . . 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 4753 1  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1647    e. wcel 1715    =/= wne 2529   A.wral 2628   _Vcvv 2873   (/)c0 3543   U_ciun 4007   Oncon0 4495   Lim wlim 4496   suc csuc 4497  (class class class)co 5981   1oc1o 6614    .o comu 6619    ^o coe 6620
This theorem is referenced by:  oewordi  6731  oeoe  6739  cantnflem2  7539
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-recs 6530  df-rdg 6565  df-1o 6621  df-oadd 6625  df-omul 6626  df-oexp 6627
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