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Theorem oe1m 6511
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
StepHypRef Expression
1 oveq2 5800 . . 3  |-  ( x  =  (/)  ->  ( 1o 
^o  x )  =  ( 1o  ^o  (/) ) )
21eqeq1d 2266 . 2  |-  ( x  =  (/)  ->  ( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  (/) )  =  1o ) )
3 oveq2 5800 . . 3  |-  ( x  =  y  ->  ( 1o  ^o  x )  =  ( 1o  ^o  y
) )
43eqeq1d 2266 . 2  |-  ( x  =  y  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  y )  =  1o ) )
5 oveq2 5800 . . 3  |-  ( x  =  suc  y  -> 
( 1o  ^o  x
)  =  ( 1o 
^o  suc  y )
)
65eqeq1d 2266 . 2  |-  ( x  =  suc  y  -> 
( ( 1o  ^o  x )  =  1o  <->  ( 1o  ^o  suc  y
)  =  1o ) )
7 oveq2 5800 . . 3  |-  ( x  =  A  ->  ( 1o  ^o  x )  =  ( 1o  ^o  A
) )
87eqeq1d 2266 . 2  |-  ( x  =  A  ->  (
( 1o  ^o  x
)  =  1o  <->  ( 1o  ^o  A )  =  1o ) )
9 1on 6454 . . 3  |-  1o  e.  On
10 oe0 6489 . . 3  |-  ( 1o  e.  On  ->  ( 1o  ^o  (/) )  =  1o )
119, 10ax-mp 10 . 2  |-  ( 1o 
^o  (/) )  =  1o
12 oesuc 6494 . . . . 5  |-  ( ( 1o  e.  On  /\  y  e.  On )  ->  ( 1o  ^o  suc  y )  =  ( ( 1o  ^o  y
)  .o  1o ) )
139, 12mpan 654 . . . 4  |-  ( y  e.  On  ->  ( 1o  ^o  suc  y )  =  ( ( 1o 
^o  y )  .o  1o ) )
14 oveq1 5799 . . . . 5  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  ( 1o  .o  1o ) )
15 om1 6508 . . . . . 6  |-  ( 1o  e.  On  ->  ( 1o  .o  1o )  =  1o )
169, 15ax-mp 10 . . . . 5  |-  ( 1o 
.o  1o )  =  1o
1714, 16syl6eq 2306 . . . 4  |-  ( ( 1o  ^o  y )  =  1o  ->  (
( 1o  ^o  y
)  .o  1o )  =  1o )
1813, 17sylan9eq 2310 . . 3  |-  ( ( y  e.  On  /\  ( 1o  ^o  y
)  =  1o )  ->  ( 1o  ^o  suc  y )  =  1o )
1918ex 425 . 2  |-  ( y  e.  On  ->  (
( 1o  ^o  y
)  =  1o  ->  ( 1o  ^o  suc  y
)  =  1o ) )
20 iuneq2 3895 . . 3  |-  ( A. y  e.  x  ( 1o  ^o  y )  =  1o  ->  U_ y  e.  x  ( 1o  ^o  y )  =  U_ y  e.  x  1o )
21 vex 2766 . . . . . 6  |-  x  e. 
_V
22 0lt1o 6471 . . . . . . . 8  |-  (/)  e.  1o
23 oelim 6501 . . . . . . . 8  |-  ( ( ( 1o  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  /\  (/)  e.  1o )  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
2422, 23mpan2 655 . . . . . . 7  |-  ( ( 1o  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
259, 24mpan 654 . . . . . 6  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( 1o  ^o  x )  = 
U_ y  e.  x  ( 1o  ^o  y
) )
2621, 25mpan 654 . . . . 5  |-  ( Lim  x  ->  ( 1o  ^o  x )  =  U_ y  e.  x  ( 1o  ^o  y ) )
2726eqeq1d 2266 . . . 4  |-  ( Lim  x  ->  ( ( 1o  ^o  x )  =  1o  <->  U_ y  e.  x  ( 1o  ^o  y
)  =  1o ) )
28 0ellim 4426 . . . . . 6  |-  ( Lim  x  ->  (/)  e.  x
)
29 ne0i 3436 . . . . . 6  |-  ( (/)  e.  x  ->  x  =/=  (/) )
30 iunconst 3887 . . . . . 6  |-  ( x  =/=  (/)  ->  U_ y  e.  x  1o  =  1o )
3128, 29, 303syl 20 . . . . 5  |-  ( Lim  x  ->  U_ y  e.  x  1o  =  1o )
3231eqeq2d 2269 . . . 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 249 . . 3  |-  ( Lim  x  ->  ( ( 1o  ^o  x )  =  1o  <->  U_ y  e.  x  ( 1o  ^o  y
)  =  U_ y  e.  x  1o )
)
3420, 33syl5ibr 214 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( 1o  ^o  y )  =  1o  ->  ( 1o  ^o  x )  =  1o ) )
352, 4, 6, 8, 11, 19, 34tfinds 4622 1  |-  ( A  e.  On  ->  ( 1o  ^o  A )  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   A.wral 2518   _Vcvv 2763   (/)c0 3430   U_ciun 3879   Oncon0 4364   Lim wlim 4365   suc csuc 4366  (class class class)co 5792   1oc1o 6440    .o comu 6445    ^o coe 6446
This theorem is referenced by:  oewordi  6557  oeoe  6565  cantnflem2  7360
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-recs 6356  df-rdg 6391  df-1o 6447  df-oadd 6451  df-omul 6452  df-oexp 6453
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