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Theorem oa0r 6718
Description: Ordinal addition with zero. Proposition 8.3 of [TakeutiZaring] p. 57. (Contributed by NM, 5-May-1995.)
Assertion
Ref Expression
oa0r  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )

Proof of Theorem oa0r
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6028 . . 3  |-  ( x  =  (/)  ->  ( (/)  +o  x )  =  (
(/)  +o  (/) ) )
2 id 20 . . 3  |-  ( x  =  (/)  ->  x  =  (/) )
31, 2eqeq12d 2401 . 2  |-  ( x  =  (/)  ->  ( (
(/)  +o  x )  =  x  <->  ( (/)  +o  (/) )  =  (/) ) )
4 oveq2 6028 . . 3  |-  ( x  =  y  ->  ( (/) 
+o  x )  =  ( (/)  +o  y
) )
5 id 20 . . 3  |-  ( x  =  y  ->  x  =  y )
64, 5eqeq12d 2401 . 2  |-  ( x  =  y  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  y
)  =  y ) )
7 oveq2 6028 . . 3  |-  ( x  =  suc  y  -> 
( (/)  +o  x )  =  ( (/)  +o  suc  y ) )
8 id 20 . . 3  |-  ( x  =  suc  y  ->  x  =  suc  y )
97, 8eqeq12d 2401 . 2  |-  ( x  =  suc  y  -> 
( ( (/)  +o  x
)  =  x  <->  ( (/)  +o  suc  y )  =  suc  y ) )
10 oveq2 6028 . . 3  |-  ( x  =  A  ->  ( (/) 
+o  x )  =  ( (/)  +o  A
) )
11 id 20 . . 3  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2401 . 2  |-  ( x  =  A  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  A
)  =  A ) )
13 0elon 4575 . . 3  |-  (/)  e.  On
14 oa0 6696 . . 3  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
1513, 14ax-mp 8 . 2  |-  ( (/)  +o  (/) )  =  (/)
16 oasuc 6704 . . . . 5  |-  ( (
(/)  e.  On  /\  y  e.  On )  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
1713, 16mpan 652 . . . 4  |-  ( y  e.  On  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
18 suceq 4587 . . . 4  |-  ( (
(/)  +o  y )  =  y  ->  suc  ( (/) 
+o  y )  =  suc  y )
1917, 18sylan9eq 2439 . . 3  |-  ( ( y  e.  On  /\  ( (/)  +o  y )  =  y )  -> 
( (/)  +o  suc  y
)  =  suc  y
)
2019ex 424 . 2  |-  ( y  e.  On  ->  (
( (/)  +o  y )  =  y  ->  ( (/) 
+o  suc  y )  =  suc  y ) )
21 iuneq2 4051 . . . 4  |-  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  U_ y  e.  x  ( (/)  +o  y
)  =  U_ y  e.  x  y )
22 uniiun 4085 . . . 4  |-  U. x  =  U_ y  e.  x  y
2321, 22syl6eqr 2437 . . 3  |-  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  U_ y  e.  x  ( (/)  +o  y
)  =  U. x
)
24 vex 2902 . . . . 5  |-  x  e. 
_V
25 oalim 6712 . . . . . 6  |-  ( (
(/)  e.  On  /\  (
x  e.  _V  /\  Lim  x ) )  -> 
( (/)  +o  x )  =  U_ y  e.  x  ( (/)  +o  y
) )
2613, 25mpan 652 . . . . 5  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( (/) 
+o  x )  = 
U_ y  e.  x  ( (/)  +o  y ) )
2724, 26mpan 652 . . . 4  |-  ( Lim  x  ->  ( (/)  +o  x
)  =  U_ y  e.  x  ( (/)  +o  y
) )
28 limuni 4582 . . . 4  |-  ( Lim  x  ->  x  =  U. x )
2927, 28eqeq12d 2401 . . 3  |-  ( Lim  x  ->  ( ( (/) 
+o  x )  =  x  <->  U_ y  e.  x  ( (/)  +o  y )  =  U. x ) )
3023, 29syl5ibr 213 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  ( (/)  +o  x
)  =  x ) )
313, 6, 9, 12, 15, 20, 30tfinds 4779 1  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2649   _Vcvv 2899   (/)c0 3571   U.cuni 3957   U_ciun 4035   Oncon0 4522   Lim wlim 4523   suc csuc 4524  (class class class)co 6020    +o coa 6657
This theorem is referenced by:  om1  6721  oaword2  6732  oeeui  6781  oaabs2  6824  cantnfp1  7570
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-recs 6569  df-rdg 6604  df-oadd 6664
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