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Theorem oa0r 6553
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 5882 . . 3  |-  ( x  =  (/)  ->  ( (/)  +o  x )  =  (
(/)  +o  (/) ) )
2 id 19 . . 3  |-  ( x  =  (/)  ->  x  =  (/) )
31, 2eqeq12d 2310 . 2  |-  ( x  =  (/)  ->  ( (
(/)  +o  x )  =  x  <->  ( (/)  +o  (/) )  =  (/) ) )
4 oveq2 5882 . . 3  |-  ( x  =  y  ->  ( (/) 
+o  x )  =  ( (/)  +o  y
) )
5 id 19 . . 3  |-  ( x  =  y  ->  x  =  y )
64, 5eqeq12d 2310 . 2  |-  ( x  =  y  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  y
)  =  y ) )
7 oveq2 5882 . . 3  |-  ( x  =  suc  y  -> 
( (/)  +o  x )  =  ( (/)  +o  suc  y ) )
8 id 19 . . 3  |-  ( x  =  suc  y  ->  x  =  suc  y )
97, 8eqeq12d 2310 . 2  |-  ( x  =  suc  y  -> 
( ( (/)  +o  x
)  =  x  <->  ( (/)  +o  suc  y )  =  suc  y ) )
10 oveq2 5882 . . 3  |-  ( x  =  A  ->  ( (/) 
+o  x )  =  ( (/)  +o  A
) )
11 id 19 . . 3  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2310 . 2  |-  ( x  =  A  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  A
)  =  A ) )
13 0elon 4461 . . 3  |-  (/)  e.  On
14 oa0 6531 . . 3  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
1513, 14ax-mp 8 . 2  |-  ( (/)  +o  (/) )  =  (/)
16 oasuc 6539 . . . . 5  |-  ( (
(/)  e.  On  /\  y  e.  On )  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
1713, 16mpan 651 . . . 4  |-  ( y  e.  On  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
18 suceq 4473 . . . 4  |-  ( (
(/)  +o  y )  =  y  ->  suc  ( (/) 
+o  y )  =  suc  y )
1917, 18sylan9eq 2348 . . 3  |-  ( ( y  e.  On  /\  ( (/)  +o  y )  =  y )  -> 
( (/)  +o  suc  y
)  =  suc  y
)
2019ex 423 . 2  |-  ( y  e.  On  ->  (
( (/)  +o  y )  =  y  ->  ( (/) 
+o  suc  y )  =  suc  y ) )
21 iuneq2 3937 . . . 4  |-  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  U_ y  e.  x  ( (/)  +o  y
)  =  U_ y  e.  x  y )
22 uniiun 3971 . . . 4  |-  U. x  =  U_ y  e.  x  y
2321, 22syl6eqr 2346 . . 3  |-  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  U_ y  e.  x  ( (/)  +o  y
)  =  U. x
)
24 vex 2804 . . . . 5  |-  x  e. 
_V
25 oalim 6547 . . . . . 6  |-  ( (
(/)  e.  On  /\  (
x  e.  _V  /\  Lim  x ) )  -> 
( (/)  +o  x )  =  U_ y  e.  x  ( (/)  +o  y
) )
2613, 25mpan 651 . . . . 5  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( (/) 
+o  x )  = 
U_ y  e.  x  ( (/)  +o  y ) )
2724, 26mpan 651 . . . 4  |-  ( Lim  x  ->  ( (/)  +o  x
)  =  U_ y  e.  x  ( (/)  +o  y
) )
28 limuni 4468 . . . 4  |-  ( Lim  x  ->  x  =  U. x )
2927, 28eqeq12d 2310 . . 3  |-  ( Lim  x  ->  ( ( (/) 
+o  x )  =  x  <->  U_ y  e.  x  ( (/)  +o  y )  =  U. x ) )
3023, 29syl5ibr 212 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  ( (/)  +o  x
)  =  x ) )
313, 6, 9, 12, 15, 20, 30tfinds 4666 1  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801   (/)c0 3468   U.cuni 3843   U_ciun 3921   Oncon0 4408   Lim wlim 4409   suc csuc 4410  (class class class)co 5874    +o coa 6492
This theorem is referenced by:  om1  6556  oaword2  6567  oeeui  6616  oaabs2  6659  cantnfp1  7399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-oadd 6499
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