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Theorem oa0r 6470
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
StepHypRef Expression
1 oveq2 5765 . . 3  |-  ( x  =  (/)  ->  ( (/)  +o  x )  =  (
(/)  +o  (/) ) )
2 id 21 . . 3  |-  ( x  =  (/)  ->  x  =  (/) )
31, 2eqeq12d 2270 . 2  |-  ( x  =  (/)  ->  ( (
(/)  +o  x )  =  x  <->  ( (/)  +o  (/) )  =  (/) ) )
4 oveq2 5765 . . 3  |-  ( x  =  y  ->  ( (/) 
+o  x )  =  ( (/)  +o  y
) )
5 id 21 . . 3  |-  ( x  =  y  ->  x  =  y )
64, 5eqeq12d 2270 . 2  |-  ( x  =  y  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  y
)  =  y ) )
7 oveq2 5765 . . 3  |-  ( x  =  suc  y  -> 
( (/)  +o  x )  =  ( (/)  +o  suc  y ) )
8 id 21 . . 3  |-  ( x  =  suc  y  ->  x  =  suc  y )
97, 8eqeq12d 2270 . 2  |-  ( x  =  suc  y  -> 
( ( (/)  +o  x
)  =  x  <->  ( (/)  +o  suc  y )  =  suc  y ) )
10 oveq2 5765 . . 3  |-  ( x  =  A  ->  ( (/) 
+o  x )  =  ( (/)  +o  A
) )
11 id 21 . . 3  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2270 . 2  |-  ( x  =  A  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  A
)  =  A ) )
13 0elon 4382 . . 3  |-  (/)  e.  On
14 oa0 6448 . . 3  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
1513, 14ax-mp 10 . 2  |-  ( (/)  +o  (/) )  =  (/)
16 oasuc 6456 . . . . 5  |-  ( (
(/)  e.  On  /\  y  e.  On )  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
1713, 16mpan 654 . . . 4  |-  ( y  e.  On  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
18 suceq 4394 . . . 4  |-  ( (
(/)  +o  y )  =  y  ->  suc  ( (/) 
+o  y )  =  suc  y )
1917, 18sylan9eq 2308 . . 3  |-  ( ( y  e.  On  /\  ( (/)  +o  y )  =  y )  -> 
( (/)  +o  suc  y
)  =  suc  y
)
2019ex 425 . 2  |-  ( y  e.  On  ->  (
( (/)  +o  y )  =  y  ->  ( (/) 
+o  suc  y )  =  suc  y ) )
21 iuneq2 3862 . . . 4  |-  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  U_ y  e.  x  ( (/)  +o  y
)  =  U_ y  e.  x  y )
22 uniiun 3896 . . . 4  |-  U. x  =  U_ y  e.  x  y
2321, 22syl6eqr 2306 . . 3  |-  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  U_ y  e.  x  ( (/)  +o  y
)  =  U. x
)
24 vex 2743 . . . . 5  |-  x  e. 
_V
25 oalim 6464 . . . . . 6  |-  ( (
(/)  e.  On  /\  (
x  e.  _V  /\  Lim  x ) )  -> 
( (/)  +o  x )  =  U_ y  e.  x  ( (/)  +o  y
) )
2613, 25mpan 654 . . . . 5  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( (/) 
+o  x )  = 
U_ y  e.  x  ( (/)  +o  y ) )
2724, 26mpan 654 . . . 4  |-  ( Lim  x  ->  ( (/)  +o  x
)  =  U_ y  e.  x  ( (/)  +o  y
) )
28 limuni 4389 . . . 4  |-  ( Lim  x  ->  x  =  U. x )
2927, 28eqeq12d 2270 . . 3  |-  ( Lim  x  ->  ( ( (/) 
+o  x )  =  x  <->  U_ y  e.  x  ( (/)  +o  y )  =  U. x ) )
3023, 29syl5ibr 214 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  ( (/)  +o  x
)  =  x ) )
313, 6, 9, 12, 15, 20, 30tfinds 4587 1  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   A.wral 2516   _Vcvv 2740   (/)c0 3397   U.cuni 3768   U_ciun 3846   Oncon0 4329   Lim wlim 4330   suc csuc 4331  (class class class)co 5757    +o coa 6409
This theorem is referenced by:  om1  6473  oaword2  6484  oeeui  6533  oaabs2  6576  cantnfp1  7316
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 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
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 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-recs 6321  df-rdg 6356  df-oadd 6416
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