MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  oa0r Structured version   Unicode version

Theorem oa0r 7178
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 6283 . . 3  |-  ( x  =  (/)  ->  ( (/)  +o  x )  =  (
(/)  +o  (/) ) )
2 id 22 . . 3  |-  ( x  =  (/)  ->  x  =  (/) )
31, 2eqeq12d 2482 . 2  |-  ( x  =  (/)  ->  ( (
(/)  +o  x )  =  x  <->  ( (/)  +o  (/) )  =  (/) ) )
4 oveq2 6283 . . 3  |-  ( x  =  y  ->  ( (/) 
+o  x )  =  ( (/)  +o  y
) )
5 id 22 . . 3  |-  ( x  =  y  ->  x  =  y )
64, 5eqeq12d 2482 . 2  |-  ( x  =  y  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  y
)  =  y ) )
7 oveq2 6283 . . 3  |-  ( x  =  suc  y  -> 
( (/)  +o  x )  =  ( (/)  +o  suc  y ) )
8 id 22 . . 3  |-  ( x  =  suc  y  ->  x  =  suc  y )
97, 8eqeq12d 2482 . 2  |-  ( x  =  suc  y  -> 
( ( (/)  +o  x
)  =  x  <->  ( (/)  +o  suc  y )  =  suc  y ) )
10 oveq2 6283 . . 3  |-  ( x  =  A  ->  ( (/) 
+o  x )  =  ( (/)  +o  A
) )
11 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2482 . 2  |-  ( x  =  A  ->  (
( (/)  +o  x )  =  x  <->  ( (/)  +o  A
)  =  A ) )
13 0elon 4924 . . 3  |-  (/)  e.  On
14 oa0 7156 . . 3  |-  ( (/)  e.  On  ->  ( (/)  +o  (/) )  =  (/) )
1513, 14ax-mp 5 . 2  |-  ( (/)  +o  (/) )  =  (/)
16 oasuc 7164 . . . . 5  |-  ( (
(/)  e.  On  /\  y  e.  On )  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
1713, 16mpan 670 . . . 4  |-  ( y  e.  On  ->  ( (/) 
+o  suc  y )  =  suc  ( (/)  +o  y
) )
18 suceq 4936 . . . 4  |-  ( (
(/)  +o  y )  =  y  ->  suc  ( (/) 
+o  y )  =  suc  y )
1917, 18sylan9eq 2521 . . 3  |-  ( ( y  e.  On  /\  ( (/)  +o  y )  =  y )  -> 
( (/)  +o  suc  y
)  =  suc  y
)
2019ex 434 . 2  |-  ( y  e.  On  ->  (
( (/)  +o  y )  =  y  ->  ( (/) 
+o  suc  y )  =  suc  y ) )
21 iuneq2 4335 . . . 4  |-  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  U_ y  e.  x  ( (/)  +o  y
)  =  U_ y  e.  x  y )
22 uniiun 4371 . . . 4  |-  U. x  =  U_ y  e.  x  y
2321, 22syl6eqr 2519 . . 3  |-  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  U_ y  e.  x  ( (/)  +o  y
)  =  U. x
)
24 vex 3109 . . . . 5  |-  x  e. 
_V
25 oalim 7172 . . . . . 6  |-  ( (
(/)  e.  On  /\  (
x  e.  _V  /\  Lim  x ) )  -> 
( (/)  +o  x )  =  U_ y  e.  x  ( (/)  +o  y
) )
2613, 25mpan 670 . . . . 5  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( (/) 
+o  x )  = 
U_ y  e.  x  ( (/)  +o  y ) )
2724, 26mpan 670 . . . 4  |-  ( Lim  x  ->  ( (/)  +o  x
)  =  U_ y  e.  x  ( (/)  +o  y
) )
28 limuni 4931 . . . 4  |-  ( Lim  x  ->  x  =  U. x )
2927, 28eqeq12d 2482 . . 3  |-  ( Lim  x  ->  ( ( (/) 
+o  x )  =  x  <->  U_ y  e.  x  ( (/)  +o  y )  =  U. x ) )
3023, 29syl5ibr 221 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( (/) 
+o  y )  =  y  ->  ( (/)  +o  x
)  =  x ) )
313, 6, 9, 12, 15, 20, 30tfinds 6665 1  |-  ( A  e.  On  ->  ( (/) 
+o  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   _Vcvv 3106   (/)c0 3778   U.cuni 4238   U_ciun 4318   Oncon0 4871   Lim wlim 4872   suc csuc 4873  (class class class)co 6275    +o coa 7117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-recs 7032  df-rdg 7066  df-oadd 7124
This theorem is referenced by:  om1  7181  oaword2  7192  oeeui  7241  oaabs2  7284  cantnfp1  8089  cantnfp1OLD  8115
  Copyright terms: Public domain W3C validator