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Theorem om1r 7210
Description: Ordinal multiplication with 1. Proposition 8.18(2) of [TakeutiZaring] p. 63. (Contributed by NM, 3-Aug-2004.)
Assertion
Ref Expression
om1r  |-  ( A  e.  On  ->  ( 1o  .o  A )  =  A )

Proof of Theorem om1r
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6304 . . 3  |-  ( x  =  (/)  ->  ( 1o 
.o  x )  =  ( 1o  .o  (/) ) )
2 id 22 . . 3  |-  ( x  =  (/)  ->  x  =  (/) )
31, 2eqeq12d 2479 . 2  |-  ( x  =  (/)  ->  ( ( 1o  .o  x )  =  x  <->  ( 1o  .o  (/) )  =  (/) ) )
4 oveq2 6304 . . 3  |-  ( x  =  y  ->  ( 1o  .o  x )  =  ( 1o  .o  y
) )
5 id 22 . . 3  |-  ( x  =  y  ->  x  =  y )
64, 5eqeq12d 2479 . 2  |-  ( x  =  y  ->  (
( 1o  .o  x
)  =  x  <->  ( 1o  .o  y )  =  y ) )
7 oveq2 6304 . . 3  |-  ( x  =  suc  y  -> 
( 1o  .o  x
)  =  ( 1o 
.o  suc  y )
)
8 id 22 . . 3  |-  ( x  =  suc  y  ->  x  =  suc  y )
97, 8eqeq12d 2479 . 2  |-  ( x  =  suc  y  -> 
( ( 1o  .o  x )  =  x  <-> 
( 1o  .o  suc  y )  =  suc  y ) )
10 oveq2 6304 . . 3  |-  ( x  =  A  ->  ( 1o  .o  x )  =  ( 1o  .o  A
) )
11 id 22 . . 3  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2479 . 2  |-  ( x  =  A  ->  (
( 1o  .o  x
)  =  x  <->  ( 1o  .o  A )  =  A ) )
13 om0x 7187 . 2  |-  ( 1o 
.o  (/) )  =  (/)
14 1on 7155 . . . . . 6  |-  1o  e.  On
15 omsuc 7194 . . . . . 6  |-  ( ( 1o  e.  On  /\  y  e.  On )  ->  ( 1o  .o  suc  y )  =  ( ( 1o  .o  y
)  +o  1o ) )
1614, 15mpan 670 . . . . 5  |-  ( y  e.  On  ->  ( 1o  .o  suc  y )  =  ( ( 1o 
.o  y )  +o  1o ) )
17 oveq1 6303 . . . . 5  |-  ( ( 1o  .o  y )  =  y  ->  (
( 1o  .o  y
)  +o  1o )  =  ( y  +o  1o ) )
1816, 17sylan9eq 2518 . . . 4  |-  ( ( y  e.  On  /\  ( 1o  .o  y
)  =  y )  ->  ( 1o  .o  suc  y )  =  ( y  +o  1o ) )
19 oa1suc 7199 . . . . 5  |-  ( y  e.  On  ->  (
y  +o  1o )  =  suc  y )
2019adantr 465 . . . 4  |-  ( ( y  e.  On  /\  ( 1o  .o  y
)  =  y )  ->  ( y  +o  1o )  =  suc  y )
2118, 20eqtrd 2498 . . 3  |-  ( ( y  e.  On  /\  ( 1o  .o  y
)  =  y )  ->  ( 1o  .o  suc  y )  =  suc  y )
2221ex 434 . 2  |-  ( y  e.  On  ->  (
( 1o  .o  y
)  =  y  -> 
( 1o  .o  suc  y )  =  suc  y ) )
23 iuneq2 4349 . . . 4  |-  ( A. y  e.  x  ( 1o  .o  y )  =  y  ->  U_ y  e.  x  ( 1o  .o  y )  =  U_ y  e.  x  y
)
24 uniiun 4385 . . . 4  |-  U. x  =  U_ y  e.  x  y
2523, 24syl6eqr 2516 . . 3  |-  ( A. y  e.  x  ( 1o  .o  y )  =  y  ->  U_ y  e.  x  ( 1o  .o  y )  =  U. x )
26 vex 3112 . . . . 5  |-  x  e. 
_V
27 omlim 7201 . . . . . 6  |-  ( ( 1o  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( 1o  .o  x )  =  U_ y  e.  x  ( 1o  .o  y ) )
2814, 27mpan 670 . . . . 5  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( 1o  .o  x )  = 
U_ y  e.  x  ( 1o  .o  y
) )
2926, 28mpan 670 . . . 4  |-  ( Lim  x  ->  ( 1o  .o  x )  =  U_ y  e.  x  ( 1o  .o  y ) )
30 limuni 4947 . . . 4  |-  ( Lim  x  ->  x  =  U. x )
3129, 30eqeq12d 2479 . . 3  |-  ( Lim  x  ->  ( ( 1o  .o  x )  =  x  <->  U_ y  e.  x  ( 1o  .o  y
)  =  U. x
) )
3225, 31syl5ibr 221 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( 1o  .o  y )  =  y  ->  ( 1o  .o  x )  =  x ) )
333, 6, 9, 12, 13, 22, 32tfinds 6693 1  |-  ( A  e.  On  ->  ( 1o  .o  A )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807   _Vcvv 3109   (/)c0 3793   U.cuni 4251   U_ciun 4332   Oncon0 4887   Lim wlim 4888   suc csuc 4889  (class class class)co 6296   1oc1o 7141    +o coa 7145    .o comu 7146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-omul 7153
This theorem is referenced by:  oe1  7211  omword2  7241
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