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Theorem om1r 6777
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 6080 . . 3  |-  ( x  =  (/)  ->  ( 1o 
.o  x )  =  ( 1o  .o  (/) ) )
2 id 20 . . 3  |-  ( x  =  (/)  ->  x  =  (/) )
31, 2eqeq12d 2449 . 2  |-  ( x  =  (/)  ->  ( ( 1o  .o  x )  =  x  <->  ( 1o  .o  (/) )  =  (/) ) )
4 oveq2 6080 . . 3  |-  ( x  =  y  ->  ( 1o  .o  x )  =  ( 1o  .o  y
) )
5 id 20 . . 3  |-  ( x  =  y  ->  x  =  y )
64, 5eqeq12d 2449 . 2  |-  ( x  =  y  ->  (
( 1o  .o  x
)  =  x  <->  ( 1o  .o  y )  =  y ) )
7 oveq2 6080 . . 3  |-  ( x  =  suc  y  -> 
( 1o  .o  x
)  =  ( 1o 
.o  suc  y )
)
8 id 20 . . 3  |-  ( x  =  suc  y  ->  x  =  suc  y )
97, 8eqeq12d 2449 . 2  |-  ( x  =  suc  y  -> 
( ( 1o  .o  x )  =  x  <-> 
( 1o  .o  suc  y )  =  suc  y ) )
10 oveq2 6080 . . 3  |-  ( x  =  A  ->  ( 1o  .o  x )  =  ( 1o  .o  A
) )
11 id 20 . . 3  |-  ( x  =  A  ->  x  =  A )
1210, 11eqeq12d 2449 . 2  |-  ( x  =  A  ->  (
( 1o  .o  x
)  =  x  <->  ( 1o  .o  A )  =  A ) )
13 om0x 6754 . 2  |-  ( 1o 
.o  (/) )  =  (/)
14 1on 6722 . . . . . 6  |-  1o  e.  On
15 omsuc 6761 . . . . . 6  |-  ( ( 1o  e.  On  /\  y  e.  On )  ->  ( 1o  .o  suc  y )  =  ( ( 1o  .o  y
)  +o  1o ) )
1614, 15mpan 652 . . . . 5  |-  ( y  e.  On  ->  ( 1o  .o  suc  y )  =  ( ( 1o 
.o  y )  +o  1o ) )
17 oveq1 6079 . . . . 5  |-  ( ( 1o  .o  y )  =  y  ->  (
( 1o  .o  y
)  +o  1o )  =  ( y  +o  1o ) )
1816, 17sylan9eq 2487 . . . 4  |-  ( ( y  e.  On  /\  ( 1o  .o  y
)  =  y )  ->  ( 1o  .o  suc  y )  =  ( y  +o  1o ) )
19 oa1suc 6766 . . . . 5  |-  ( y  e.  On  ->  (
y  +o  1o )  =  suc  y )
2019adantr 452 . . . 4  |-  ( ( y  e.  On  /\  ( 1o  .o  y
)  =  y )  ->  ( y  +o  1o )  =  suc  y )
2118, 20eqtrd 2467 . . 3  |-  ( ( y  e.  On  /\  ( 1o  .o  y
)  =  y )  ->  ( 1o  .o  suc  y )  =  suc  y )
2221ex 424 . 2  |-  ( y  e.  On  ->  (
( 1o  .o  y
)  =  y  -> 
( 1o  .o  suc  y )  =  suc  y ) )
23 iuneq2 4101 . . . 4  |-  ( A. y  e.  x  ( 1o  .o  y )  =  y  ->  U_ y  e.  x  ( 1o  .o  y )  =  U_ y  e.  x  y
)
24 uniiun 4136 . . . 4  |-  U. x  =  U_ y  e.  x  y
2523, 24syl6eqr 2485 . . 3  |-  ( A. y  e.  x  ( 1o  .o  y )  =  y  ->  U_ y  e.  x  ( 1o  .o  y )  =  U. x )
26 vex 2951 . . . . 5  |-  x  e. 
_V
27 omlim 6768 . . . . . 6  |-  ( ( 1o  e.  On  /\  ( x  e.  _V  /\ 
Lim  x ) )  ->  ( 1o  .o  x )  =  U_ y  e.  x  ( 1o  .o  y ) )
2814, 27mpan 652 . . . . 5  |-  ( ( x  e.  _V  /\  Lim  x )  ->  ( 1o  .o  x )  = 
U_ y  e.  x  ( 1o  .o  y
) )
2926, 28mpan 652 . . . 4  |-  ( Lim  x  ->  ( 1o  .o  x )  =  U_ y  e.  x  ( 1o  .o  y ) )
30 limuni 4633 . . . 4  |-  ( Lim  x  ->  x  =  U. x )
3129, 30eqeq12d 2449 . . 3  |-  ( Lim  x  ->  ( ( 1o  .o  x )  =  x  <->  U_ y  e.  x  ( 1o  .o  y
)  =  U. x
) )
3225, 31syl5ibr 213 . 2  |-  ( Lim  x  ->  ( A. y  e.  x  ( 1o  .o  y )  =  y  ->  ( 1o  .o  x )  =  x ) )
333, 6, 9, 12, 13, 22, 32tfinds 4830 1  |-  ( A  e.  On  ->  ( 1o  .o  A )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948   (/)c0 3620   U.cuni 4007   U_ciun 4085   Oncon0 4573   Lim wlim 4574   suc csuc 4575  (class class class)co 6072   1oc1o 6708    +o coa 6712    .o comu 6713
This theorem is referenced by:  oe1  6778  omword2  6808
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4692
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4837  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-ov 6075  df-oprab 6076  df-mpt2 6077  df-1st 6340  df-2nd 6341  df-recs 6624  df-rdg 6659  df-1o 6715  df-oadd 6719  df-omul 6720
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