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Theorem oaabs 6824
Description: Ordinal addition absorbs a natural number added to the left of a transfinite number. Proposition 8.10 of [TakeutiZaring] p. 59. (Contributed by NM, 9-Dec-2004.) (Proof shortened by Mario Carneiro, 29-May-2015.)
Assertion
Ref Expression
oaabs  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( A  +o  B )  =  B )

Proof of Theorem oaabs
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 ssexg 4291 . . . . . . . . 9  |-  ( ( om  C_  B  /\  B  e.  On )  ->  om  e.  _V )
21ex 424 . . . . . . . 8  |-  ( om  C_  B  ->  ( B  e.  On  ->  om  e.  _V ) )
3 omelon2 4798 . . . . . . . 8  |-  ( om  e.  _V  ->  om  e.  On )
42, 3syl6com 33 . . . . . . 7  |-  ( B  e.  On  ->  ( om  C_  B  ->  om  e.  On ) )
54imp 419 . . . . . 6  |-  ( ( B  e.  On  /\  om  C_  B )  ->  om  e.  On )
65adantll 695 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  om  e.  On )
7 simplr 732 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  B  e.  On )
86, 7jca 519 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( om  e.  On  /\  B  e.  On ) )
9 oawordeu 6735 . . . 4  |-  ( ( ( om  e.  On  /\  B  e.  On )  /\  om  C_  B
)  ->  E! x  e.  On  ( om  +o  x )  =  B )
108, 9sylancom 649 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  E! x  e.  On  ( om  +o  x )  =  B )
11 reurex 2866 . . 3  |-  ( E! x  e.  On  ( om  +o  x )  =  B  ->  E. x  e.  On  ( om  +o  x )  =  B )
1210, 11syl 16 . 2  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  E. x  e.  On  ( om  +o  x )  =  B )
13 nnon 4792 . . . . . . 7  |-  ( A  e.  om  ->  A  e.  On )
1413ad3antrrr 711 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  A  e.  On )
156adantr 452 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  om  e.  On )
16 simpr 448 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  x  e.  On )
17 oaass 6741 . . . . . 6  |-  ( ( A  e.  On  /\  om  e.  On  /\  x  e.  On )  ->  (
( A  +o  om )  +o  x )  =  ( A  +o  ( om  +o  x ) ) )
1814, 15, 16, 17syl3anc 1184 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( A  +o  om )  +o  x )  =  ( A  +o  ( om 
+o  x ) ) )
19 simpll 731 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  A  e.  om )
20 oaabslem 6823 . . . . . . . 8  |-  ( ( om  e.  On  /\  A  e.  om )  ->  ( A  +o  om )  =  om )
216, 19, 20syl2anc 643 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( A  +o  om )  =  om )
2221adantr 452 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( A  +o  om )  =  om )
2322oveq1d 6036 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( A  +o  om )  +o  x )  =  ( om  +o  x ) )
2418, 23eqtr3d 2422 . . . 4  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( A  +o  ( om  +o  x ) )  =  ( om  +o  x
) )
25 oveq2 6029 . . . . 5  |-  ( ( om  +o  x )  =  B  ->  ( A  +o  ( om  +o  x ) )  =  ( A  +o  B
) )
26 id 20 . . . . 5  |-  ( ( om  +o  x )  =  B  ->  ( om  +o  x )  =  B )
2725, 26eqeq12d 2402 . . . 4  |-  ( ( om  +o  x )  =  B  ->  (
( A  +o  ( om  +o  x ) )  =  ( om  +o  x )  <->  ( A  +o  B )  =  B ) )
2824, 27syl5ibcom 212 . . 3  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( om  +o  x )  =  B  ->  ( A  +o  B )  =  B ) )
2928rexlimdva 2774 . 2  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( E. x  e.  On  ( om  +o  x )  =  B  ->  ( A  +o  B )  =  B ) )
3012, 29mpd 15 1  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( A  +o  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651   E!wreu 2652   _Vcvv 2900    C_ wss 3264   Oncon0 4523   omcom 4786  (class class class)co 6021    +o coa 6658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-recs 6570  df-rdg 6605  df-oadd 6665
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