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Theorem oaabs 6642
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 4160 . . . . . . . . 9  |-  ( ( om  C_  B  /\  B  e.  On )  ->  om  e.  _V )
21ex 423 . . . . . . . 8  |-  ( om  C_  B  ->  ( B  e.  On  ->  om  e.  _V ) )
3 omelon2 4668 . . . . . . . 8  |-  ( om  e.  _V  ->  om  e.  On )
42, 3syl6com 31 . . . . . . 7  |-  ( B  e.  On  ->  ( om  C_  B  ->  om  e.  On ) )
54imp 418 . . . . . 6  |-  ( ( B  e.  On  /\  om  C_  B )  ->  om  e.  On )
65adantll 694 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  om  e.  On )
7 simplr 731 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  B  e.  On )
86, 7jca 518 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( om  e.  On  /\  B  e.  On ) )
9 oawordeu 6553 . . . 4  |-  ( ( ( om  e.  On  /\  B  e.  On )  /\  om  C_  B
)  ->  E! x  e.  On  ( om  +o  x )  =  B )
108, 9sylancom 648 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  E! x  e.  On  ( om  +o  x )  =  B )
11 reurex 2754 . . 3  |-  ( E! x  e.  On  ( om  +o  x )  =  B  ->  E. x  e.  On  ( om  +o  x )  =  B )
1210, 11syl 15 . 2  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  E. x  e.  On  ( om  +o  x )  =  B )
13 nnon 4662 . . . . . . 7  |-  ( A  e.  om  ->  A  e.  On )
1413ad3antrrr 710 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  A  e.  On )
156adantr 451 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  om  e.  On )
16 simpr 447 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  x  e.  On )
17 oaass 6559 . . . . . 6  |-  ( ( A  e.  On  /\  om  e.  On  /\  x  e.  On )  ->  (
( A  +o  om )  +o  x )  =  ( A  +o  ( om  +o  x ) ) )
1814, 15, 16, 17syl3anc 1182 . . . . 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 730 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  A  e.  om )
20 oaabslem 6641 . . . . . . . 8  |-  ( ( om  e.  On  /\  A  e.  om )  ->  ( A  +o  om )  =  om )
216, 19, 20syl2anc 642 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( A  +o  om )  =  om )
2221adantr 451 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( A  +o  om )  =  om )
2322oveq1d 5873 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( A  +o  om )  +o  x )  =  ( om  +o  x ) )
2418, 23eqtr3d 2317 . . . 4  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( A  +o  ( om  +o  x ) )  =  ( om  +o  x
) )
25 oveq2 5866 . . . . 5  |-  ( ( om  +o  x )  =  B  ->  ( A  +o  ( om  +o  x ) )  =  ( A  +o  B
) )
26 id 19 . . . . 5  |-  ( ( om  +o  x )  =  B  ->  ( om  +o  x )  =  B )
2725, 26eqeq12d 2297 . . . 4  |-  ( ( om  +o  x )  =  B  ->  (
( A  +o  ( om  +o  x ) )  =  ( om  +o  x )  <->  ( A  +o  B )  =  B ) )
2824, 27syl5ibcom 211 . . 3  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( om  +o  x )  =  B  ->  ( A  +o  B )  =  B ) )
2928rexlimdva 2667 . 2  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( E. x  e.  On  ( om  +o  x )  =  B  ->  ( A  +o  B )  =  B ) )
3012, 29mpd 14 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 358    = wceq 1623    e. wcel 1684   E.wrex 2544   E!wreu 2545   _Vcvv 2788    C_ wss 3152   Oncon0 4392   omcom 4656  (class class class)co 5858    +o coa 6476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-recs 6388  df-rdg 6423  df-oadd 6483
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