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Theorem oaabs 6658
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 4176 . . . . . . . . 9  |-  ( ( om  C_  B  /\  B  e.  On )  ->  om  e.  _V )
21ex 423 . . . . . . . 8  |-  ( om  C_  B  ->  ( B  e.  On  ->  om  e.  _V ) )
3 omelon2 4684 . . . . . . . 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 6569 . . . 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 2767 . . 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 4678 . . . . . . 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 6575 . . . . . 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 6657 . . . . . . . 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 5889 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( A  +o  om )  +o  x )  =  ( om  +o  x ) )
2418, 23eqtr3d 2330 . . . 4  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( A  +o  ( om  +o  x ) )  =  ( om  +o  x
) )
25 oveq2 5882 . . . . 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 2310 . . . 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 2680 . 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 1632    e. wcel 1696   E.wrex 2557   E!wreu 2558   _Vcvv 2801    C_ wss 3165   Oncon0 4408   omcom 4672  (class class class)co 5874    +o coa 6492
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-recs 6404  df-rdg 6439  df-oadd 6499
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