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Theorem oaabs 6575
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
StepHypRef Expression
1 ssexg 4100 . . . . . . . . 9  |-  ( ( om  C_  B  /\  B  e.  On )  ->  om  e.  _V )
21ex 425 . . . . . . . 8  |-  ( om  C_  B  ->  ( B  e.  On  ->  om  e.  _V ) )
3 omelon2 4605 . . . . . . . 8  |-  ( om  e.  _V  ->  om  e.  On )
42, 3syl6com 33 . . . . . . 7  |-  ( B  e.  On  ->  ( om  C_  B  ->  om  e.  On ) )
54imp 420 . . . . . 6  |-  ( ( B  e.  On  /\  om  C_  B )  ->  om  e.  On )
65adantll 697 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  om  e.  On )
7 simplr 734 . . . . 5  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  B  e.  On )
86, 7jca 520 . . . 4  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( om  e.  On  /\  B  e.  On ) )
9 oawordeu 6486 . . . 4  |-  ( ( ( om  e.  On  /\  B  e.  On )  /\  om  C_  B
)  ->  E! x  e.  On  ( om  +o  x )  =  B )
108, 9sylancom 651 . . 3  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  E! x  e.  On  ( om  +o  x )  =  B )
11 reurex 2904 . . 3  |-  ( E! x  e.  On  ( om  +o  x )  =  B  ->  E. x  e.  On  ( om  +o  x )  =  B )
1210, 11syl 17 . 2  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  E. x  e.  On  ( om  +o  x )  =  B )
13 nnon 4599 . . . . . . 7  |-  ( A  e.  om  ->  A  e.  On )
1413ad3antrrr 713 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  A  e.  On )
156adantr 453 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  om  e.  On )
16 simpr 449 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  x  e.  On )
17 oaass 6492 . . . . . 6  |-  ( ( A  e.  On  /\  om  e.  On  /\  x  e.  On )  ->  (
( A  +o  om )  +o  x )  =  ( A  +o  ( om  +o  x ) ) )
1814, 15, 16, 17syl3anc 1187 . . . . 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 733 . . . . . . . 8  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  A  e.  om )
20 oaabslem 6574 . . . . . . . 8  |-  ( ( om  e.  On  /\  A  e.  om )  ->  ( A  +o  om )  =  om )
216, 19, 20syl2anc 645 . . . . . . 7  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( A  +o  om )  =  om )
2221adantr 453 . . . . . 6  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( A  +o  om )  =  om )
2322oveq1d 5772 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( A  +o  om )  +o  x )  =  ( om  +o  x ) )
2418, 23eqtr3d 2290 . . . 4  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( A  +o  ( om  +o  x ) )  =  ( om  +o  x
) )
25 oveq2 5765 . . . . 5  |-  ( ( om  +o  x )  =  B  ->  ( A  +o  ( om  +o  x ) )  =  ( A  +o  B
) )
26 id 21 . . . . 5  |-  ( ( om  +o  x )  =  B  ->  ( om  +o  x )  =  B )
2725, 26eqeq12d 2270 . . . 4  |-  ( ( om  +o  x )  =  B  ->  (
( A  +o  ( om  +o  x ) )  =  ( om  +o  x )  <->  ( A  +o  B )  =  B ) )
2824, 27syl5ibcom 213 . . 3  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( om  +o  x )  =  B  ->  ( A  +o  B )  =  B ) )
2928rexlimdva 2638 . 2  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( E. x  e.  On  ( om  +o  x )  =  B  ->  ( A  +o  B )  =  B ) )
3012, 29mpd 16 1  |-  ( ( ( A  e.  om  /\  B  e.  On )  /\  om  C_  B
)  ->  ( A  +o  B )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 6    /\ wa 360    = wceq 1619    e. wcel 1621   E.wrex 2517   E!wreu 2518   _Vcvv 2740    C_ wss 3094   Oncon0 4329   omcom 4593  (class class class)co 5757    +o coa 6409
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4071  ax-sep 4081  ax-nul 4089  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-reu 2522  df-rab 2523  df-v 2742  df-sbc 2936  df-csb 3024  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-iun 3848  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-lim 4334  df-suc 4335  df-om 4594  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-ov 5760  df-oprab 5761  df-mpt2 5762  df-recs 6321  df-rdg 6356  df-oadd 6416
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