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Theorem oaabs 6850
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 4313 . . . . . . . . 9  |-  ( ( om  C_  B  /\  B  e.  On )  ->  om  e.  _V )
21ex 424 . . . . . . . 8  |-  ( om  C_  B  ->  ( B  e.  On  ->  om  e.  _V ) )
3 omelon2 4820 . . . . . . . 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 6761 . . . 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 2886 . . 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 4814 . . . . . . 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 6767 . . . . . 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 6849 . . . . . . . 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 6059 . . . . 5  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( ( A  +o  om )  +o  x )  =  ( om  +o  x ) )
2418, 23eqtr3d 2442 . . . 4  |-  ( ( ( ( A  e. 
om  /\  B  e.  On )  /\  om  C_  B
)  /\  x  e.  On )  ->  ( A  +o  ( om  +o  x ) )  =  ( om  +o  x
) )
25 oveq2 6052 . . . . 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 2422 . . . 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 2794 . 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 1721   E.wrex 2671   E!wreu 2672   _Vcvv 2920    C_ wss 3284   Oncon0 4545   omcom 4808  (class class class)co 6044    +o coa 6684
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rmo 2678  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-pss 3300  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-tp 3786  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-tr 4267  df-eprel 4458  df-id 4462  df-po 4467  df-so 4468  df-fr 4505  df-we 4507  df-ord 4548  df-on 4549  df-lim 4550  df-suc 4551  df-om 4809  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-recs 6596  df-rdg 6631  df-oadd 6691
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