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Theorem nnarcl 7257
Description: Reverse closure law for addition of natural numbers. Exercise 1 of [TakeutiZaring] p. 62 and its converse. (Contributed by NM, 12-Dec-2004.)
Assertion
Ref Expression
nnarcl  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  <->  ( A  e.  om  /\  B  e.  om )
) )

Proof of Theorem nnarcl
StepHypRef Expression
1 oaword1 7193 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
2 eloni 4877 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
3 ordom 6682 . . . . . . 7  |-  Ord  om
4 ordtr2 4911 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( ( A  C_  ( A  +o  B )  /\  ( A  +o  B )  e. 
om )  ->  A  e.  om ) )
52, 3, 4sylancl 660 . . . . . 6  |-  ( A  e.  On  ->  (
( A  C_  ( A  +o  B )  /\  ( A  +o  B
)  e.  om )  ->  A  e.  om )
)
65expd 434 . . . . 5  |-  ( A  e.  On  ->  ( A  C_  ( A  +o  B )  ->  (
( A  +o  B
)  e.  om  ->  A  e.  om ) ) )
76adantr 463 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  ( A  +o  B )  -> 
( ( A  +o  B )  e.  om  ->  A  e.  om )
) )
81, 7mpd 15 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  A  e.  om )
)
9 oaword2 7194 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  B  C_  ( A  +o  B ) )
109ancoms 451 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  C_  ( A  +o  B ) )
11 eloni 4877 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
12 ordtr2 4911 . . . . . . 7  |-  ( ( Ord  B  /\  Ord  om )  ->  ( ( B  C_  ( A  +o  B )  /\  ( A  +o  B )  e. 
om )  ->  B  e.  om ) )
1311, 3, 12sylancl 660 . . . . . 6  |-  ( B  e.  On  ->  (
( B  C_  ( A  +o  B )  /\  ( A  +o  B
)  e.  om )  ->  B  e.  om )
)
1413expd 434 . . . . 5  |-  ( B  e.  On  ->  ( B  C_  ( A  +o  B )  ->  (
( A  +o  B
)  e.  om  ->  B  e.  om ) ) )
1514adantl 464 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  ( A  +o  B )  -> 
( ( A  +o  B )  e.  om  ->  B  e.  om )
) )
1610, 15mpd 15 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  B  e.  om )
)
178, 16jcad 531 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  ( A  e.  om  /\  B  e.  om )
) )
18 nnacl 7252 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  B
)  e.  om )
1917, 18impbid1 203 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  <->  ( A  e.  om  /\  B  e.  om )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    e. wcel 1823    C_ wss 3461   Ord word 4866   Oncon0 4867  (class class class)co 6270   omcom 6673    +o coa 7119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-recs 7034  df-rdg 7068  df-oadd 7126
This theorem is referenced by: (None)
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