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Theorem nnarcl 7158
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 7094 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
2 eloni 4830 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
3 ordom 6588 . . . . . . 7  |-  Ord  om
4 ordtr2 4864 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( ( A  C_  ( A  +o  B )  /\  ( A  +o  B )  e. 
om )  ->  A  e.  om ) )
52, 3, 4sylancl 662 . . . . . 6  |-  ( A  e.  On  ->  (
( A  C_  ( A  +o  B )  /\  ( A  +o  B
)  e.  om )  ->  A  e.  om )
)
65expd 436 . . . . 5  |-  ( A  e.  On  ->  ( A  C_  ( A  +o  B )  ->  (
( A  +o  B
)  e.  om  ->  A  e.  om ) ) )
76adantr 465 . . . 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 7095 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  B  C_  ( A  +o  B ) )
109ancoms 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  C_  ( A  +o  B ) )
11 eloni 4830 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
12 ordtr2 4864 . . . . . . 7  |-  ( ( Ord  B  /\  Ord  om )  ->  ( ( B  C_  ( A  +o  B )  /\  ( A  +o  B )  e. 
om )  ->  B  e.  om ) )
1311, 3, 12sylancl 662 . . . . . 6  |-  ( B  e.  On  ->  (
( B  C_  ( A  +o  B )  /\  ( A  +o  B
)  e.  om )  ->  B  e.  om )
)
1413expd 436 . . . . 5  |-  ( B  e.  On  ->  ( B  C_  ( A  +o  B )  ->  (
( A  +o  B
)  e.  om  ->  B  e.  om ) ) )
1514adantl 466 . . . 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 533 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  ( A  e.  om  /\  B  e.  om )
) )
18 nnacl 7153 . 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 369    e. wcel 1758    C_ wss 3429   Ord word 4819   Oncon0 4820  (class class class)co 6193   omcom 6579    +o coa 7020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-recs 6935  df-rdg 6969  df-oadd 7027
This theorem is referenced by: (None)
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