MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nnarcl Unicode version

Theorem nnarcl 6609
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 6545 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  A  C_  ( A  +o  B ) )
2 eloni 4401 . . . . . . 7  |-  ( A  e.  On  ->  Ord  A )
3 ordom 4664 . . . . . . 7  |-  Ord  om
4 ordtr2 4435 . . . . . . 7  |-  ( ( Ord  A  /\  Ord  om )  ->  ( ( A  C_  ( A  +o  B )  /\  ( A  +o  B )  e. 
om )  ->  A  e.  om ) )
52, 3, 4sylancl 646 . . . . . 6  |-  ( A  e.  On  ->  (
( A  C_  ( A  +o  B )  /\  ( A  +o  B
)  e.  om )  ->  A  e.  om )
)
65exp3a 427 . . . . 5  |-  ( A  e.  On  ->  ( A  C_  ( A  +o  B )  ->  (
( A  +o  B
)  e.  om  ->  A  e.  om ) ) )
76adantr 453 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( A  C_  ( A  +o  B )  -> 
( ( A  +o  B )  e.  om  ->  A  e.  om )
) )
81, 7mpd 16 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  A  e.  om )
)
9 oaword2 6546 . . . . 5  |-  ( ( B  e.  On  /\  A  e.  On )  ->  B  C_  ( A  +o  B ) )
109ancoms 441 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  B  C_  ( A  +o  B ) )
11 eloni 4401 . . . . . . 7  |-  ( B  e.  On  ->  Ord  B )
12 ordtr2 4435 . . . . . . 7  |-  ( ( Ord  B  /\  Ord  om )  ->  ( ( B  C_  ( A  +o  B )  /\  ( A  +o  B )  e. 
om )  ->  B  e.  om ) )
1311, 3, 12sylancl 646 . . . . . 6  |-  ( B  e.  On  ->  (
( B  C_  ( A  +o  B )  /\  ( A  +o  B
)  e.  om )  ->  B  e.  om )
)
1413exp3a 427 . . . . 5  |-  ( B  e.  On  ->  ( B  C_  ( A  +o  B )  ->  (
( A  +o  B
)  e.  om  ->  B  e.  om ) ) )
1514adantl 454 . . . 4  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( B  C_  ( A  +o  B )  -> 
( ( A  +o  B )  e.  om  ->  B  e.  om )
) )
1610, 15mpd 16 . . 3  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  B  e.  om )
)
178, 16jcad 521 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  ->  ( A  e.  om  /\  B  e.  om )
) )
18 nnacl 6604 . 2  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( A  +o  B
)  e.  om )
1917, 18impbid1 196 1  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( ( A  +o  B )  e.  om  <->  ( A  e.  om  /\  B  e.  om )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    e. wcel 1688    C_ wss 3153   Ord word 4390   Oncon0 4391   omcom 4655  (class class class)co 5819    +o coa 6471
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1538  ax-5 1549  ax-17 1608  ax-9 1641  ax-8 1648  ax-13 1690  ax-14 1692  ax-6 1707  ax-7 1712  ax-11 1719  ax-12 1869  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
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 1534  df-nf 1537  df-sb 1636  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-lim 4396  df-suc 4397  df-om 4656  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-ov 5822  df-oprab 5823  df-mpt2 5824  df-recs 6383  df-rdg 6418  df-oadd 6478
  Copyright terms: Public domain W3C validator