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Theorem nn0opthi 12394
Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. We can represent an ordered pair of nonnegative integers  A and  B by  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B ). If two such ordered pairs are equal, their first elements are equal and their second elements are equal. Contrast this ordered pair representation with the standard one df-op 3979 that works for any set. (Contributed by Raph Levien, 10-Dec-2002.) (Proof shortened by Scott Fenton, 8-Sep-2010.)
Hypotheses
Ref Expression
nn0opth.1  |-  A  e. 
NN0
nn0opth.2  |-  B  e. 
NN0
nn0opth.3  |-  C  e. 
NN0
nn0opth.4  |-  D  e. 
NN0
Assertion
Ref Expression
nn0opthi  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  <->  ( A  =  C  /\  B  =  D ) )

Proof of Theorem nn0opthi
StepHypRef Expression
1 nn0opth.1 . . . . . . . . . 10  |-  A  e. 
NN0
2 nn0opth.2 . . . . . . . . . 10  |-  B  e. 
NN0
31, 2nn0addcli 10874 . . . . . . . . 9  |-  ( A  +  B )  e. 
NN0
43nn0rei 10847 . . . . . . . 8  |-  ( A  +  B )  e.  RR
5 nn0opth.3 . . . . . . . . . 10  |-  C  e. 
NN0
6 nn0opth.4 . . . . . . . . . 10  |-  D  e. 
NN0
75, 6nn0addcli 10874 . . . . . . . . 9  |-  ( C  +  D )  e. 
NN0
87nn0rei 10847 . . . . . . . 8  |-  ( C  +  D )  e.  RR
94, 8lttri2i 9730 . . . . . . 7  |-  ( ( A  +  B )  =/=  ( C  +  D )  <->  ( ( A  +  B )  <  ( C  +  D
)  \/  ( C  +  D )  < 
( A  +  B
) ) )
101, 2, 7, 6nn0opthlem2 12393 . . . . . . . . 9  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( C  +  D )  x.  ( C  +  D )
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
1110necomd 2674 . . . . . . . 8  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
125, 6, 3, 2nn0opthlem2 12393 . . . . . . . 8  |-  ( ( C  +  D )  <  ( A  +  B )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
1311, 12jaoi 377 . . . . . . 7  |-  ( ( ( A  +  B
)  <  ( C  +  D )  \/  ( C  +  D )  <  ( A  +  B
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
149, 13sylbi 195 . . . . . 6  |-  ( ( A  +  B )  =/=  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
1514necon4i 2647 . . . . 5  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  +  B )  =  ( C  +  D ) )
16 id 22 . . . . . . . 8  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
1715, 15oveq12d 6296 . . . . . . . . 9  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  (
( A  +  B
)  x.  ( A  +  B ) )  =  ( ( C  +  D )  x.  ( C  +  D
) ) )
1817oveq1d 6293 . . . . . . . 8  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  D )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
1916, 18eqtr4d 2446 . . . . . . 7  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  D ) )
203nn0cni 10848 . . . . . . . . 9  |-  ( A  +  B )  e.  CC
2120, 20mulcli 9631 . . . . . . . 8  |-  ( ( A  +  B )  x.  ( A  +  B ) )  e.  CC
222nn0cni 10848 . . . . . . . 8  |-  B  e.  CC
236nn0cni 10848 . . . . . . . 8  |-  D  e.  CC
2421, 22, 23addcani 9807 . . . . . . 7  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  D )  <->  B  =  D )
2519, 24sylib 196 . . . . . 6  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  B  =  D )
2625oveq2d 6294 . . . . 5  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( C  +  B )  =  ( C  +  D ) )
2715, 26eqtr4d 2446 . . . 4  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  +  B )  =  ( C  +  B ) )
281nn0cni 10848 . . . . 5  |-  A  e.  CC
295nn0cni 10848 . . . . 5  |-  C  e.  CC
3028, 29, 22addcan2i 9808 . . . 4  |-  ( ( A  +  B )  =  ( C  +  B )  <->  A  =  C )
3127, 30sylib 196 . . 3  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  A  =  C )
3231, 25jca 530 . 2  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  =  C  /\  B  =  D )
)
33 oveq12 6287 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  B
)  =  ( C  +  D ) )
3433, 33oveq12d 6296 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  +  B )  x.  ( A  +  B )
)  =  ( ( C  +  D )  x.  ( C  +  D ) ) )
35 simpr 459 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  B  =  D )
3634, 35oveq12d 6296 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D ) )
3732, 36impbii 187 1  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   class class class wbr 4395  (class class class)co 6278    + caddc 9525    x. cmul 9527    < clt 9658   NN0cn0 10836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-om 6684  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-nn 10577  df-2 10635  df-n0 10837  df-z 10906  df-uz 11128  df-seq 12152  df-exp 12211
This theorem is referenced by:  nn0opth2i  12395
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