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Theorem nn0opthi 11563
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 3823 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 10257 . . . . . . . . 9  |-  ( A  +  B )  e. 
NN0
43nn0rei 10232 . . . . . . . 8  |-  ( A  +  B )  e.  RR
5 nn0opth.3 . . . . . . . . . 10  |-  C  e. 
NN0
6 nn0opth.4 . . . . . . . . . 10  |-  D  e. 
NN0
75, 6nn0addcli 10257 . . . . . . . . 9  |-  ( C  +  D )  e. 
NN0
87nn0rei 10232 . . . . . . . 8  |-  ( C  +  D )  e.  RR
94, 8lttri2i 9187 . . . . . . 7  |-  ( ( A  +  B )  =/=  ( C  +  D )  <->  ( ( A  +  B )  <  ( C  +  D
)  \/  ( C  +  D )  < 
( A  +  B
) ) )
101, 2, 7, 6nn0opthlem2 11562 . . . . . . . . 9  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( C  +  D )  x.  ( C  +  D )
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
1110necomd 2687 . . . . . . . 8  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
125, 6, 3, 2nn0opthlem2 11562 . . . . . . . 8  |-  ( ( C  +  D )  <  ( A  +  B )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
1311, 12jaoi 369 . . . . . . 7  |-  ( ( ( A  +  B
)  <  ( C  +  D )  \/  ( C  +  D )  <  ( A  +  B
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
149, 13sylbi 188 . . . . . 6  |-  ( ( A  +  B )  =/=  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
1514necon4i 2664 . . . . 5  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  +  B )  =  ( C  +  D ) )
16 id 20 . . . . . . . 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 6099 . . . . . . . . 9  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  (
( A  +  B
)  x.  ( A  +  B ) )  =  ( ( C  +  D )  x.  ( C  +  D
) ) )
1817oveq1d 6096 . . . . . . . 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 2471 . . . . . . 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 10233 . . . . . . . . 9  |-  ( A  +  B )  e.  CC
2120, 20mulcli 9095 . . . . . . . 8  |-  ( ( A  +  B )  x.  ( A  +  B ) )  e.  CC
222nn0cni 10233 . . . . . . . 8  |-  B  e.  CC
236nn0cni 10233 . . . . . . . 8  |-  D  e.  CC
2421, 22, 23addcani 9259 . . . . . . 7  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  D )  <->  B  =  D )
2519, 24sylib 189 . . . . . 6  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  B  =  D )
2625oveq2d 6097 . . . . 5  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( C  +  B )  =  ( C  +  D ) )
2715, 26eqtr4d 2471 . . . 4  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  +  B )  =  ( C  +  B ) )
281nn0cni 10233 . . . . 5  |-  A  e.  CC
295nn0cni 10233 . . . . 5  |-  C  e.  CC
3028, 29, 22addcan2i 9260 . . . 4  |-  ( ( A  +  B )  =  ( C  +  B )  <->  A  =  C )
3127, 30sylib 189 . . 3  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  A  =  C )
3231, 25jca 519 . 2  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  =  C  /\  B  =  D )
)
33 oveq12 6090 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  B
)  =  ( C  +  D ) )
3433, 33oveq12d 6099 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  +  B )  x.  ( A  +  B )
)  =  ( ( C  +  D )  x.  ( C  +  D ) ) )
35 simpr 448 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  B  =  D )
3634, 35oveq12d 6099 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D ) )
3732, 36impbii 181 1  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  <->  ( A  =  C  /\  B  =  D ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   class class class wbr 4212  (class class class)co 6081    + caddc 8993    x. cmul 8995    < clt 9120   NN0cn0 10221
This theorem is referenced by:  nn0opth2i  11564
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-n0 10222  df-z 10283  df-uz 10489  df-seq 11324  df-exp 11383
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