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Theorem nn0opthi 11251
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 3623 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 9968 . . . . . . . . 9  |-  ( A  +  B )  e. 
NN0
43nn0rei 9943 . . . . . . . 8  |-  ( A  +  B )  e.  RR
5 nn0opth.3 . . . . . . . . . 10  |-  C  e. 
NN0
6 nn0opth.4 . . . . . . . . . 10  |-  D  e. 
NN0
75, 6nn0addcli 9968 . . . . . . . . 9  |-  ( C  +  D )  e. 
NN0
87nn0rei 9943 . . . . . . . 8  |-  ( C  +  D )  e.  RR
94, 8lttri2i 8900 . . . . . . 7  |-  ( ( A  +  B )  =/=  ( C  +  D )  <->  ( ( A  +  B )  <  ( C  +  D
)  \/  ( C  +  D )  < 
( A  +  B
) ) )
101, 2, 7, 6nn0opthlem2 11250 . . . . . . . . 9  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( C  +  D )  x.  ( C  +  D )
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
1110necomd 2504 . . . . . . . 8  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
125, 6, 3, 2nn0opthlem2 11250 . . . . . . . 8  |-  ( ( C  +  D )  <  ( A  +  B )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
1311, 12jaoi 370 . . . . . . 7  |-  ( ( ( A  +  B
)  <  ( C  +  D )  \/  ( C  +  D )  <  ( A  +  B
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
149, 13sylbi 189 . . . . . 6  |-  ( ( A  +  B )  =/=  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
1514necon4i 2481 . . . . 5  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  +  B )  =  ( C  +  D ) )
16 id 21 . . . . . . . 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 5810 . . . . . . . . 9  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  (
( A  +  B
)  x.  ( A  +  B ) )  =  ( ( C  +  D )  x.  ( C  +  D
) ) )
1817oveq1d 5807 . . . . . . . 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 2293 . . . . . . 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 9944 . . . . . . . . 9  |-  ( A  +  B )  e.  CC
2120, 20mulcli 8810 . . . . . . . 8  |-  ( ( A  +  B )  x.  ( A  +  B ) )  e.  CC
222nn0cni 9944 . . . . . . . 8  |-  B  e.  CC
236nn0cni 9944 . . . . . . . 8  |-  D  e.  CC
2421, 22, 23addcani 8973 . . . . . . 7  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  D )  <->  B  =  D )
2519, 24sylib 190 . . . . . 6  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  B  =  D )
2625oveq2d 5808 . . . . 5  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( C  +  B )  =  ( C  +  D ) )
2715, 26eqtr4d 2293 . . . 4  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  +  B )  =  ( C  +  B ) )
281nn0cni 9944 . . . . 5  |-  A  e.  CC
295nn0cni 9944 . . . . 5  |-  C  e.  CC
3028, 29, 22addcan2i 8974 . . . 4  |-  ( ( A  +  B )  =  ( C  +  B )  <->  A  =  C )
3127, 30sylib 190 . . 3  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  A  =  C )
3231, 25jca 520 . 2  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  =  C  /\  B  =  D )
)
33 oveq12 5801 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  B
)  =  ( C  +  D ) )
3433, 33oveq12d 5810 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  +  B )  x.  ( A  +  B )
)  =  ( ( C  +  D )  x.  ( C  +  D ) ) )
35 simpr 449 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  B  =  D )
3634, 35oveq12d 5810 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D ) )
3732, 36impbii 182 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 178    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621    =/= wne 2421   class class class wbr 3997  (class class class)co 5792    + caddc 8708    x. cmul 8710    < clt 8835   NN0cn0 9932
This theorem is referenced by:  nn0opth2i  11252
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782
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 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-iun 3881  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-2nd 6057  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-er 6628  df-en 6832  df-dom 6833  df-sdom 6834  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-n0 9933  df-z 9992  df-uz 10198  df-seq 11013  df-exp 11071
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