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Theorem nn0opthi 11279
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 3650 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 9996 . . . . . . . . 9  |-  ( A  +  B )  e. 
NN0
43nn0rei 9971 . . . . . . . 8  |-  ( A  +  B )  e.  RR
5 nn0opth.3 . . . . . . . . . 10  |-  C  e. 
NN0
6 nn0opth.4 . . . . . . . . . 10  |-  D  e. 
NN0
75, 6nn0addcli 9996 . . . . . . . . 9  |-  ( C  +  D )  e. 
NN0
87nn0rei 9971 . . . . . . . 8  |-  ( C  +  D )  e.  RR
94, 8lttri2i 8927 . . . . . . 7  |-  ( ( A  +  B )  =/=  ( C  +  D )  <->  ( ( A  +  B )  <  ( C  +  D
)  \/  ( C  +  D )  < 
( A  +  B
) ) )
101, 2, 7, 6nn0opthlem2 11278 . . . . . . . . 9  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( C  +  D )  x.  ( C  +  D )
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
1110necomd 2530 . . . . . . . 8  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
125, 6, 3, 2nn0opthlem2 11278 . . . . . . . 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 2507 . . . . 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 5837 . . . . . . . . 9  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  (
( A  +  B
)  x.  ( A  +  B ) )  =  ( ( C  +  D )  x.  ( C  +  D
) ) )
1817oveq1d 5834 . . . . . . . 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 2319 . . . . . . 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 9972 . . . . . . . . 9  |-  ( A  +  B )  e.  CC
2120, 20mulcli 8837 . . . . . . . 8  |-  ( ( A  +  B )  x.  ( A  +  B ) )  e.  CC
222nn0cni 9972 . . . . . . . 8  |-  B  e.  CC
236nn0cni 9972 . . . . . . . 8  |-  D  e.  CC
2421, 22, 23addcani 9000 . . . . . . 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 5835 . . . . 5  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( C  +  B )  =  ( C  +  D ) )
2715, 26eqtr4d 2319 . . . 4  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  +  B )  =  ( C  +  B ) )
281nn0cni 9972 . . . . 5  |-  A  e.  CC
295nn0cni 9972 . . . . 5  |-  C  e.  CC
3028, 29, 22addcan2i 9001 . . . 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 5828 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  B
)  =  ( C  +  D ) )
3433, 33oveq12d 5837 . . 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 5837 . 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 1624    e. wcel 1685    =/= wne 2447   class class class wbr 4024  (class class class)co 5819    + caddc 8735    x. cmul 8737    < clt 8862   NN0cn0 9960
This theorem is referenced by:  nn0opth2i  11280
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511  ax-cnex 8788  ax-resscn 8789  ax-1cn 8790  ax-icn 8791  ax-addcl 8792  ax-addrcl 8793  ax-mulcl 8794  ax-mulrcl 8795  ax-mulcom 8796  ax-addass 8797  ax-mulass 8798  ax-distr 8799  ax-i2m1 8800  ax-1ne0 8801  ax-1rid 8802  ax-rnegex 8803  ax-rrecex 8804  ax-cnre 8805  ax-pre-lttri 8806  ax-pre-lttrn 8807  ax-pre-ltadd 8808  ax-pre-mulgt0 8809
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-nel 2450  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-2nd 6084  df-iota 6252  df-riota 6299  df-recs 6383  df-rdg 6418  df-er 6655  df-en 6859  df-dom 6860  df-sdom 6861  df-pnf 8864  df-mnf 8865  df-xr 8866  df-ltxr 8867  df-le 8868  df-sub 9034  df-neg 9035  df-nn 9742  df-2 9799  df-n0 9961  df-z 10020  df-uz 10226  df-seq 11041  df-exp 11099
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