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Theorem nn0opthd 9816
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 3425 that works for any set. (Contributed by Jim Kingdon, 31-Oct-2021.)
Hypotheses
Ref Expression
nn0opthd.1  |-  ( ph  ->  A  e.  NN0 )
nn0opthd.2  |-  ( ph  ->  B  e.  NN0 )
nn0opthd.3  |-  ( ph  ->  C  e.  NN0 )
nn0opthd.4  |-  ( ph  ->  D  e.  NN0 )
Assertion
Ref Expression
nn0opthd  |-  ( ph  ->  ( ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D )  <-> 
( A  =  C  /\  B  =  D ) ) )

Proof of Theorem nn0opthd
StepHypRef Expression
1 nn0opthd.1 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  e.  NN0 )
2 nn0opthd.2 . . . . . . . . . . . . . . 15  |-  ( ph  ->  B  e.  NN0 )
3 nn0opthd.3 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  C  e.  NN0 )
4 nn0opthd.4 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  D  e.  NN0 )
53, 4nn0addcld 8482 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  +  D
)  e.  NN0 )
61, 2, 5, 4nn0opthlem2d 9815 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( A  +  B )  <  ( C  +  D )  ->  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  =/=  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B ) ) )
76imp 122 . . . . . . . . . . . . 13  |-  ( (
ph  /\  ( A  +  B )  <  ( C  +  D )
)  ->  ( (
( C  +  D
)  x.  ( C  +  D ) )  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
) )
87necomd 2335 . . . . . . . . . . . 12  |-  ( (
ph  /\  ( A  +  B )  <  ( C  +  D )
)  ->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )
98ex 113 . . . . . . . . . . 11  |-  ( ph  ->  ( ( A  +  B )  <  ( C  +  D )  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =/=  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D ) ) )
101, 2nn0addcld 8482 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  B
)  e.  NN0 )
113, 4, 10, 2nn0opthlem2d 9815 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  +  D )  <  ( A  +  B )  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =/=  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D ) ) )
129, 11jaod 670 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A  +  B )  < 
( C  +  D
)  \/  ( C  +  D )  < 
( A  +  B
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) ) )
1310nn0red 8479 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  B
)  e.  RR )
145nn0red 8479 . . . . . . . . . . 11  |-  ( ph  ->  ( C  +  D
)  e.  RR )
15 reaplt 7825 . . . . . . . . . . 11  |-  ( ( ( A  +  B
)  e.  RR  /\  ( C  +  D
)  e.  RR )  ->  ( ( A  +  B ) #  ( C  +  D )  <-> 
( ( A  +  B )  <  ( C  +  D )  \/  ( C  +  D
)  <  ( A  +  B ) ) ) )
1613, 14, 15syl2anc 403 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B ) #  ( C  +  D )  <->  ( ( A  +  B )  <  ( C  +  D
)  \/  ( C  +  D )  < 
( A  +  B
) ) ) )
1710, 10nn0mulcld 8483 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A  +  B )  x.  ( A  +  B )
)  e.  NN0 )
1817, 2nn0addcld 8482 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  NN0 )
1918nn0zd 8618 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  ZZ )
205, 5nn0mulcld 8483 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  +  D )  x.  ( C  +  D )
)  e.  NN0 )
2120, 4nn0addcld 8482 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  e.  NN0 )
2221nn0zd 8618 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  e.  ZZ )
23 zapne 8573 . . . . . . . . . . 11  |-  ( ( ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  ZZ  /\  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  e.  ZZ )  ->  ( ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B ) #  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D )  <-> 
( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =/=  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D ) ) )
2419, 22, 23syl2anc 403 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) #  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D )  <->  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) ) )
2512, 16, 243imtr4d 201 . . . . . . . . 9  |-  ( ph  ->  ( ( A  +  B ) #  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B ) #  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) ) )
2625con3d 594 . . . . . . . 8  |-  ( ph  ->  ( -.  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B ) #  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D )  ->  -.  ( A  +  B ) #  ( C  +  D ) ) )
2718nn0cnd 8480 . . . . . . . . 9  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  CC )
2821nn0cnd 8480 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  e.  CC )
29 apti 7859 . . . . . . . . 9  |-  ( ( ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  CC  /\  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  e.  CC )  ->  ( ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  <->  -.  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B ) #  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D ) ) )
3027, 28, 29syl2anc 403 . . . . . . . 8  |-  ( ph  ->  ( ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D )  <->  -.  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
) #  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) ) )
3110nn0cnd 8480 . . . . . . . . 9  |-  ( ph  ->  ( A  +  B
)  e.  CC )
325nn0cnd 8480 . . . . . . . . 9  |-  ( ph  ->  ( C  +  D
)  e.  CC )
33 apti 7859 . . . . . . . . 9  |-  ( ( ( A  +  B
)  e.  CC  /\  ( C  +  D
)  e.  CC )  ->  ( ( A  +  B )  =  ( C  +  D
)  <->  -.  ( A  +  B ) #  ( C  +  D ) ) )
3431, 32, 33syl2anc 403 . . . . . . . 8  |-  ( ph  ->  ( ( A  +  B )  =  ( C  +  D )  <->  -.  ( A  +  B
) #  ( C  +  D ) ) )
3526, 30, 343imtr4d 201 . . . . . . 7  |-  ( ph  ->  ( ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D )  ->  ( A  +  B )  =  ( C  +  D ) ) )
3635imp 122 . . . . . 6  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  ( A  +  B )  =  ( C  +  D ) )
37 simpr 108 . . . . . . . . 9  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
3836, 36oveq12d 5582 . . . . . . . . . 10  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  (
( A  +  B
)  x.  ( A  +  B ) )  =  ( ( C  +  D )  x.  ( C  +  D
) ) )
3938oveq1d 5579 . . . . . . . . 9  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  D )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
4037, 39eqtr4d 2118 . . . . . . . 8  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  D ) )
4131, 31mulcld 7271 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  x.  ( A  +  B )
)  e.  CC )
422nn0cnd 8480 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
434nn0cnd 8480 . . . . . . . . . 10  |-  ( ph  ->  D  e.  CC )
4441, 42, 43addcand 7429 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  ( ( ( A  +  B )  x.  ( A  +  B )
)  +  D )  <-> 
B  =  D ) )
4544adantr 270 . . . . . . . 8  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  (
( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  D )  <->  B  =  D ) )
4640, 45mpbid 145 . . . . . . 7  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  B  =  D )
4746oveq2d 5580 . . . . . 6  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  ( C  +  B )  =  ( C  +  D ) )
4836, 47eqtr4d 2118 . . . . 5  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  ( A  +  B )  =  ( C  +  B ) )
491nn0cnd 8480 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
503nn0cnd 8480 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
5149, 50, 42addcan2d 7430 . . . . . 6  |-  ( ph  ->  ( ( A  +  B )  =  ( C  +  B )  <-> 
A  =  C ) )
5251adantr 270 . . . . 5  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  (
( A  +  B
)  =  ( C  +  B )  <->  A  =  C ) )
5348, 52mpbid 145 . . . 4  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  A  =  C )
5453, 46jca 300 . . 3  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  ( A  =  C  /\  B  =  D )
)
5554ex 113 . 2  |-  ( ph  ->  ( ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D )  ->  ( A  =  C  /\  B  =  D ) ) )
56 oveq12 5573 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  B
)  =  ( C  +  D ) )
5756, 56oveq12d 5582 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( A  +  B )  x.  ( A  +  B )
)  =  ( ( C  +  D )  x.  ( C  +  D ) ) )
58 simpr 108 . . 3  |-  ( ( A  =  C  /\  B  =  D )  ->  B  =  D )
5957, 58oveq12d 5582 . 2  |-  ( ( A  =  C  /\  B  =  D )  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D ) )
6055, 59impbid1 140 1  |-  ( ph  ->  ( ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D )  <-> 
( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103    \/ wo 662    = wceq 1285    e. wcel 1434    =/= wne 2249   class class class wbr 3805  (class class class)co 5564   CCcc 7111   RRcr 7112    + caddc 7116    x. cmul 7118    < clt 7285   # cap 7818   NN0cn0 8425   ZZcz 8502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-iinf 4357  ax-cnex 7199  ax-resscn 7200  ax-1cn 7201  ax-1re 7202  ax-icn 7203  ax-addcl 7204  ax-addrcl 7205  ax-mulcl 7206  ax-mulrcl 7207  ax-addcom 7208  ax-mulcom 7209  ax-addass 7210  ax-mulass 7211  ax-distr 7212  ax-i2m1 7213  ax-0lt1 7214  ax-1rid 7215  ax-0id 7216  ax-rnegex 7217  ax-precex 7218  ax-cnre 7219  ax-pre-ltirr 7220  ax-pre-ltwlin 7221  ax-pre-lttrn 7222  ax-pre-apti 7223  ax-pre-ltadd 7224  ax-pre-mulgt0 7225  ax-pre-mulext 7226
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-po 4079  df-iso 4080  df-iord 4149  df-on 4151  df-ilim 4152  df-suc 4154  df-iom 4360  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-riota 5520  df-ov 5567  df-oprab 5568  df-mpt2 5569  df-1st 5819  df-2nd 5820  df-recs 5975  df-frec 6061  df-pnf 7287  df-mnf 7288  df-xr 7289  df-ltxr 7290  df-le 7291  df-sub 7418  df-neg 7419  df-reap 7812  df-ap 7819  df-div 7898  df-inn 8177  df-2 8235  df-n0 8426  df-z 8503  df-uz 8771  df-iseq 9592  df-iexp 9643
This theorem is referenced by:  nn0opth2d  9817
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