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Theorem nn0opthd 10126
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 3455 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 8728 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( C  +  D
)  e.  NN0 )
61, 2, 5, 4nn0opthlem2d 10125 . . . . . . . . . . . . . 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 2341 . . . . . . . . . . . 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 8728 . . . . . . . . . . . 12  |-  ( ph  ->  ( A  +  B
)  e.  NN0 )
113, 4, 10, 2nn0opthlem2d 10125 . . . . . . . . . . 11  |-  ( ph  ->  ( ( C  +  D )  <  ( A  +  B )  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  =/=  ( ( ( C  +  D
)  x.  ( C  +  D ) )  +  D ) ) )
129, 11jaod 672 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( A  +  B )  < 
( C  +  D
)  \/  ( C  +  D )  < 
( A  +  B
) )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) ) )
1310nn0red 8725 . . . . . . . . . . 11  |-  ( ph  ->  ( A  +  B
)  e.  RR )
145nn0red 8725 . . . . . . . . . . 11  |-  ( ph  ->  ( C  +  D
)  e.  RR )
15 reaplt 8063 . . . . . . . . . . 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 8729 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( A  +  B )  x.  ( A  +  B )
)  e.  NN0 )
1817, 2nn0addcld 8728 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  NN0 )
1918nn0zd 8864 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  ZZ )
205, 5nn0mulcld 8729 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( C  +  D )  x.  ( C  +  D )
)  e.  NN0 )
2120, 4nn0addcld 8728 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  e.  NN0 )
2221nn0zd 8864 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  e.  ZZ )
23 zapne 8819 . . . . . . . . . . 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 596 . . . . . . . 8  |-  ( ph  ->  ( -.  ( ( ( A  +  B
)  x.  ( A  +  B ) )  +  B ) #  ( ( ( C  +  D )  x.  ( C  +  D )
)  +  D )  ->  -.  ( A  +  B ) #  ( C  +  D ) ) )
2718nn0cnd 8726 . . . . . . . . 9  |-  ( ph  ->  ( ( ( A  +  B )  x.  ( A  +  B
) )  +  B
)  e.  CC )
2821nn0cnd 8726 . . . . . . . . 9  |-  ( ph  ->  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
)  e.  CC )
29 apti 8097 . . . . . . . . 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 8726 . . . . . . . . 9  |-  ( ph  ->  ( A  +  B
)  e.  CC )
325nn0cnd 8726 . . . . . . . . 9  |-  ( ph  ->  ( C  +  D
)  e.  CC )
33 apti 8097 . . . . . . . . 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 5670 . . . . . . . . . 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 5667 . . . . . . . . 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 2123 . . . . . . . 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 7506 . . . . . . . . . 10  |-  ( ph  ->  ( ( A  +  B )  x.  ( A  +  B )
)  e.  CC )
422nn0cnd 8726 . . . . . . . . . 10  |-  ( ph  ->  B  e.  CC )
434nn0cnd 8726 . . . . . . . . . 10  |-  ( ph  ->  D  e.  CC )
4441, 42, 43addcand 7664 . . . . . . . . 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 5668 . . . . . 6  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  ( C  +  B )  =  ( C  +  D ) )
4836, 47eqtr4d 2123 . . . . 5  |-  ( (
ph  /\  ( (
( A  +  B
)  x.  ( A  +  B ) )  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D
) )  +  D
) )  ->  ( A  +  B )  =  ( C  +  B ) )
491nn0cnd 8726 . . . . . . 7  |-  ( ph  ->  A  e.  CC )
503nn0cnd 8726 . . . . . . 7  |-  ( ph  ->  C  e.  CC )
5149, 50, 42addcan2d 7665 . . . . . 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 5661 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  B
)  =  ( C  +  D ) )
5756, 56oveq12d 5670 . . 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 5670 . 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 664    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3845  (class class class)co 5652   CCcc 7346   RRcr 7347    + caddc 7351    x. cmul 7353    < clt 7520   # cap 8056   NN0cn0 8671   ZZcz 8748
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-nul 3965  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-iinf 4403  ax-cnex 7434  ax-resscn 7435  ax-1cn 7436  ax-1re 7437  ax-icn 7438  ax-addcl 7439  ax-addrcl 7440  ax-mulcl 7441  ax-mulrcl 7442  ax-addcom 7443  ax-mulcom 7444  ax-addass 7445  ax-mulass 7446  ax-distr 7447  ax-i2m1 7448  ax-0lt1 7449  ax-1rid 7450  ax-0id 7451  ax-rnegex 7452  ax-precex 7453  ax-cnre 7454  ax-pre-ltirr 7455  ax-pre-ltwlin 7456  ax-pre-lttrn 7457  ax-pre-apti 7458  ax-pre-ltadd 7459  ax-pre-mulgt0 7460  ax-pre-mulext 7461
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3394  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-int 3689  df-iun 3732  df-br 3846  df-opab 3900  df-mpt 3901  df-tr 3937  df-id 4120  df-po 4123  df-iso 4124  df-iord 4193  df-on 4195  df-ilim 4196  df-suc 4198  df-iom 4406  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-1st 5911  df-2nd 5912  df-recs 6070  df-frec 6156  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525  df-le 7526  df-sub 7653  df-neg 7654  df-reap 8050  df-ap 8057  df-div 8138  df-inn 8421  df-2 8479  df-n0 8672  df-z 8749  df-uz 9018  df-iseq 9849  df-seq3 9850  df-exp 9951
This theorem is referenced by:  nn0opth2d  10127
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