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Theorem nn0opthi 11163
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 3553 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 9880 . . . . . . . . 9  |-  ( A  +  B )  e. 
NN0
43nn0rei 9855 . . . . . . . 8  |-  ( A  +  B )  e.  RR
5 nn0opth.3 . . . . . . . . . 10  |-  C  e. 
NN0
6 nn0opth.4 . . . . . . . . . 10  |-  D  e. 
NN0
75, 6nn0addcli 9880 . . . . . . . . 9  |-  ( C  +  D )  e. 
NN0
87nn0rei 9855 . . . . . . . 8  |-  ( C  +  D )  e.  RR
94, 8lttri2i 8812 . . . . . . 7  |-  ( ( A  +  B )  =/=  ( C  +  D )  <->  ( ( A  +  B )  <  ( C  +  D
)  \/  ( C  +  D )  < 
( A  +  B
) ) )
101, 2, 7, 6nn0opthlem2 11162 . . . . . . . . 9  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( C  +  D )  x.  ( C  +  D )
)  +  D )  =/=  ( ( ( A  +  B )  x.  ( A  +  B ) )  +  B ) )
1110necomd 2495 . . . . . . . 8  |-  ( ( A  +  B )  <  ( C  +  D )  ->  (
( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =/=  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D ) )
125, 6, 3, 2nn0opthlem2 11162 . . . . . . . 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 2472 . . . . 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 5728 . . . . . . . . 9  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  (
( A  +  B
)  x.  ( A  +  B ) )  =  ( ( C  +  D )  x.  ( C  +  D
) ) )
1817oveq1d 5725 . . . . . . . 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 2288 . . . . . . 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 9856 . . . . . . . . 9  |-  ( A  +  B )  e.  CC
2120, 20mulcli 8722 . . . . . . . 8  |-  ( ( A  +  B )  x.  ( A  +  B ) )  e.  CC
222nn0cni 9856 . . . . . . . 8  |-  B  e.  CC
236nn0cni 9856 . . . . . . . 8  |-  D  e.  CC
2421, 22, 23addcani 8885 . . . . . . 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 5726 . . . . 5  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( C  +  B )  =  ( C  +  D ) )
2715, 26eqtr4d 2288 . . . 4  |-  ( ( ( ( A  +  B )  x.  ( A  +  B )
)  +  B )  =  ( ( ( C  +  D )  x.  ( C  +  D ) )  +  D )  ->  ( A  +  B )  =  ( C  +  B ) )
281nn0cni 9856 . . . . 5  |-  A  e.  CC
295nn0cni 9856 . . . . 5  |-  C  e.  CC
3028, 29, 22addcan2i 8886 . . . 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 5719 . . . 4  |-  ( ( A  =  C  /\  B  =  D )  ->  ( A  +  B
)  =  ( C  +  D ) )
3433, 33oveq12d 5728 . . 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 5728 . 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 2412   class class class wbr 3920  (class class class)co 5710    + caddc 8620    x. cmul 8622    < clt 8747   NN0cn0 9844
This theorem is referenced by:  nn0opth2i  11164
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-n0 9845  df-z 9904  df-uz 10110  df-seq 10925  df-exp 10983
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