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Theorem nn0opth2 12332
Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 12330. (Contributed by NM, 22-Jul-2004.)
Assertion
Ref Expression
nn0opth2  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( C  e.  NN0  /\  D  e.  NN0 )
)  ->  ( (
( ( A  +  B ) ^ 2 )  +  B )  =  ( ( ( C  +  D ) ^ 2 )  +  D )  <->  ( A  =  C  /\  B  =  D ) ) )

Proof of Theorem nn0opth2
StepHypRef Expression
1 oveq1 6302 . . . . . 6  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( A  +  B
)  =  ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) )
21oveq1d 6310 . . . . 5  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( A  +  B ) ^ 2 )  =  ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  B
) ^ 2 ) )
32oveq1d 6310 . . . 4  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( ( A  +  B ) ^
2 )  +  B
)  =  ( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) ^ 2 )  +  B ) )
43eqeq1d 2469 . . 3  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( ( ( A  +  B ) ^ 2 )  +  B )  =  ( ( ( C  +  D ) ^ 2 )  +  D )  <-> 
( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) ^
2 )  +  B
)  =  ( ( ( C  +  D
) ^ 2 )  +  D ) ) )
5 eqeq1 2471 . . . 4  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( A  =  C  <-> 
if ( A  e. 
NN0 ,  A , 
0 )  =  C ) )
65anbi1d 704 . . 3  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( A  =  C  /\  B  =  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  B  =  D ) ) )
74, 6bibi12d 321 . 2  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( ( ( ( A  +  B
) ^ 2 )  +  B )  =  ( ( ( C  +  D ) ^
2 )  +  D
)  <->  ( A  =  C  /\  B  =  D ) )  <->  ( (
( ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) ^
2 )  +  B
)  =  ( ( ( C  +  D
) ^ 2 )  +  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  B  =  D ) ) ) )
8 oveq2 6303 . . . . . 6  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  -> 
( if ( A  e.  NN0 ,  A ,  0 )  +  B )  =  ( if ( A  e. 
NN0 ,  A , 
0 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) )
98oveq1d 6310 . . . . 5  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  -> 
( ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) ^
2 )  =  ( ( if ( A  e.  NN0 ,  A ,  0 )  +  if ( B  e. 
NN0 ,  B , 
0 ) ) ^
2 ) )
10 id 22 . . . . 5  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  ->  B  =  if ( B  e.  NN0 ,  B ,  0 ) )
119, 10oveq12d 6313 . . . 4  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  -> 
( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) ^
2 )  +  B
)  =  ( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  if ( B  e. 
NN0 ,  B , 
0 ) ) ^
2 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) )
1211eqeq1d 2469 . . 3  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  -> 
( ( ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  B
) ^ 2 )  +  B )  =  ( ( ( C  +  D ) ^
2 )  +  D
)  <->  ( ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) ^ 2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( C  +  D ) ^ 2 )  +  D ) ) )
13 eqeq1 2471 . . . 4  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  -> 
( B  =  D  <-> 
if ( B  e. 
NN0 ,  B , 
0 )  =  D ) )
1413anbi2d 703 . . 3  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  -> 
( ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  B  =  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  if ( B  e.  NN0 ,  B ,  0 )  =  D ) ) )
1512, 14bibi12d 321 . 2  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  -> 
( ( ( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) ^ 2 )  +  B )  =  ( ( ( C  +  D ) ^ 2 )  +  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  B  =  D ) )  <->  ( (
( ( if ( A  e.  NN0 ,  A ,  0 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) ^ 2 )  +  if ( B  e. 
NN0 ,  B , 
0 ) )  =  ( ( ( C  +  D ) ^
2 )  +  D
)  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  if ( B  e.  NN0 ,  B ,  0 )  =  D ) ) ) )
16 oveq1 6302 . . . . . 6  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( C  +  D
)  =  ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) )
1716oveq1d 6310 . . . . 5  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( C  +  D ) ^ 2 )  =  ( ( if ( C  e. 
NN0 ,  C , 
0 )  +  D
) ^ 2 ) )
1817oveq1d 6310 . . . 4  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( ( C  +  D ) ^
2 )  +  D
)  =  ( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) ^ 2 )  +  D ) )
1918eqeq2d 2481 . . 3  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) ^ 2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( C  +  D ) ^ 2 )  +  D )  <->  ( (
( if ( A  e.  NN0 ,  A ,  0 )  +  if ( B  e. 
NN0 ,  B , 
0 ) ) ^
2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) ^ 2 )  +  D ) ) )
20 eqeq2 2482 . . . 4  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( if ( A  e.  NN0 ,  A ,  0 )  =  C  <->  if ( A  e. 
NN0 ,  A , 
0 )  =  if ( C  e.  NN0 ,  C ,  0 ) ) )
2120anbi1d 704 . . 3  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  if ( B  e.  NN0 ,  B ,  0 )  =  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  if ( C  e.  NN0 ,  C ,  0 )  /\  if ( B  e.  NN0 ,  B ,  0 )  =  D ) ) )
2219, 21bibi12d 321 . 2  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( ( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  if ( B  e. 
NN0 ,  B , 
0 ) ) ^
2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( C  +  D
) ^ 2 )  +  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  if ( B  e.  NN0 ,  B ,  0 )  =  D ) )  <-> 
( ( ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) ^ 2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( if ( C  e. 
NN0 ,  C , 
0 )  +  D
) ^ 2 )  +  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  if ( C  e.  NN0 ,  C ,  0 )  /\  if ( B  e.  NN0 ,  B ,  0 )  =  D ) ) ) )
23 oveq2 6303 . . . . . 6  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  -> 
( if ( C  e.  NN0 ,  C ,  0 )  +  D )  =  ( if ( C  e. 
NN0 ,  C , 
0 )  +  if ( D  e.  NN0 ,  D ,  0 ) ) )
2423oveq1d 6310 . . . . 5  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  -> 
( ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) ^
2 )  =  ( ( if ( C  e.  NN0 ,  C ,  0 )  +  if ( D  e. 
NN0 ,  D , 
0 ) ) ^
2 ) )
25 id 22 . . . . 5  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  ->  D  =  if ( D  e.  NN0 ,  D ,  0 ) )
2624, 25oveq12d 6313 . . . 4  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  -> 
( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) ^
2 )  +  D
)  =  ( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  if ( D  e. 
NN0 ,  D , 
0 ) ) ^
2 )  +  if ( D  e.  NN0 ,  D ,  0 ) ) )
2726eqeq2d 2481 . . 3  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  -> 
( ( ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) ^ 2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( if ( C  e. 
NN0 ,  C , 
0 )  +  D
) ^ 2 )  +  D )  <->  ( (
( if ( A  e.  NN0 ,  A ,  0 )  +  if ( B  e. 
NN0 ,  B , 
0 ) ) ^
2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  if ( D  e. 
NN0 ,  D , 
0 ) ) ^
2 )  +  if ( D  e.  NN0 ,  D ,  0 ) ) ) )
28 eqeq2 2482 . . . 4  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  -> 
( if ( B  e.  NN0 ,  B ,  0 )  =  D  <->  if ( B  e. 
NN0 ,  B , 
0 )  =  if ( D  e.  NN0 ,  D ,  0 ) ) )
2928anbi2d 703 . . 3  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  -> 
( ( if ( A  e.  NN0 ,  A ,  0 )  =  if ( C  e.  NN0 ,  C ,  0 )  /\  if ( B  e.  NN0 ,  B ,  0 )  =  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  if ( C  e.  NN0 ,  C ,  0 )  /\  if ( B  e.  NN0 ,  B ,  0 )  =  if ( D  e.  NN0 ,  D ,  0 ) ) ) )
3027, 29bibi12d 321 . 2  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  -> 
( ( ( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  if ( B  e. 
NN0 ,  B , 
0 ) ) ^
2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) ^ 2 )  +  D )  <-> 
( if ( A  e.  NN0 ,  A ,  0 )  =  if ( C  e. 
NN0 ,  C , 
0 )  /\  if ( B  e.  NN0 ,  B ,  0 )  =  D ) )  <-> 
( ( ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) ^ 2 )  +  if ( B  e.  NN0 ,  B ,  0 ) )  =  ( ( ( if ( C  e. 
NN0 ,  C , 
0 )  +  if ( D  e.  NN0 ,  D ,  0 ) ) ^ 2 )  +  if ( D  e.  NN0 ,  D ,  0 ) )  <-> 
( if ( A  e.  NN0 ,  A ,  0 )  =  if ( C  e. 
NN0 ,  C , 
0 )  /\  if ( B  e.  NN0 ,  B ,  0 )  =  if ( D  e.  NN0 ,  D ,  0 ) ) ) ) )
31 0nn0 10822 . . . 4  |-  0  e.  NN0
3231elimel 4008 . . 3  |-  if ( A  e.  NN0 ,  A ,  0 )  e.  NN0
3331elimel 4008 . . 3  |-  if ( B  e.  NN0 ,  B ,  0 )  e.  NN0
3431elimel 4008 . . 3  |-  if ( C  e.  NN0 ,  C ,  0 )  e.  NN0
3531elimel 4008 . . 3  |-  if ( D  e.  NN0 ,  D ,  0 )  e.  NN0
3632, 33, 34, 35nn0opth2i 12331 . 2  |-  ( ( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  if ( B  e.  NN0 ,  B ,  0 ) ) ^ 2 )  +  if ( B  e. 
NN0 ,  B , 
0 ) )  =  ( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  if ( D  e.  NN0 ,  D ,  0 ) ) ^ 2 )  +  if ( D  e. 
NN0 ,  D , 
0 ) )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  if ( C  e.  NN0 ,  C ,  0 )  /\  if ( B  e.  NN0 ,  B ,  0 )  =  if ( D  e.  NN0 ,  D ,  0 ) ) )
377, 15, 22, 30, 36dedth4h 4000 1  |-  ( ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( C  e.  NN0  /\  D  e.  NN0 )
)  ->  ( (
( ( A  +  B ) ^ 2 )  +  B )  =  ( ( ( C  +  D ) ^ 2 )  +  D )  <->  ( A  =  C  /\  B  =  D ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   ifcif 3945  (class class class)co 6295   0cc0 9504    + caddc 9507   2c2 10597   NN0cn0 10807   ^cexp 12146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-n0 10808  df-z 10877  df-uz 11095  df-seq 12088  df-exp 12147
This theorem is referenced by:  xpnnenOLD  13821
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