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Theorem nn0opth2 11287
Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 11285. (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 5865 . . . . . 6  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( A  +  B
)  =  ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) )
21oveq1d 5873 . . . . 5  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( A  +  B ) ^ 2 )  =  ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  B
) ^ 2 ) )
32oveq1d 5873 . . . 4  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( ( A  +  B ) ^
2 )  +  B
)  =  ( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) ^ 2 )  +  B ) )
43eqeq1d 2291 . . 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 2289 . . . 4  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( A  =  C  <-> 
if ( A  e. 
NN0 ,  A , 
0 )  =  C ) )
65anbi1d 685 . . 3  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( A  =  C  /\  B  =  D )  <->  ( if ( A  e.  NN0 ,  A ,  0 )  =  C  /\  B  =  D ) ) )
74, 6bibi12d 312 . 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 5866 . . . . . 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 5873 . . . . 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 19 . . . . 5  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  ->  B  =  if ( B  e.  NN0 ,  B ,  0 ) )
119, 10oveq12d 5876 . . . 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 2291 . . 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 2289 . . . 4  |-  ( B  =  if ( B  e.  NN0 ,  B ,  0 )  -> 
( B  =  D  <-> 
if ( B  e. 
NN0 ,  B , 
0 )  =  D ) )
1413anbi2d 684 . . 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 312 . 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 5865 . . . . . 6  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( C  +  D
)  =  ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) )
1716oveq1d 5873 . . . . 5  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( C  +  D ) ^ 2 )  =  ( ( if ( C  e. 
NN0 ,  C , 
0 )  +  D
) ^ 2 ) )
1817oveq1d 5873 . . . 4  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( ( C  +  D ) ^
2 )  +  D
)  =  ( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) ^ 2 )  +  D ) )
1918eqeq2d 2294 . . 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 2292 . . . 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 685 . . 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 312 . 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 5866 . . . . . 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 5873 . . . . 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 19 . . . . 5  |-  ( D  =  if ( D  e.  NN0 ,  D ,  0 )  ->  D  =  if ( D  e.  NN0 ,  D ,  0 ) )
2624, 25oveq12d 5876 . . . 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 2294 . . 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 2292 . . . 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 684 . . 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 312 . 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 9980 . . . 4  |-  0  e.  NN0
3231elimel 3617 . . 3  |-  if ( A  e.  NN0 ,  A ,  0 )  e.  NN0
3331elimel 3617 . . 3  |-  if ( B  e.  NN0 ,  B ,  0 )  e.  NN0
3431elimel 3617 . . 3  |-  if ( C  e.  NN0 ,  C ,  0 )  e.  NN0
3531elimel 3617 . . 3  |-  if ( D  e.  NN0 ,  D ,  0 )  e.  NN0
3632, 33, 34, 35nn0opth2i 11286 . 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 3609 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 set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   ifcif 3565  (class class class)co 5858   0cc0 8737    + caddc 8740   2c2 9795   NN0cn0 9965   ^cexp 11104
This theorem is referenced by:  xpnnenOLD  12488
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-n0 9966  df-z 10025  df-uz 10231  df-seq 11047  df-exp 11105
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