MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nn0opth2 Structured version   Unicode version

Theorem nn0opth2 12355
Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 12353. (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 6303 . . . . . 6  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( A  +  B
)  =  ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) )
21oveq1d 6311 . . . . 5  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( A  +  B ) ^ 2 )  =  ( ( if ( A  e. 
NN0 ,  A , 
0 )  +  B
) ^ 2 ) )
32oveq1d 6311 . . . 4  |-  ( A  =  if ( A  e.  NN0 ,  A ,  0 )  -> 
( ( ( A  +  B ) ^
2 )  +  B
)  =  ( ( ( if ( A  e.  NN0 ,  A ,  0 )  +  B ) ^ 2 )  +  B ) )
43eqeq1d 2459 . . 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 2461 . . . 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 6304 . . . . . 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 6311 . . . . 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 6314 . . . 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 2459 . . 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 2461 . . . 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 6303 . . . . . 6  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( C  +  D
)  =  ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) )
1716oveq1d 6311 . . . . 5  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( C  +  D ) ^ 2 )  =  ( ( if ( C  e. 
NN0 ,  C , 
0 )  +  D
) ^ 2 ) )
1817oveq1d 6311 . . . 4  |-  ( C  =  if ( C  e.  NN0 ,  C ,  0 )  -> 
( ( ( C  +  D ) ^
2 )  +  D
)  =  ( ( ( if ( C  e.  NN0 ,  C ,  0 )  +  D ) ^ 2 )  +  D ) )
1918eqeq2d 2471 . . 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 2472 . . . 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 6304 . . . . . 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 6311 . . . . 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 6314 . . . 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 2471 . . 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 2472 . . . 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 10831 . . . 4  |-  0  e.  NN0
3231elimel 4007 . . 3  |-  if ( A  e.  NN0 ,  A ,  0 )  e.  NN0
3331elimel 4007 . . 3  |-  if ( B  e.  NN0 ,  B ,  0 )  e.  NN0
3431elimel 4007 . . 3  |-  if ( C  e.  NN0 ,  C ,  0 )  e.  NN0
3531elimel 4007 . . 3  |-  if ( D  e.  NN0 ,  D ,  0 )  e.  NN0
3632, 33, 34, 35nn0opth2i 12354 . 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 3999 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 1395    e. wcel 1819   ifcif 3944  (class class class)co 6296   0cc0 9509    + caddc 9512   2c2 10606   NN0cn0 10816   ^cexp 12169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-2 10615  df-n0 10817  df-z 10886  df-uz 11107  df-seq 12111  df-exp 12170
This theorem is referenced by:  xpnnenOLD  13955
  Copyright terms: Public domain W3C validator