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Theorem xpnnenOLD 13603
Description: The Cartesian product of the set of positive integers with itself is equinumerous to the set of positive integers. The key idea is to use nn0opth2 12160 to show that the mapping from positive integers  z and  w to  ( ( z  +  w ) ^ 2 )  +  w is one-to-one. (Contributed by NM, 1-Aug-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
xpnnenOLD  |-  ( NN 
X.  NN )  ~~  NN

Proof of Theorem xpnnenOLD
Dummy variables  x  y  z  w  v  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nnex 10432 . . 3  |-  NN  e.  _V
2 elxp5 6626 . . . . 5  |-  ( x  e.  ( NN  X.  NN )  <->  ( x  = 
<. |^| |^| x ,  U. ran  { x } >.  /\  ( |^| |^| x  e.  NN  /\  U. ran  { x }  e.  NN ) ) )
3 nnaddcl 10448 . . . . . . . 8  |-  ( (
|^| |^| x  e.  NN  /\ 
U. ran  { x }  e.  NN )  ->  ( |^| |^| x  +  U. ran  { x } )  e.  NN )
4 2nn0 10700 . . . . . . . 8  |-  2  e.  NN0
5 nnexpcl 11988 . . . . . . . 8  |-  ( ( ( |^| |^| x  +  U. ran  { x } )  e.  NN  /\  2  e.  NN0 )  ->  ( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  e.  NN )
63, 4, 5sylancl 662 . . . . . . 7  |-  ( (
|^| |^| x  e.  NN  /\ 
U. ran  { x }  e.  NN )  ->  ( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  e.  NN )
7 nnaddcl 10448 . . . . . . 7  |-  ( ( ( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  e.  NN  /\  U.
ran  { x }  e.  NN )  ->  ( ( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  +  U. ran  { x } )  e.  NN )
86, 7sylancom 667 . . . . . 6  |-  ( (
|^| |^| x  e.  NN  /\ 
U. ran  { x }  e.  NN )  ->  ( ( ( |^| |^| x  +  U. ran  { x } ) ^
2 )  +  U. ran  { x } )  e.  NN )
98adantl 466 . . . . 5  |-  ( ( x  =  <. |^| |^| x ,  U. ran  { x } >.  /\  ( |^| |^| x  e.  NN  /\  U.
ran  { x }  e.  NN ) )  ->  (
( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  +  U. ran  { x } )  e.  NN )
102, 9sylbi 195 . . . 4  |-  ( x  e.  ( NN  X.  NN )  ->  ( ( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  +  U. ran  { x } )  e.  NN )
11 elxp2 4959 . . . . 5  |-  ( x  e.  ( NN  X.  NN )  <->  E. z  e.  NN  E. w  e.  NN  x  =  <. z ,  w >. )
12 elxp2 4959 . . . . 5  |-  ( y  e.  ( NN  X.  NN )  <->  E. v  e.  NN  E. u  e.  NN  y  =  <. v ,  u >. )
13 inteq 4232 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. z ,  w >.  ->  |^| x  =  |^| <.
z ,  w >. )
1413inteqd 4234 . . . . . . . . . . . . . . . . 17  |-  ( x  =  <. z ,  w >.  ->  |^| |^| x  =  |^| |^|
<. z ,  w >. )
15 vex 3074 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
16 vex 3074 . . . . . . . . . . . . . . . . . 18  |-  w  e. 
_V
1715, 16op1stb 4663 . . . . . . . . . . . . . . . . 17  |-  |^| |^| <. z ,  w >.  =  z
1814, 17syl6eq 2508 . . . . . . . . . . . . . . . 16  |-  ( x  =  <. z ,  w >.  ->  |^| |^| x  =  z )
19 sneq 3988 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  <. z ,  w >.  ->  { x }  =  { <. z ,  w >. } )
2019rneqd 5168 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. z ,  w >.  ->  ran  { x }  =  ran  { <. z ,  w >. } )
2120unieqd 4202 . . . . . . . . . . . . . . . . 17  |-  ( x  =  <. z ,  w >.  ->  U. ran  { x }  =  U. ran  { <. z ,  w >. } )
2215, 16op2nda 5425 . . . . . . . . . . . . . . . . 17  |-  U. ran  {
<. z ,  w >. }  =  w
2321, 22syl6eq 2508 . . . . . . . . . . . . . . . 16  |-  ( x  =  <. z ,  w >.  ->  U. ran  { x }  =  w )
2418, 23oveq12d 6211 . . . . . . . . . . . . . . 15  |-  ( x  =  <. z ,  w >.  ->  ( |^| |^| x  +  U. ran  { x } )  =  ( z  +  w ) )
2524oveq1d 6208 . . . . . . . . . . . . . 14  |-  ( x  =  <. z ,  w >.  ->  ( ( |^| |^| x  +  U. ran  { x } ) ^
2 )  =  ( ( z  +  w
) ^ 2 ) )
2625, 23oveq12d 6211 . . . . . . . . . . . . 13  |-  ( x  =  <. z ,  w >.  ->  ( ( (
|^| |^| x  +  U. ran  { x } ) ^ 2 )  + 
U. ran  { x } )  =  ( ( ( z  +  w ) ^ 2 )  +  w ) )
27 inteq 4232 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  <. v ,  u >.  ->  |^| y  =  |^| <.
v ,  u >. )
2827inteqd 4234 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. v ,  u >.  ->  |^| |^| y  =  |^| |^|
<. v ,  u >. )
29 vex 3074 . . . . . . . . . . . . . . . . . 18  |-  v  e. 
_V
30 vex 3074 . . . . . . . . . . . . . . . . . 18  |-  u  e. 
_V
3129, 30op1stb 4663 . . . . . . . . . . . . . . . . 17  |-  |^| |^| <. v ,  u >.  =  v
3228, 31syl6eq 2508 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. v ,  u >.  ->  |^| |^| y  =  v )
33 sneq 3988 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  <. v ,  u >.  ->  { y }  =  { <. v ,  u >. } )
3433rneqd 5168 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  <. v ,  u >.  ->  ran  { y }  =  ran  { <. v ,  u >. } )
3534unieqd 4202 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. v ,  u >.  ->  U. ran  { y }  =  U. ran  {
<. v ,  u >. } )
3629, 30op2nda 5425 . . . . . . . . . . . . . . . . 17  |-  U. ran  {
<. v ,  u >. }  =  u
3735, 36syl6eq 2508 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. v ,  u >.  ->  U. ran  { y }  =  u )
3832, 37oveq12d 6211 . . . . . . . . . . . . . . 15  |-  ( y  =  <. v ,  u >.  ->  ( |^| |^| y  +  U. ran  { y } )  =  ( v  +  u ) )
3938oveq1d 6208 . . . . . . . . . . . . . 14  |-  ( y  =  <. v ,  u >.  ->  ( ( |^| |^| y  +  U. ran  { y } ) ^
2 )  =  ( ( v  +  u
) ^ 2 ) )
4039, 37oveq12d 6211 . . . . . . . . . . . . 13  |-  ( y  =  <. v ,  u >.  ->  ( ( (
|^| |^| y  +  U. ran  { y } ) ^ 2 )  + 
U. ran  { y } )  =  ( ( ( v  +  u ) ^ 2 )  +  u ) )
4126, 40eqeqan12d 2474 . . . . . . . . . . . 12  |-  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  ->  ( (
( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  +  U. ran  { x } )  =  ( ( ( |^| |^| y  +  U. ran  { y } ) ^
2 )  +  U. ran  { y } )  <-> 
( ( ( z  +  w ) ^
2 )  +  w
)  =  ( ( ( v  +  u
) ^ 2 )  +  u ) ) )
42 nnnn0 10690 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  z  e.  NN0 )
43 nnnn0 10690 . . . . . . . . . . . . . 14  |-  ( w  e.  NN  ->  w  e.  NN0 )
4442, 43anim12i 566 . . . . . . . . . . . . 13  |-  ( ( z  e.  NN  /\  w  e.  NN )  ->  ( z  e.  NN0  /\  w  e.  NN0 )
)
45 nnnn0 10690 . . . . . . . . . . . . . 14  |-  ( v  e.  NN  ->  v  e.  NN0 )
46 nnnn0 10690 . . . . . . . . . . . . . 14  |-  ( u  e.  NN  ->  u  e.  NN0 )
4745, 46anim12i 566 . . . . . . . . . . . . 13  |-  ( ( v  e.  NN  /\  u  e.  NN )  ->  ( v  e.  NN0  /\  u  e.  NN0 )
)
48 nn0opth2 12160 . . . . . . . . . . . . 13  |-  ( ( ( z  e.  NN0  /\  w  e.  NN0 )  /\  ( v  e.  NN0  /\  u  e.  NN0 )
)  ->  ( (
( ( z  +  w ) ^ 2 )  +  w )  =  ( ( ( v  +  u ) ^ 2 )  +  u )  <->  ( z  =  v  /\  w  =  u ) ) )
4944, 47, 48syl2an 477 . . . . . . . . . . . 12  |-  ( ( ( z  e.  NN  /\  w  e.  NN )  /\  ( v  e.  NN  /\  u  e.  NN ) )  -> 
( ( ( ( z  +  w ) ^ 2 )  +  w )  =  ( ( ( v  +  u ) ^ 2 )  +  u )  <-> 
( z  =  v  /\  w  =  u ) ) )
5041, 49sylan9bbr 700 . . . . . . . . . . 11  |-  ( ( ( ( z  e.  NN  /\  w  e.  NN )  /\  (
v  e.  NN  /\  u  e.  NN )
)  /\  ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )
)  ->  ( (
( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  +  U. ran  { x } )  =  ( ( ( |^| |^| y  +  U. ran  { y } ) ^
2 )  +  U. ran  { y } )  <-> 
( z  =  v  /\  w  =  u ) ) )
51 eqeq12 2470 . . . . . . . . . . . . 13  |-  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  ->  ( x  =  y  <->  <. z ,  w >.  =  <. v ,  u >. ) )
5215, 16opth 4667 . . . . . . . . . . . . 13  |-  ( <.
z ,  w >.  = 
<. v ,  u >.  <->  (
z  =  v  /\  w  =  u )
)
5351, 52syl6bb 261 . . . . . . . . . . . 12  |-  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  ->  ( x  =  y  <->  ( z  =  v  /\  w  =  u ) ) )
5453adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( z  e.  NN  /\  w  e.  NN )  /\  (
v  e.  NN  /\  u  e.  NN )
)  /\  ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )
)  ->  ( x  =  y  <->  ( z  =  v  /\  w  =  u ) ) )
5550, 54bitr4d 256 . . . . . . . . . 10  |-  ( ( ( ( z  e.  NN  /\  w  e.  NN )  /\  (
v  e.  NN  /\  u  e.  NN )
)  /\  ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )
)  ->  ( (
( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  +  U. ran  { x } )  =  ( ( ( |^| |^| y  +  U. ran  { y } ) ^
2 )  +  U. ran  { y } )  <-> 
x  =  y ) )
5655exp43 612 . . . . . . . . 9  |-  ( ( z  e.  NN  /\  w  e.  NN )  ->  ( ( v  e.  NN  /\  u  e.  NN )  ->  (
x  =  <. z ,  w >.  ->  ( y  =  <. v ,  u >.  ->  ( ( ( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  +  U. ran  { x } )  =  ( ( ( |^| |^| y  +  U. ran  { y } ) ^
2 )  +  U. ran  { y } )  <-> 
x  =  y ) ) ) ) )
5756com23 78 . . . . . . . 8  |-  ( ( z  e.  NN  /\  w  e.  NN )  ->  ( x  =  <. z ,  w >.  ->  (
( v  e.  NN  /\  u  e.  NN )  ->  ( y  = 
<. v ,  u >.  -> 
( ( ( (
|^| |^| x  +  U. ran  { x } ) ^ 2 )  + 
U. ran  { x } )  =  ( ( ( |^| |^| y  +  U. ran  { y } ) ^ 2 )  +  U. ran  { y } )  <->  x  =  y ) ) ) ) )
5857rexlimivv 2945 . . . . . . 7  |-  ( E. z  e.  NN  E. w  e.  NN  x  =  <. z ,  w >.  ->  ( ( v  e.  NN  /\  u  e.  NN )  ->  (
y  =  <. v ,  u >.  ->  ( ( ( ( |^| |^| x  +  U. ran  { x } ) ^ 2 )  +  U. ran  { x } )  =  ( ( ( |^| |^| y  +  U. ran  { y } ) ^
2 )  +  U. ran  { y } )  <-> 
x  =  y ) ) ) )
5958rexlimdvv 2946 . . . . . 6  |-  ( E. z  e.  NN  E. w  e.  NN  x  =  <. z ,  w >.  ->  ( E. v  e.  NN  E. u  e.  NN  y  =  <. v ,  u >.  ->  (
( ( ( |^| |^| x  +  U. ran  { x } ) ^
2 )  +  U. ran  { x } )  =  ( ( (
|^| |^| y  +  U. ran  { y } ) ^ 2 )  + 
U. ran  { y } )  <->  x  =  y ) ) )
6059imp 429 . . . . 5  |-  ( ( E. z  e.  NN  E. w  e.  NN  x  =  <. z ,  w >.  /\  E. v  e.  NN  E. u  e.  NN  y  =  <. v ,  u >. )  ->  ( ( ( (
|^| |^| x  +  U. ran  { x } ) ^ 2 )  + 
U. ran  { x } )  =  ( ( ( |^| |^| y  +  U. ran  { y } ) ^ 2 )  +  U. ran  { y } )  <->  x  =  y ) )
6111, 12, 60syl2anb 479 . . . 4  |-  ( ( x  e.  ( NN 
X.  NN )  /\  y  e.  ( NN  X.  NN ) )  -> 
( ( ( (
|^| |^| x  +  U. ran  { x } ) ^ 2 )  + 
U. ran  { x } )  =  ( ( ( |^| |^| y  +  U. ran  { y } ) ^ 2 )  +  U. ran  { y } )  <->  x  =  y ) )
6210, 61dom2 7455 . . 3  |-  ( NN  e.  _V  ->  ( NN  X.  NN )  ~<_  NN )
631, 62ax-mp 5 . 2  |-  ( NN 
X.  NN )  ~<_  NN
64 1nn 10437 . . . . . 6  |-  1  e.  NN
6564elexi 3081 . . . . 5  |-  1  e.  _V
661, 65xpsnen 7498 . . . 4  |-  ( NN 
X.  { 1 } )  ~~  NN
6766ensymi 7462 . . 3  |-  NN  ~~  ( NN  X.  { 1 } )
681, 1xpex 6611 . . . 4  |-  ( NN 
X.  NN )  e. 
_V
69 ssid 3476 . . . . 5  |-  NN  C_  NN
70 snssi 4118 . . . . . 6  |-  ( 1  e.  NN  ->  { 1 }  C_  NN )
7164, 70ax-mp 5 . . . . 5  |-  { 1 }  C_  NN
72 xpss12 5046 . . . . 5  |-  ( ( NN  C_  NN  /\  {
1 }  C_  NN )  ->  ( NN  X.  { 1 } ) 
C_  ( NN  X.  NN ) )
7369, 71, 72mp2an 672 . . . 4  |-  ( NN 
X.  { 1 } )  C_  ( NN  X.  NN )
74 ssdomg 7458 . . . 4  |-  ( ( NN  X.  NN )  e.  _V  ->  (
( NN  X.  {
1 } )  C_  ( NN  X.  NN )  ->  ( NN  X.  { 1 } )  ~<_  ( NN  X.  NN ) ) )
7568, 73, 74mp2 9 . . 3  |-  ( NN 
X.  { 1 } )  ~<_  ( NN  X.  NN )
76 endomtr 7470 . . 3  |-  ( ( NN  ~~  ( NN 
X.  { 1 } )  /\  ( NN 
X.  { 1 } )  ~<_  ( NN  X.  NN ) )  ->  NN  ~<_  ( NN  X.  NN ) )
7767, 75, 76mp2an 672 . 2  |-  NN  ~<_  ( NN 
X.  NN )
78 sbth 7534 . 2  |-  ( ( ( NN  X.  NN )  ~<_  NN  /\  NN  ~<_  ( NN 
X.  NN ) )  ->  ( NN  X.  NN )  ~~  NN )
7963, 77, 78mp2an 672 1  |-  ( NN 
X.  NN )  ~~  NN
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2796   _Vcvv 3071    C_ wss 3429   {csn 3978   <.cop 3984   U.cuni 4192   |^|cint 4229   class class class wbr 4393    X. cxp 4939   ran crn 4942  (class class class)co 6193    ~~ cen 7410    ~<_ cdom 7411   1c1 9387    + caddc 9389   NNcn 10426   2c2 10475   NN0cn0 10683   ^cexp 11975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-int 4230  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-2 10484  df-n0 10684  df-z 10751  df-uz 10966  df-seq 11917  df-exp 11976
This theorem is referenced by: (None)
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