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Theorem xpnnenOLD 13955
Description: TODO-NM: Should this theorem be kept (and renamed, maybe with suffix ALT) or deleted? 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 12355 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 10562 . . 3  |-  NN  e.  _V
2 elxp5 6744 . . . . 5  |-  ( x  e.  ( NN  X.  NN )  <->  ( x  = 
<. |^| |^| x ,  U. ran  { x } >.  /\  ( |^| |^| x  e.  NN  /\  U. ran  { x }  e.  NN ) ) )
3 nnaddcl 10578 . . . . . . . 8  |-  ( (
|^| |^| x  e.  NN  /\ 
U. ran  { x }  e.  NN )  ->  ( |^| |^| x  +  U. ran  { x } )  e.  NN )
4 2nn0 10833 . . . . . . . 8  |-  2  e.  NN0
5 nnexpcl 12182 . . . . . . . 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 10578 . . . . . . 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 5026 . . . . 5  |-  ( x  e.  ( NN  X.  NN )  <->  E. z  e.  NN  E. w  e.  NN  x  =  <. z ,  w >. )
12 elxp2 5026 . . . . 5  |-  ( y  e.  ( NN  X.  NN )  <->  E. v  e.  NN  E. u  e.  NN  y  =  <. v ,  u >. )
13 inteq 4291 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. z ,  w >.  ->  |^| x  =  |^| <.
z ,  w >. )
1413inteqd 4293 . . . . . . . . . . . . . . . . 17  |-  ( x  =  <. z ,  w >.  ->  |^| |^| x  =  |^| |^|
<. z ,  w >. )
15 vex 3112 . . . . . . . . . . . . . . . . . 18  |-  z  e. 
_V
16 vex 3112 . . . . . . . . . . . . . . . . . 18  |-  w  e. 
_V
1715, 16op1stb 4726 . . . . . . . . . . . . . . . . 17  |-  |^| |^| <. z ,  w >.  =  z
1814, 17syl6eq 2514 . . . . . . . . . . . . . . . 16  |-  ( x  =  <. z ,  w >.  ->  |^| |^| x  =  z )
19 sneq 4042 . . . . . . . . . . . . . . . . . . 19  |-  ( x  =  <. z ,  w >.  ->  { x }  =  { <. z ,  w >. } )
2019rneqd 5240 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  <. z ,  w >.  ->  ran  { x }  =  ran  { <. z ,  w >. } )
2120unieqd 4261 . . . . . . . . . . . . . . . . 17  |-  ( x  =  <. z ,  w >.  ->  U. ran  { x }  =  U. ran  { <. z ,  w >. } )
2215, 16op2nda 5499 . . . . . . . . . . . . . . . . 17  |-  U. ran  {
<. z ,  w >. }  =  w
2321, 22syl6eq 2514 . . . . . . . . . . . . . . . 16  |-  ( x  =  <. z ,  w >.  ->  U. ran  { x }  =  w )
2418, 23oveq12d 6314 . . . . . . . . . . . . . . 15  |-  ( x  =  <. z ,  w >.  ->  ( |^| |^| x  +  U. ran  { x } )  =  ( z  +  w ) )
2524oveq1d 6311 . . . . . . . . . . . . . 14  |-  ( x  =  <. z ,  w >.  ->  ( ( |^| |^| x  +  U. ran  { x } ) ^
2 )  =  ( ( z  +  w
) ^ 2 ) )
2625, 23oveq12d 6314 . . . . . . . . . . . . 13  |-  ( x  =  <. z ,  w >.  ->  ( ( (
|^| |^| x  +  U. ran  { x } ) ^ 2 )  + 
U. ran  { x } )  =  ( ( ( z  +  w ) ^ 2 )  +  w ) )
27 inteq 4291 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  <. v ,  u >.  ->  |^| y  =  |^| <.
v ,  u >. )
2827inteqd 4293 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. v ,  u >.  ->  |^| |^| y  =  |^| |^|
<. v ,  u >. )
29 vex 3112 . . . . . . . . . . . . . . . . . 18  |-  v  e. 
_V
30 vex 3112 . . . . . . . . . . . . . . . . . 18  |-  u  e. 
_V
3129, 30op1stb 4726 . . . . . . . . . . . . . . . . 17  |-  |^| |^| <. v ,  u >.  =  v
3228, 31syl6eq 2514 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. v ,  u >.  ->  |^| |^| y  =  v )
33 sneq 4042 . . . . . . . . . . . . . . . . . . 19  |-  ( y  =  <. v ,  u >.  ->  { y }  =  { <. v ,  u >. } )
3433rneqd 5240 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  <. v ,  u >.  ->  ran  { y }  =  ran  { <. v ,  u >. } )
3534unieqd 4261 . . . . . . . . . . . . . . . . 17  |-  ( y  =  <. v ,  u >.  ->  U. ran  { y }  =  U. ran  {
<. v ,  u >. } )
3629, 30op2nda 5499 . . . . . . . . . . . . . . . . 17  |-  U. ran  {
<. v ,  u >. }  =  u
3735, 36syl6eq 2514 . . . . . . . . . . . . . . . 16  |-  ( y  =  <. v ,  u >.  ->  U. ran  { y }  =  u )
3832, 37oveq12d 6314 . . . . . . . . . . . . . . 15  |-  ( y  =  <. v ,  u >.  ->  ( |^| |^| y  +  U. ran  { y } )  =  ( v  +  u ) )
3938oveq1d 6311 . . . . . . . . . . . . . 14  |-  ( y  =  <. v ,  u >.  ->  ( ( |^| |^| y  +  U. ran  { y } ) ^
2 )  =  ( ( v  +  u
) ^ 2 ) )
4039, 37oveq12d 6314 . . . . . . . . . . . . 13  |-  ( y  =  <. v ,  u >.  ->  ( ( (
|^| |^| y  +  U. ran  { y } ) ^ 2 )  + 
U. ran  { y } )  =  ( ( ( v  +  u ) ^ 2 )  +  u ) )
4126, 40eqeqan12d 2480 . . . . . . . . . . . 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 10823 . . . . . . . . . . . . . 14  |-  ( z  e.  NN  ->  z  e.  NN0 )
43 nnnn0 10823 . . . . . . . . . . . . . 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 10823 . . . . . . . . . . . . . 14  |-  ( v  e.  NN  ->  v  e.  NN0 )
46 nnnn0 10823 . . . . . . . . . . . . . 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 12355 . . . . . . . . . . . . 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 2476 . . . . . . . . . . . . 13  |-  ( ( x  =  <. z ,  w >.  /\  y  =  <. v ,  u >. )  ->  ( x  =  y  <->  <. z ,  w >.  =  <. v ,  u >. ) )
5215, 16opth 4730 . . . . . . . . . . . . 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 2954 . . . . . . 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 2955 . . . . . 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 7577 . . 3  |-  ( NN  e.  _V  ->  ( NN  X.  NN )  ~<_  NN )
631, 62ax-mp 5 . 2  |-  ( NN 
X.  NN )  ~<_  NN
64 1nn 10567 . . . . . 6  |-  1  e.  NN
6564elexi 3119 . . . . 5  |-  1  e.  _V
661, 65xpsnen 7620 . . . 4  |-  ( NN 
X.  { 1 } )  ~~  NN
6766ensymi 7584 . . 3  |-  NN  ~~  ( NN  X.  { 1 } )
681, 1xpex 6603 . . . 4  |-  ( NN 
X.  NN )  e. 
_V
69 ssid 3518 . . . . 5  |-  NN  C_  NN
70 snssi 4176 . . . . . 6  |-  ( 1  e.  NN  ->  { 1 }  C_  NN )
7164, 70ax-mp 5 . . . . 5  |-  { 1 }  C_  NN
72 xpss12 5117 . . . . 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 7580 . . . 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 7592 . . 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 7656 . 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 1395    e. wcel 1819   E.wrex 2808   _Vcvv 3109    C_ wss 3471   {csn 4032   <.cop 4038   U.cuni 4251   |^|cint 4288   class class class wbr 4456    X. cxp 5006   ran crn 5009  (class class class)co 6296    ~~ cen 7532    ~<_ cdom 7533   1c1 9510    + caddc 9512   NNcn 10556   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-rep 4568  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-int 4289  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: (None)
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