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Theorem expclzlem 12172
Description: Closure law for integer exponentiation. (Contributed by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expclzlem  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  ( CC  \  {
0 } ) )

Proof of Theorem expclzlem
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4141 . . . 4  |-  ( A  e.  ( CC  \  { 0 } )  <-> 
( A  e.  CC  /\  A  =/=  0 ) )
2 difss 3617 . . . . . 6  |-  ( CC 
\  { 0 } )  C_  CC
3 eldifsn 4141 . . . . . . 7  |-  ( x  e.  ( CC  \  { 0 } )  <-> 
( x  e.  CC  /\  x  =/=  0 ) )
4 eldifsn 4141 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  <-> 
( y  e.  CC  /\  y  =/=  0 ) )
5 mulcl 9565 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  y  e.  CC )  ->  ( x  x.  y
)  e.  CC )
65ad2ant2r 744 . . . . . . . 8  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  CC )
7 mulne0 10187 . . . . . . . 8  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  =/=  0 )
8 eldifsn 4141 . . . . . . . 8  |-  ( ( x  x.  y )  e.  ( CC  \  { 0 } )  <-> 
( ( x  x.  y )  e.  CC  /\  ( x  x.  y
)  =/=  0 ) )
96, 7, 8sylanbrc 662 . . . . . . 7  |-  ( ( ( x  e.  CC  /\  x  =/=  0 )  /\  ( y  e.  CC  /\  y  =/=  0 ) )  -> 
( x  x.  y
)  e.  ( CC 
\  { 0 } ) )
103, 4, 9syl2anb 477 . . . . . 6  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( x  x.  y )  e.  ( CC  \  { 0 } ) )
11 ax-1cn 9539 . . . . . . 7  |-  1  e.  CC
12 ax-1ne0 9550 . . . . . . 7  |-  1  =/=  0
13 eldifsn 4141 . . . . . . 7  |-  ( 1  e.  ( CC  \  { 0 } )  <-> 
( 1  e.  CC  /\  1  =/=  0 ) )
1411, 12, 13mpbir2an 918 . . . . . 6  |-  1  e.  ( CC  \  {
0 } )
15 reccl 10210 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 1  /  x
)  e.  CC )
16 recne0 10216 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( 1  /  x
)  =/=  0 )
1715, 16jca 530 . . . . . . . 8  |-  ( ( x  e.  CC  /\  x  =/=  0 )  -> 
( ( 1  /  x )  e.  CC  /\  ( 1  /  x
)  =/=  0 ) )
18 eldifsn 4141 . . . . . . . 8  |-  ( ( 1  /  x )  e.  ( CC  \  { 0 } )  <-> 
( ( 1  /  x )  e.  CC  /\  ( 1  /  x
)  =/=  0 ) )
1917, 3, 183imtr4i 266 . . . . . . 7  |-  ( x  e.  ( CC  \  { 0 } )  ->  ( 1  /  x )  e.  ( CC  \  { 0 } ) )
2019adantr 463 . . . . . 6  |-  ( ( x  e.  ( CC 
\  { 0 } )  /\  x  =/=  0 )  ->  (
1  /  x )  e.  ( CC  \  { 0 } ) )
212, 10, 14, 20expcl2lem 12160 . . . . 5  |-  ( ( A  e.  ( CC 
\  { 0 } )  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  ( CC  \  {
0 } ) )
22213expia 1196 . . . 4  |-  ( ( A  e.  ( CC 
\  { 0 } )  /\  A  =/=  0 )  ->  ( N  e.  ZZ  ->  ( A ^ N )  e.  ( CC  \  { 0 } ) ) )
231, 22sylanbr 471 . . 3  |-  ( ( ( A  e.  CC  /\  A  =/=  0 )  /\  A  =/=  0
)  ->  ( N  e.  ZZ  ->  ( A ^ N )  e.  ( CC  \  { 0 } ) ) )
2423anabss3 821 . 2  |-  ( ( A  e.  CC  /\  A  =/=  0 )  -> 
( N  e.  ZZ  ->  ( A ^ N
)  e.  ( CC 
\  { 0 } ) ) )
25243impia 1191 1  |-  ( ( A  e.  CC  /\  A  =/=  0  /\  N  e.  ZZ )  ->  ( A ^ N )  e.  ( CC  \  {
0 } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    e. wcel 1823    =/= wne 2649    \ cdif 3458   {csn 4016  (class class class)co 6270   CCcc 9479   0cc0 9481   1c1 9482    x. cmul 9486    / cdiv 10202   ZZcz 10860   ^cexp 12148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-seq 12090  df-exp 12149
This theorem is referenced by:  expclz  12173  expne0i  12180  expghm  18708
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