MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  m1expaddsub Structured version   Unicode version

Theorem m1expaddsub 16124
Description: Addition and subtraction of parities are the same. (Contributed by Stefan O'Rear, 27-Aug-2015.)
Assertion
Ref Expression
m1expaddsub  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  -  Y
) )  =  (
-u 1 ^ ( X  +  Y )
) )

Proof of Theorem m1expaddsub
StepHypRef Expression
1 m1expcl 12006 . . . . . 6  |-  ( X  e.  ZZ  ->  ( -u 1 ^ X )  e.  ZZ )
21zcnd 10860 . . . . 5  |-  ( X  e.  ZZ  ->  ( -u 1 ^ X )  e.  CC )
32adantr 465 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ X )  e.  CC )
4 m1expcl 12006 . . . . . 6  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  e.  ZZ )
54zcnd 10860 . . . . 5  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  e.  CC )
65adantl 466 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ Y )  e.  CC )
7 neg1cn 10537 . . . . . 6  |-  -u 1  e.  CC
8 neg1ne0 10539 . . . . . 6  |-  -u 1  =/=  0
9 expne0i 12014 . . . . . 6  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  Y  e.  ZZ )  ->  ( -u 1 ^ Y )  =/=  0
)
107, 8, 9mp3an12 1305 . . . . 5  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  =/=  0 )
1110adantl 466 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ Y )  =/=  0
)
123, 6, 11divrecd 10222 . . 3  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  /  ( -u 1 ^ Y ) )  =  ( (
-u 1 ^ X
)  x.  ( 1  /  ( -u 1 ^ Y ) ) ) )
13 m1expcl2 12005 . . . . . 6  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  e.  { -u 1 ,  1 } )
14 elpri 4006 . . . . . 6  |-  ( (
-u 1 ^ Y
)  e.  { -u
1 ,  1 }  ->  ( ( -u
1 ^ Y )  =  -u 1  \/  ( -u 1 ^ Y )  =  1 ) )
15 ax-1cn 9452 . . . . . . . . . 10  |-  1  e.  CC
16 ax-1ne0 9463 . . . . . . . . . 10  |-  1  =/=  0
17 divneg2 10167 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
1815, 15, 16, 17mp3an 1315 . . . . . . . . 9  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
19 1div1e1 10136 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
2019negeqi 9715 . . . . . . . . 9  |-  -u (
1  /  1 )  =  -u 1
2118, 20eqtr3i 2485 . . . . . . . 8  |-  ( 1  /  -u 1 )  = 
-u 1
22 oveq2 6209 . . . . . . . 8  |-  ( (
-u 1 ^ Y
)  =  -u 1  ->  ( 1  /  ( -u 1 ^ Y ) )  =  ( 1  /  -u 1 ) )
23 id 22 . . . . . . . 8  |-  ( (
-u 1 ^ Y
)  =  -u 1  ->  ( -u 1 ^ Y )  =  -u
1 )
2421, 22, 233eqtr4a 2521 . . . . . . 7  |-  ( (
-u 1 ^ Y
)  =  -u 1  ->  ( 1  /  ( -u 1 ^ Y ) )  =  ( -u
1 ^ Y ) )
25 oveq2 6209 . . . . . . . 8  |-  ( (
-u 1 ^ Y
)  =  1  -> 
( 1  /  ( -u 1 ^ Y ) )  =  ( 1  /  1 ) )
26 id 22 . . . . . . . 8  |-  ( (
-u 1 ^ Y
)  =  1  -> 
( -u 1 ^ Y
)  =  1 )
2719, 25, 263eqtr4a 2521 . . . . . . 7  |-  ( (
-u 1 ^ Y
)  =  1  -> 
( 1  /  ( -u 1 ^ Y ) )  =  ( -u
1 ^ Y ) )
2824, 27jaoi 379 . . . . . 6  |-  ( ( ( -u 1 ^ Y )  =  -u
1  \/  ( -u
1 ^ Y )  =  1 )  -> 
( 1  /  ( -u 1 ^ Y ) )  =  ( -u
1 ^ Y ) )
2913, 14, 283syl 20 . . . . 5  |-  ( Y  e.  ZZ  ->  (
1  /  ( -u
1 ^ Y ) )  =  ( -u
1 ^ Y ) )
3029adantl 466 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( 1  /  ( -u 1 ^ Y ) )  =  ( -u
1 ^ Y ) )
3130oveq2d 6217 . . 3  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  x.  (
1  /  ( -u
1 ^ Y ) ) )  =  ( ( -u 1 ^ X )  x.  ( -u 1 ^ Y ) ) )
3212, 31eqtrd 2495 . 2  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  /  ( -u 1 ^ Y ) )  =  ( (
-u 1 ^ X
)  x.  ( -u
1 ^ Y ) ) )
33 expsub 12029 . . 3  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( -u 1 ^ ( X  -  Y )
)  =  ( (
-u 1 ^ X
)  /  ( -u
1 ^ Y ) ) )
347, 8, 33mpanl12 682 . 2  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  -  Y
) )  =  ( ( -u 1 ^ X )  /  ( -u 1 ^ Y ) ) )
35 expaddz 12026 . . 3  |-  ( ( ( -u 1  e.  CC  /\  -u 1  =/=  0 )  /\  ( X  e.  ZZ  /\  Y  e.  ZZ ) )  -> 
( -u 1 ^ ( X  +  Y )
)  =  ( (
-u 1 ^ X
)  x.  ( -u
1 ^ Y ) ) )
367, 8, 35mpanl12 682 . 2  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  +  Y
) )  =  ( ( -u 1 ^ X )  x.  ( -u 1 ^ Y ) ) )
3732, 34, 363eqtr4d 2505 1  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  -  Y
) )  =  (
-u 1 ^ ( X  +  Y )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2648   {cpr 3988  (class class class)co 6201   CCcc 9392   0cc0 9394   1c1 9395    + caddc 9397    x. cmul 9399    - cmin 9707   -ucneg 9708    / cdiv 10105   ZZcz 10758   ^cexp 11983
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-om 6588  df-2nd 6689  df-recs 6943  df-rdg 6977  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-n0 10692  df-z 10759  df-uz 10974  df-seq 11925  df-exp 11984
This theorem is referenced by:  psgnuni  16125
  Copyright terms: Public domain W3C validator