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Theorem m1expaddsub 16316
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 12152 . . . . . 6  |-  ( X  e.  ZZ  ->  ( -u 1 ^ X )  e.  ZZ )
21zcnd 10963 . . . . 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 12152 . . . . . 6  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  e.  ZZ )
54zcnd 10963 . . . . 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 10635 . . . . . 6  |-  -u 1  e.  CC
8 neg1ne0 10637 . . . . . 6  |-  -u 1  =/=  0
9 expne0i 12160 . . . . . 6  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  Y  e.  ZZ )  ->  ( -u 1 ^ Y )  =/=  0
)
107, 8, 9mp3an12 1314 . . . . 5  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  =/=  0 )
1110adantl 466 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ Y )  =/=  0
)
123, 6, 11divrecd 10319 . . 3  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  /  ( -u 1 ^ Y ) )  =  ( (
-u 1 ^ X
)  x.  ( 1  /  ( -u 1 ^ Y ) ) ) )
13 m1expcl2 12151 . . . . . 6  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  e.  { -u 1 ,  1 } )
14 elpri 4047 . . . . . 6  |-  ( (
-u 1 ^ Y
)  e.  { -u
1 ,  1 }  ->  ( ( -u
1 ^ Y )  =  -u 1  \/  ( -u 1 ^ Y )  =  1 ) )
15 ax-1cn 9546 . . . . . . . . . 10  |-  1  e.  CC
16 ax-1ne0 9557 . . . . . . . . . 10  |-  1  =/=  0
17 divneg2 10264 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
1815, 15, 16, 17mp3an 1324 . . . . . . . . 9  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
19 1div1e1 10233 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
2019negeqi 9809 . . . . . . . . 9  |-  -u (
1  /  1 )  =  -u 1
2118, 20eqtr3i 2498 . . . . . . . 8  |-  ( 1  /  -u 1 )  = 
-u 1
22 oveq2 6290 . . . . . . . 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 2534 . . . . . . 7  |-  ( (
-u 1 ^ Y
)  =  -u 1  ->  ( 1  /  ( -u 1 ^ Y ) )  =  ( -u
1 ^ Y ) )
25 oveq2 6290 . . . . . . . 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 2534 . . . . . . 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 6298 . . 3  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  x.  (
1  /  ( -u
1 ^ Y ) ) )  =  ( ( -u 1 ^ X )  x.  ( -u 1 ^ Y ) ) )
3212, 31eqtrd 2508 . 2  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  /  ( -u 1 ^ Y ) )  =  ( (
-u 1 ^ X
)  x.  ( -u
1 ^ Y ) ) )
33 expsub 12175 . . 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 12172 . . 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 2518 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 1379    e. wcel 1767    =/= wne 2662   {cpr 4029  (class class class)co 6282   CCcc 9486   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    - cmin 9801   -ucneg 9802    / cdiv 10202   ZZcz 10860   ^cexp 12129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-seq 12071  df-exp 12130
This theorem is referenced by:  psgnuni  16317
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