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Theorem m1expaddsub 16845
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 12231 . . . . . 6  |-  ( X  e.  ZZ  ->  ( -u 1 ^ X )  e.  ZZ )
21zcnd 11008 . . . . 5  |-  ( X  e.  ZZ  ->  ( -u 1 ^ X )  e.  CC )
32adantr 463 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ X )  e.  CC )
4 m1expcl 12231 . . . . . 6  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  e.  ZZ )
54zcnd 11008 . . . . 5  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  e.  CC )
65adantl 464 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ Y )  e.  CC )
7 neg1cn 10679 . . . . . 6  |-  -u 1  e.  CC
8 neg1ne0 10681 . . . . . 6  |-  -u 1  =/=  0
9 expne0i 12240 . . . . . 6  |-  ( (
-u 1  e.  CC  /\  -u 1  =/=  0  /\  Y  e.  ZZ )  ->  ( -u 1 ^ Y )  =/=  0
)
107, 8, 9mp3an12 1316 . . . . 5  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  =/=  0 )
1110adantl 464 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ Y )  =/=  0
)
123, 6, 11divrecd 10363 . . 3  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  /  ( -u 1 ^ Y ) )  =  ( (
-u 1 ^ X
)  x.  ( 1  /  ( -u 1 ^ Y ) ) ) )
13 m1expcl2 12230 . . . . . 6  |-  ( Y  e.  ZZ  ->  ( -u 1 ^ Y )  e.  { -u 1 ,  1 } )
14 elpri 3991 . . . . . 6  |-  ( (
-u 1 ^ Y
)  e.  { -u
1 ,  1 }  ->  ( ( -u
1 ^ Y )  =  -u 1  \/  ( -u 1 ^ Y )  =  1 ) )
15 ax-1cn 9579 . . . . . . . . . 10  |-  1  e.  CC
16 ax-1ne0 9590 . . . . . . . . . 10  |-  1  =/=  0
17 divneg2 10308 . . . . . . . . . 10  |-  ( ( 1  e.  CC  /\  1  e.  CC  /\  1  =/=  0 )  ->  -u (
1  /  1 )  =  ( 1  /  -u 1 ) )
1815, 15, 16, 17mp3an 1326 . . . . . . . . 9  |-  -u (
1  /  1 )  =  ( 1  /  -u 1 )
19 1div1e1 10277 . . . . . . . . . 10  |-  ( 1  /  1 )  =  1
2019negeqi 9848 . . . . . . . . 9  |-  -u (
1  /  1 )  =  -u 1
2118, 20eqtr3i 2433 . . . . . . . 8  |-  ( 1  /  -u 1 )  = 
-u 1
22 oveq2 6285 . . . . . . . 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 2469 . . . . . . 7  |-  ( (
-u 1 ^ Y
)  =  -u 1  ->  ( 1  /  ( -u 1 ^ Y ) )  =  ( -u
1 ^ Y ) )
25 oveq2 6285 . . . . . . . 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 2469 . . . . . . 7  |-  ( (
-u 1 ^ Y
)  =  1  -> 
( 1  /  ( -u 1 ^ Y ) )  =  ( -u
1 ^ Y ) )
2824, 27jaoi 377 . . . . . 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 464 . . . 4  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( 1  /  ( -u 1 ^ Y ) )  =  ( -u
1 ^ Y ) )
3130oveq2d 6293 . . 3  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  x.  (
1  /  ( -u
1 ^ Y ) ) )  =  ( ( -u 1 ^ X )  x.  ( -u 1 ^ Y ) ) )
3212, 31eqtrd 2443 . 2  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( ( -u 1 ^ X )  /  ( -u 1 ^ Y ) )  =  ( (
-u 1 ^ X
)  x.  ( -u
1 ^ Y ) ) )
33 expsub 12256 . . 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 680 . 2  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  -  Y
) )  =  ( ( -u 1 ^ X )  /  ( -u 1 ^ Y ) ) )
35 expaddz 12252 . . 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 680 . 2  |-  ( ( X  e.  ZZ  /\  Y  e.  ZZ )  ->  ( -u 1 ^ ( X  +  Y
) )  =  ( ( -u 1 ^ X )  x.  ( -u 1 ^ Y ) ) )
3732, 34, 363eqtr4d 2453 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 366    /\ wa 367    = wceq 1405    e. wcel 1842    =/= wne 2598   {cpr 3973  (class class class)co 6277   CCcc 9519   0cc0 9521   1c1 9522    + caddc 9524    x. cmul 9526    - cmin 9840   -ucneg 9841    / cdiv 10246   ZZcz 10904   ^cexp 12208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-div 10247  df-nn 10576  df-n0 10836  df-z 10905  df-uz 11127  df-seq 12150  df-exp 12209
This theorem is referenced by:  psgnuni  16846  41prothprmlem2  37845
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