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Theorem m1expaddsub 17918
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 12883 . . . . . 6 |- ( X e. ZZ -> ( -u 1 ^ X ) e. ZZ )
21zcnd 11483 . . . . 5 |- ( X e. ZZ -> ( -u 1 ^ X ) e. CC )
32adantr 481 . . . 4 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( -u 1 ^ X ) e. CC )
4 m1expcl 12883 . . . . . 6 |- ( Y e. ZZ -> ( -u 1 ^ Y ) e. ZZ )
54zcnd 11483 . . . . 5 |- ( Y e. ZZ -> ( -u 1 ^ Y ) e. CC )
65adantl 482 . . . 4 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( -u 1 ^ Y ) e. CC )
7 neg1cn 11124 . . . . . 6 |- -u 1 e. CC
8 neg1ne0 11126 . . . . . 6 |- -u 1 =/= 0
9 expne0i 12892 . . . . . 6 |- ( ( -u 1 e. CC /\ -u 1 =/= 0 /\ Y e. ZZ ) -> ( -u 1 ^ Y ) =/= 0 )
107, 8, 9mp3an12 1414 . . . . 5 |- ( Y e. ZZ -> ( -u 1 ^ Y ) =/= 0 )
1110adantl 482 . . . 4 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( -u 1 ^ Y ) =/= 0 )
123, 6, 11divrecd 10804 . . 3 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( ( -u 1 ^ X ) / ( -u 1 ^ Y ) ) = ( ( -u 1 ^ X ) x. ( 1 / ( -u 1 ^ Y ) ) ) )
13 m1expcl2 12882 . . . . . 6 |- ( Y e. ZZ -> ( -u 1 ^ Y ) e. { -u 1 , 1 } )
14 elpri 4197 . . . . . 6 |- ( ( -u 1 ^ Y ) e. { -u 1 , 1 } -> ( ( -u 1 ^ Y ) = -u 1 \/ ( -u 1 ^ Y ) = 1 ) )
15 ax-1cn 9994 . . . . . . . . . 10 |- 1 e. CC
16 ax-1ne0 10005 . . . . . . . . . 10 |- 1 =/= 0
17 divneg2 10749 . . . . . . . . . 10 |- ( ( 1 e. CC /\ 1 e. CC /\ 1 =/= 0 ) -> -u ( 1 / 1 ) = ( 1 / -u 1 ) )
1815, 15, 16, 17mp3an 1424 . . . . . . . . 9 |- -u ( 1 / 1 ) = ( 1 / -u 1 )
19 1div1e1 10717 . . . . . . . . . 10 |- ( 1 / 1 ) = 1
2019negeqi 10274 . . . . . . . . 9 |- -u ( 1 / 1 ) = -u 1
2118, 20eqtr3i 2646 . . . . . . . 8 |- ( 1 / -u 1 ) = -u 1
22 oveq2 6658 . . . . . . . 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 2682 . . . . . . 7 |- ( ( -u 1 ^ Y ) = -u 1 -> ( 1 / ( -u 1 ^ Y ) ) = ( -u 1 ^ Y ) )
25 oveq2 6658 . . . . . . . 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 2682 . . . . . . 7 |- ( ( -u 1 ^ Y ) = 1 -> ( 1 / ( -u 1 ^ Y ) ) = ( -u 1 ^ Y ) )
2824, 27jaoi 394 . . . . . 6 |- ( ( ( -u 1 ^ Y ) = -u 1 \/ ( -u 1 ^ Y ) = 1 ) -> ( 1 / ( -u 1 ^ Y ) ) = ( -u 1 ^ Y ) )
2913, 14, 283syl 18 . . . . 5 |- ( Y e. ZZ -> ( 1 / ( -u 1 ^ Y ) ) = ( -u 1 ^ Y ) )
3029adantl 482 . . . 4 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( 1 / ( -u 1 ^ Y ) ) = ( -u 1 ^ Y ) )
3130oveq2d 6666 . . 3 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( ( -u 1 ^ X ) x. ( 1 / ( -u 1 ^ Y ) ) ) = ( ( -u 1 ^ X ) x. ( -u 1 ^ Y ) ) )
3212, 31eqtrd 2656 . 2 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( ( -u 1 ^ X ) / ( -u 1 ^ Y ) ) = ( ( -u 1 ^ X ) x. ( -u 1 ^ Y ) ) )
33 expsub 12908 . . 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 718 . 2 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( -u 1 ^ ( X - Y ) ) = ( ( -u 1 ^ X ) / ( -u 1 ^ Y ) ) )
35 expaddz 12904 . . 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 718 . 2 |- ( ( X e. ZZ /\ Y e. ZZ ) -> ( -u 1 ^ ( X + Y ) ) = ( ( -u 1 ^ X ) x. ( -u 1 ^ Y ) ) )
3732, 34, 363eqtr4d 2666 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 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   {cpr 4179  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   ZZcz 11377   ^cexp 12860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861
This theorem is referenced by:  psgnuni  17919  41prothprmlem2  41535
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