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Mirrors > Home > MPE Home > Th. List > m1expe | Structured version Visualization version GIF version |
Description: Exponentiation of -1 by an even power. Variant of m1expeven 13943. (Contributed by AV, 25-Jun-2021.) |
Ref | Expression |
---|---|
m1expe | โข (2 โฅ ๐ โ (-1โ๐) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | even2n 16158 | . 2 โข (2 โฅ ๐ โ โ๐ โ โค (2 ยท ๐) = ๐) | |
2 | oveq2 7357 | . . . . 5 โข (๐ = (2 ยท ๐) โ (-1โ๐) = (-1โ(2 ยท ๐))) | |
3 | 2 | eqcoms 2745 | . . . 4 โข ((2 ยท ๐) = ๐ โ (-1โ๐) = (-1โ(2 ยท ๐))) |
4 | m1expeven 13943 | . . . 4 โข (๐ โ โค โ (-1โ(2 ยท ๐)) = 1) | |
5 | 3, 4 | sylan9eqr 2799 | . . 3 โข ((๐ โ โค โง (2 ยท ๐) = ๐) โ (-1โ๐) = 1) |
6 | 5 | rexlimiva 3142 | . 2 โข (โ๐ โ โค (2 ยท ๐) = ๐ โ (-1โ๐) = 1) |
7 | 1, 6 | sylbi 216 | 1 โข (2 โฅ ๐ โ (-1โ๐) = 1) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 โwrex 3071 class class class wbr 5103 (class class class)co 7349 1c1 10985 ยท cmul 10989 -cneg 11319 2c2 12141 โคcz 12432 โcexp 13895 โฅ cdvds 16070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7662 ax-cnex 11040 ax-resscn 11041 ax-1cn 11042 ax-icn 11043 ax-addcl 11044 ax-addrcl 11045 ax-mulcl 11046 ax-mulrcl 11047 ax-mulcom 11048 ax-addass 11049 ax-mulass 11050 ax-distr 11051 ax-i2m1 11052 ax-1ne0 11053 ax-1rid 11054 ax-rnegex 11055 ax-rrecex 11056 ax-cnre 11057 ax-pre-lttri 11058 ax-pre-lttrn 11059 ax-pre-ltadd 11060 ax-pre-mulgt0 11061 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5528 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5585 df-we 5587 df-xp 5636 df-rel 5637 df-cnv 5638 df-co 5639 df-dm 5640 df-rn 5641 df-res 5642 df-ima 5643 df-pred 6249 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6443 df-fun 6493 df-fn 6494 df-f 6495 df-f1 6496 df-fo 6497 df-f1o 6498 df-fv 6499 df-riota 7305 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7793 df-2nd 7912 df-frecs 8179 df-wrecs 8210 df-recs 8284 df-rdg 8323 df-er 8581 df-en 8817 df-dom 8818 df-sdom 8819 df-pnf 11124 df-mnf 11125 df-xr 11126 df-ltxr 11127 df-le 11128 df-sub 11320 df-neg 11321 df-div 11746 df-nn 12087 df-2 12149 df-n0 12347 df-z 12433 df-uz 12696 df-seq 13835 df-exp 13896 df-dvds 16071 |
This theorem is referenced by: oddpwp1fsum 16208 |
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