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Mirrors > Home > MPE Home > Th. List > efcan | Structured version Visualization version GIF version |
Description: Cancellation law for exponential function. Equation 27 of [Rudin] p. 164. (Contributed by NM, 13-Jan-2006.) |
Ref | Expression |
---|---|
efcan | โข (๐ด โ โ โ ((expโ๐ด) ยท (expโ-๐ด)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 11490 | . . 3 โข (๐ด โ โ โ -๐ด โ โ) | |
2 | efadd 16070 | . . 3 โข ((๐ด โ โ โง -๐ด โ โ) โ (expโ(๐ด + -๐ด)) = ((expโ๐ด) ยท (expโ-๐ด))) | |
3 | 1, 2 | mpdan 686 | . 2 โข (๐ด โ โ โ (expโ(๐ด + -๐ด)) = ((expโ๐ด) ยท (expโ-๐ด))) |
4 | negid 11537 | . . . 4 โข (๐ด โ โ โ (๐ด + -๐ด) = 0) | |
5 | 4 | fveq2d 6901 | . . 3 โข (๐ด โ โ โ (expโ(๐ด + -๐ด)) = (expโ0)) |
6 | ef0 16067 | . . 3 โข (expโ0) = 1 | |
7 | 5, 6 | eqtrdi 2784 | . 2 โข (๐ด โ โ โ (expโ(๐ด + -๐ด)) = 1) |
8 | 3, 7 | eqtr3d 2770 | 1 โข (๐ด โ โ โ ((expโ๐ด) ยท (expโ-๐ด)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 โcfv 6548 (class class class)co 7420 โcc 11136 0cc0 11138 1c1 11139 + caddc 11141 ยท cmul 11143 -cneg 11475 expce 16037 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-er 8724 df-pm 8847 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-ico 13362 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 |
This theorem is referenced by: efne0 16073 efneg 16074 efsub 16076 tanval3 16110 reeff1o 26383 efne0d 41908 |
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