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Mirrors > Home > ILE Home > Th. List > efsub | GIF version |
Description: Difference of exponents law for exponential function. (Contributed by Steve Rodriguez, 25-Nov-2007.) |
Ref | Expression |
---|---|
efsub | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 − 𝐵)) = ((exp‘𝐴) / (exp‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | efcl 11643 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) ∈ ℂ) | |
2 | 1 | 3ad2ant1 1018 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘𝐴) ∈ ℂ) |
3 | efcl 11643 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (exp‘𝐵) ∈ ℂ) | |
4 | 3 | 3ad2ant2 1019 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘𝐵) ∈ ℂ) |
5 | efap0 11656 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (exp‘𝐵) # 0) | |
6 | 5 | 3ad2ant2 1019 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘𝐵) # 0) |
7 | 2, 4, 6 | divrecapd 8726 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((exp‘𝐴) / (exp‘𝐵)) = ((exp‘𝐴) · (1 / (exp‘𝐵)))) |
8 | 7 | 3anidm23 1297 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((exp‘𝐴) / (exp‘𝐵)) = ((exp‘𝐴) · (1 / (exp‘𝐵)))) |
9 | efcan 11655 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((exp‘𝐵) · (exp‘-𝐵)) = 1) | |
10 | 9 | eqcomd 2183 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → 1 = ((exp‘𝐵) · (exp‘-𝐵))) |
11 | 1cnd 7951 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → 1 ∈ ℂ) | |
12 | negcl 8134 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
13 | efcl 11643 | . . . . . . . 8 ⊢ (-𝐵 ∈ ℂ → (exp‘-𝐵) ∈ ℂ) | |
14 | 12, 13 | syl 14 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (exp‘-𝐵) ∈ ℂ) |
15 | 11, 14, 3, 5 | divmulap2d 8757 | . . . . . 6 ⊢ (𝐵 ∈ ℂ → ((1 / (exp‘𝐵)) = (exp‘-𝐵) ↔ 1 = ((exp‘𝐵) · (exp‘-𝐵)))) |
16 | 10, 15 | mpbird 167 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (1 / (exp‘𝐵)) = (exp‘-𝐵)) |
17 | 16 | oveq2d 5884 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((exp‘𝐴) · (1 / (exp‘𝐵))) = ((exp‘𝐴) · (exp‘-𝐵))) |
18 | 17 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((exp‘𝐴) · (1 / (exp‘𝐵))) = ((exp‘𝐴) · (exp‘-𝐵))) |
19 | efadd 11654 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ -𝐵 ∈ ℂ) → (exp‘(𝐴 + -𝐵)) = ((exp‘𝐴) · (exp‘-𝐵))) | |
20 | 12, 19 | sylan2 286 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 + -𝐵)) = ((exp‘𝐴) · (exp‘-𝐵))) |
21 | 18, 20 | eqtr4d 2213 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((exp‘𝐴) · (1 / (exp‘𝐵))) = (exp‘(𝐴 + -𝐵))) |
22 | negsub 8182 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + -𝐵) = (𝐴 − 𝐵)) | |
23 | 22 | fveq2d 5514 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 + -𝐵)) = (exp‘(𝐴 − 𝐵))) |
24 | 8, 21, 23 | 3eqtrrd 2215 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (exp‘(𝐴 − 𝐵)) = ((exp‘𝐴) / (exp‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 ℂcc 7787 0cc0 7789 1c1 7790 + caddc 7792 · cmul 7794 − cmin 8105 -cneg 8106 # cap 8515 / cdiv 8605 expce 11621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 ax-arch 7908 ax-caucvg 7909 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-disj 3978 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-isom 5220 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-irdg 6364 df-frec 6385 df-1o 6410 df-oadd 6414 df-er 6528 df-en 6734 df-dom 6735 df-fin 6736 df-sup 6976 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-n0 9153 df-z 9230 df-uz 9505 df-q 9596 df-rp 9628 df-ico 9868 df-fz 9983 df-fzo 10116 df-seqfrec 10419 df-exp 10493 df-fac 10677 df-bc 10699 df-ihash 10727 df-cj 10822 df-re 10823 df-im 10824 df-rsqrt 10978 df-abs 10979 df-clim 11258 df-sumdc 11333 df-ef 11627 |
This theorem is referenced by: reeff1oleme 13826 relogdiv 13924 |
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