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Mirrors > Home > ILE Home > Th. List > recvalap | GIF version |
Description: Reciprocal expressed with a real denominator. (Contributed by Jim Kingdon, 13-Aug-2021.) |
Ref | Expression |
---|---|
recvalap | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) = ((∗‘𝐴) / ((abs‘𝐴)↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjcl 10828 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
2 | 1 | adantr 276 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (∗‘𝐴) ∈ ℂ) |
3 | simpl 109 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → 𝐴 ∈ ℂ) | |
4 | 2, 3 | mulcomd 7956 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((∗‘𝐴) · 𝐴) = (𝐴 · (∗‘𝐴))) |
5 | absvalsq 11033 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | |
6 | 5 | adantr 276 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) |
7 | 4, 6 | eqtr4d 2213 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((∗‘𝐴) · 𝐴) = ((abs‘𝐴)↑2)) |
8 | abscl 11031 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
9 | 8 | adantr 276 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) ∈ ℝ) |
10 | 9 | recnd 7963 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) ∈ ℂ) |
11 | 10 | sqcld 10624 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((abs‘𝐴)↑2) ∈ ℂ) |
12 | cjap0 10887 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 # 0 ↔ (∗‘𝐴) # 0)) | |
13 | 12 | biimpa 296 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (∗‘𝐴) # 0) |
14 | 11, 2, 3, 13 | divmulapd 8745 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((((abs‘𝐴)↑2) / (∗‘𝐴)) = 𝐴 ↔ ((∗‘𝐴) · 𝐴) = ((abs‘𝐴)↑2))) |
15 | 7, 14 | mpbird 167 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (((abs‘𝐴)↑2) / (∗‘𝐴)) = 𝐴) |
16 | 15 | oveq2d 5884 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (((abs‘𝐴)↑2) / (∗‘𝐴))) = (1 / 𝐴)) |
17 | abs00ap 11042 | . . . . 5 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) # 0 ↔ 𝐴 # 0)) | |
18 | 17 | biimpar 297 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (abs‘𝐴) # 0) |
19 | sqap0 10559 | . . . . 5 ⊢ ((abs‘𝐴) ∈ ℂ → (((abs‘𝐴)↑2) # 0 ↔ (abs‘𝐴) # 0)) | |
20 | 10, 19 | syl 14 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (((abs‘𝐴)↑2) # 0 ↔ (abs‘𝐴) # 0)) |
21 | 18, 20 | mpbird 167 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → ((abs‘𝐴)↑2) # 0) |
22 | 11, 2, 21, 13 | recdivapd 8740 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / (((abs‘𝐴)↑2) / (∗‘𝐴))) = ((∗‘𝐴) / ((abs‘𝐴)↑2))) |
23 | 16, 22 | eqtr3d 2212 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 # 0) → (1 / 𝐴) = ((∗‘𝐴) / ((abs‘𝐴)↑2))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 ℂcc 7787 ℝcr 7788 0cc0 7789 1c1 7790 · cmul 7794 # cap 8515 / cdiv 8605 2c2 8946 ↑cexp 10492 ∗ccj 10819 abscabs 10977 |
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-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-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-recs 6299 df-frec 6385 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-rp 9628 df-seqfrec 10419 df-exp 10493 df-cj 10822 df-re 10823 df-im 10824 df-rsqrt 10978 df-abs 10979 |
This theorem is referenced by: (None) |
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