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Mirrors > Home > MPE Home > Th. List > dvmptim | Structured version Visualization version GIF version |
Description: Function-builder for derivative, imaginary part. (Contributed by Mario Carneiro, 1-Sep-2014.) |
Ref | Expression |
---|---|
dvmptcj.a | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
dvmptcj.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) |
dvmptcj.da | ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) |
Ref | Expression |
---|---|
dvmptim | ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reelprrecn 11250 | . . . 4 ⊢ ℝ ∈ {ℝ, ℂ} | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
3 | dvmptcj.a | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) | |
4 | 3 | cjcld 15201 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∗‘𝐴) ∈ ℂ) |
5 | 3, 4 | subcld 11621 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐴 − (∗‘𝐴)) ∈ ℂ) |
6 | dvmptcj.b | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) | |
7 | dvmptcj.da | . . . . 5 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) | |
8 | 2, 3, 6, 7 | dvmptcl 25982 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
9 | 8 | cjcld 15201 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∗‘𝐵) ∈ ℂ) |
10 | 8, 9 | subcld 11621 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝐵 − (∗‘𝐵)) ∈ ℂ) |
11 | 3, 6, 7 | dvmptcj 25991 | . . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵))) |
12 | 2, 3, 6, 7, 4, 9, 11 | dvmptsub 25990 | . . 3 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (𝐴 − (∗‘𝐴)))) = (𝑥 ∈ 𝑋 ↦ (𝐵 − (∗‘𝐵)))) |
13 | 2mulicn 12487 | . . . . 5 ⊢ (2 · i) ∈ ℂ | |
14 | 2muline0 12488 | . . . . 5 ⊢ (2 · i) ≠ 0 | |
15 | 13, 14 | reccli 11995 | . . . 4 ⊢ (1 / (2 · i)) ∈ ℂ |
16 | 15 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / (2 · i)) ∈ ℂ) |
17 | 2, 5, 10, 12, 16 | dvmptcmul 25987 | . 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ ((1 / (2 · i)) · (𝐴 − (∗‘𝐴))))) = (𝑥 ∈ 𝑋 ↦ ((1 / (2 · i)) · (𝐵 − (∗‘𝐵))))) |
18 | imval2 15156 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (ℑ‘𝐴) = ((𝐴 − (∗‘𝐴)) / (2 · i))) | |
19 | 3, 18 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℑ‘𝐴) = ((𝐴 − (∗‘𝐴)) / (2 · i))) |
20 | divrec2 11940 | . . . . . . 7 ⊢ (((𝐴 − (∗‘𝐴)) ∈ ℂ ∧ (2 · i) ∈ ℂ ∧ (2 · i) ≠ 0) → ((𝐴 − (∗‘𝐴)) / (2 · i)) = ((1 / (2 · i)) · (𝐴 − (∗‘𝐴)))) | |
21 | 13, 14, 20 | mp3an23 1450 | . . . . . 6 ⊢ ((𝐴 − (∗‘𝐴)) ∈ ℂ → ((𝐴 − (∗‘𝐴)) / (2 · i)) = ((1 / (2 · i)) · (𝐴 − (∗‘𝐴)))) |
22 | 5, 21 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐴 − (∗‘𝐴)) / (2 · i)) = ((1 / (2 · i)) · (𝐴 − (∗‘𝐴)))) |
23 | 19, 22 | eqtrd 2766 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℑ‘𝐴) = ((1 / (2 · i)) · (𝐴 − (∗‘𝐴)))) |
24 | 23 | mpteq2dva 5253 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐴)) = (𝑥 ∈ 𝑋 ↦ ((1 / (2 · i)) · (𝐴 − (∗‘𝐴))))) |
25 | 24 | oveq2d 7440 | . 2 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐴))) = (ℝ D (𝑥 ∈ 𝑋 ↦ ((1 / (2 · i)) · (𝐴 − (∗‘𝐴)))))) |
26 | imval2 15156 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (ℑ‘𝐵) = ((𝐵 − (∗‘𝐵)) / (2 · i))) | |
27 | 8, 26 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℑ‘𝐵) = ((𝐵 − (∗‘𝐵)) / (2 · i))) |
28 | divrec2 11940 | . . . . . 6 ⊢ (((𝐵 − (∗‘𝐵)) ∈ ℂ ∧ (2 · i) ∈ ℂ ∧ (2 · i) ≠ 0) → ((𝐵 − (∗‘𝐵)) / (2 · i)) = ((1 / (2 · i)) · (𝐵 − (∗‘𝐵)))) | |
29 | 13, 14, 28 | mp3an23 1450 | . . . . 5 ⊢ ((𝐵 − (∗‘𝐵)) ∈ ℂ → ((𝐵 − (∗‘𝐵)) / (2 · i)) = ((1 / (2 · i)) · (𝐵 − (∗‘𝐵)))) |
30 | 10, 29 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐵 − (∗‘𝐵)) / (2 · i)) = ((1 / (2 · i)) · (𝐵 − (∗‘𝐵)))) |
31 | 27, 30 | eqtrd 2766 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (ℑ‘𝐵) = ((1 / (2 · i)) · (𝐵 − (∗‘𝐵)))) |
32 | 31 | mpteq2dva 5253 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐵)) = (𝑥 ∈ 𝑋 ↦ ((1 / (2 · i)) · (𝐵 − (∗‘𝐵))))) |
33 | 17, 25, 32 | 3eqtr4d 2776 | 1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (ℑ‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 {cpr 4635 ↦ cmpt 5236 ‘cfv 6554 (class class class)co 7424 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 ici 11160 · cmul 11163 − cmin 11494 / cdiv 11921 2c2 12319 ∗ccj 15101 ℑcim 15103 D cdv 25883 |
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 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 ax-addf 11237 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-pm 8858 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-fi 9454 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-ioo 13382 df-icc 13385 df-fz 13539 df-fzo 13682 df-seq 14022 df-exp 14082 df-hash 14348 df-cj 15104 df-re 15105 df-im 15106 df-sqrt 15240 df-abs 15241 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-starv 17281 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-unif 17289 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-pt 17459 df-prds 17462 df-xrs 17517 df-qtop 17522 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-met 21337 df-bl 21338 df-mopn 21339 df-fbas 21340 df-fg 21341 df-cnfld 21344 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-cld 23014 df-ntr 23015 df-cls 23016 df-nei 23093 df-lp 23131 df-perf 23132 df-cn 23222 df-cnp 23223 df-haus 23310 df-tx 23557 df-hmeo 23750 df-fil 23841 df-fm 23933 df-flim 23934 df-flf 23935 df-xms 24317 df-ms 24318 df-tms 24319 df-cncf 24889 df-limc 25886 df-dv 25887 |
This theorem is referenced by: (None) |
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