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Mirrors > Home > MPE Home > Th. List > dvmptcj | Structured version Visualization version GIF version |
Description: Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
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 |
---|---|
dvmptcj | ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptcj.a | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) | |
2 | 1 | fmpttd 6749 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
3 | dvmptcj.da | . . . . . 6 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) | |
4 | 3 | dmeqd 5667 | . . . . 5 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵)) |
5 | dvmptcj.b | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) | |
6 | 5 | ralrimiva 3151 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉) |
7 | dmmptg 5978 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋) | |
8 | 6, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋) |
9 | 4, 8 | eqtrd 2833 | . . . 4 ⊢ (𝜑 → dom (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋) |
10 | dvbsss 24187 | . . . 4 ⊢ dom (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ⊆ ℝ | |
11 | 9, 10 | syl6eqssr 3949 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ℝ) |
12 | dvcj 24234 | . . 3 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ ∧ 𝑋 ⊆ ℝ) → (ℝ D (∗ ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (∗ ∘ (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)))) | |
13 | 2, 11, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → (ℝ D (∗ ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (∗ ∘ (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)))) |
14 | cjf 14301 | . . . . 5 ⊢ ∗:ℂ⟶ℂ | |
15 | 14 | a1i 11 | . . . 4 ⊢ (𝜑 → ∗:ℂ⟶ℂ) |
16 | 15, 1 | cofmpt 6764 | . . 3 ⊢ (𝜑 → (∗ ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) |
17 | 16 | oveq2d 7039 | . 2 ⊢ (𝜑 → (ℝ D (∗ ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴)))) |
18 | reelprrecn 10482 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
20 | 19, 1, 5, 3 | dvmptcl 24243 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
21 | 15 | feqmptd 6608 | . . 3 ⊢ (𝜑 → ∗ = (𝑦 ∈ ℂ ↦ (∗‘𝑦))) |
22 | fveq2 6545 | . . 3 ⊢ (𝑦 = 𝐵 → (∗‘𝑦) = (∗‘𝐵)) | |
23 | 20, 3, 21, 22 | fmptco 6761 | . 2 ⊢ (𝜑 → (∗ ∘ (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵))) |
24 | 13, 17, 23 | 3eqtr3d 2841 | 1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∀wral 3107 ⊆ wss 3865 {cpr 4480 ↦ cmpt 5047 dom cdm 5450 ∘ ccom 5454 ⟶wf 6228 ‘cfv 6232 (class class class)co 7023 ℂcc 10388 ℝcr 10389 ∗ccj 14293 D cdv 24148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-cnex 10446 ax-resscn 10447 ax-1cn 10448 ax-icn 10449 ax-addcl 10450 ax-addrcl 10451 ax-mulcl 10452 ax-mulrcl 10453 ax-mulcom 10454 ax-addass 10455 ax-mulass 10456 ax-distr 10457 ax-i2m1 10458 ax-1ne0 10459 ax-1rid 10460 ax-rnegex 10461 ax-rrecex 10462 ax-cnre 10463 ax-pre-lttri 10464 ax-pre-lttrn 10465 ax-pre-ltadd 10466 ax-pre-mulgt0 10467 ax-pre-sup 10468 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-nel 3093 df-ral 3112 df-rex 3113 df-reu 3114 df-rmo 3115 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-iun 4833 df-iin 4834 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-pred 6030 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-riota 6984 df-ov 7026 df-oprab 7027 df-mpo 7028 df-om 7444 df-1st 7552 df-2nd 7553 df-wrecs 7805 df-recs 7867 df-rdg 7905 df-1o 7960 df-oadd 7964 df-er 8146 df-map 8265 df-pm 8266 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-fi 8728 df-sup 8759 df-inf 8760 df-pnf 10530 df-mnf 10531 df-xr 10532 df-ltxr 10533 df-le 10534 df-sub 10725 df-neg 10726 df-div 11152 df-nn 11493 df-2 11554 df-3 11555 df-4 11556 df-5 11557 df-6 11558 df-7 11559 df-8 11560 df-9 11561 df-n0 11752 df-z 11836 df-dec 11953 df-uz 12098 df-q 12202 df-rp 12244 df-xneg 12361 df-xadd 12362 df-xmul 12363 df-ioo 12596 df-icc 12599 df-fz 12747 df-seq 13224 df-exp 13284 df-cj 14296 df-re 14297 df-im 14298 df-sqrt 14432 df-abs 14433 df-struct 16318 df-ndx 16319 df-slot 16320 df-base 16322 df-plusg 16411 df-mulr 16412 df-starv 16413 df-tset 16417 df-ple 16418 df-ds 16420 df-unif 16421 df-rest 16529 df-topn 16530 df-topgen 16550 df-psmet 20223 df-xmet 20224 df-met 20225 df-bl 20226 df-mopn 20227 df-fbas 20228 df-fg 20229 df-cnfld 20232 df-top 21190 df-topon 21207 df-topsp 21229 df-bases 21242 df-cld 21315 df-ntr 21316 df-cls 21317 df-nei 21394 df-lp 21432 df-perf 21433 df-cn 21523 df-cnp 21524 df-haus 21611 df-fil 22142 df-fm 22234 df-flim 22235 df-flf 22236 df-xms 22617 df-ms 22618 df-cncf 23173 df-limc 24151 df-dv 24152 |
This theorem is referenced by: dvmptre 24253 dvmptim 24254 |
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