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Mirrors > Home > ILE Home > Th. List > dvmptcjx | GIF version |
Description: Function-builder for derivative, conjugate rule. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Jim Kingdon, 24-May-2024.) |
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)) |
dvmptcjx.x | ⊢ (𝜑 → 𝑋 ⊆ ℝ) |
Ref | Expression |
---|---|
dvmptcjx | ⊢ (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 5615 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
3 | dvmptcjx.x | . . 3 ⊢ (𝜑 → 𝑋 ⊆ ℝ) | |
4 | dvcj 13012 | . . 3 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ ∧ 𝑋 ⊆ ℝ) → (ℝ D (∗ ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (∗ ∘ (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)))) | |
5 | 2, 3, 4 | syl2anc 409 | . 2 ⊢ (𝜑 → (ℝ D (∗ ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (∗ ∘ (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)))) |
6 | cjf 10724 | . . . . 5 ⊢ ∗:ℂ⟶ℂ | |
7 | 6 | a1i 9 | . . . 4 ⊢ (𝜑 → ∗:ℂ⟶ℂ) |
8 | 7, 1 | cofmpt 5629 | . . 3 ⊢ (𝜑 → (∗ ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) |
9 | 8 | oveq2d 5830 | . 2 ⊢ (𝜑 → (ℝ D (∗ ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴)))) |
10 | reelprrecn 7846 | . . . . 5 ⊢ ℝ ∈ {ℝ, ℂ} | |
11 | 10 | a1i 9 | . . . 4 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
12 | dvmptcj.b | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉) | |
13 | dvmptcj.da | . . . 4 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵)) | |
14 | 11, 1, 12, 13, 3 | dvmptclx 13019 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ ℂ) |
15 | 7 | feqmptd 5514 | . . 3 ⊢ (𝜑 → ∗ = (𝑦 ∈ ℂ ↦ (∗‘𝑦))) |
16 | fveq2 5461 | . . 3 ⊢ (𝑦 = 𝐵 → (∗‘𝑦) = (∗‘𝐵)) | |
17 | 14, 13, 15, 16 | fmptco 5626 | . 2 ⊢ (𝜑 → (∗ ∘ (ℝ D (𝑥 ∈ 𝑋 ↦ 𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵))) |
18 | 5, 9, 17 | 3eqtr3d 2195 | 1 ⊢ (𝜑 → (ℝ D (𝑥 ∈ 𝑋 ↦ (∗‘𝐴))) = (𝑥 ∈ 𝑋 ↦ (∗‘𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 2125 ⊆ wss 3098 {cpr 3557 ↦ cmpt 4021 ∘ ccom 4583 ⟶wf 5159 ‘cfv 5163 (class class class)co 5814 ℂcc 7709 ℝcr 7710 ∗ccj 10716 D cdv 12963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 ax-caucvg 7831 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-isom 5172 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-1st 6078 df-2nd 6079 df-recs 6242 df-frec 6328 df-map 6584 df-pm 6585 df-sup 6916 df-inf 6917 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-3 8872 df-4 8873 df-n0 9070 df-z 9147 df-uz 9419 df-q 9507 df-rp 9539 df-xneg 9657 df-xadd 9658 df-ioo 9774 df-seqfrec 10323 df-exp 10397 df-cj 10719 df-re 10720 df-im 10721 df-rsqrt 10875 df-abs 10876 df-rest 12292 df-topgen 12311 df-psmet 12326 df-xmet 12327 df-met 12328 df-bl 12329 df-mopn 12330 df-top 12335 df-topon 12348 df-bases 12380 df-ntr 12435 df-cn 12527 df-cnp 12528 df-cncf 12897 df-limced 12964 df-dvap 12965 |
This theorem is referenced by: (None) |
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