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Mirrors > Home > ILE Home > Th. List > dvaddxx | GIF version |
Description: The sum rule for derivatives at a point. For the (more general) relation version, see dvaddxxbr 12834. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Jim Kingdon, 25-Nov-2023.) |
Ref | Expression |
---|---|
dvadd.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvadd.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
dvaddxx.g | ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) |
dvadd.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvadd.df | ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) |
dvadd.dg | ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺)) |
Ref | Expression |
---|---|
dvaddxx | ⊢ (𝜑 → ((𝑆 D (𝐹 ∘𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvadd.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | cnex 7744 | . . . . . 6 ⊢ ℂ ∈ V | |
3 | 2 | a1i 9 | . . . . 5 ⊢ (𝜑 → ℂ ∈ V) |
4 | addcl 7745 | . . . . . . 7 ⊢ ((𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ) → (𝑢 + 𝑣) ∈ ℂ) | |
5 | 4 | adantl 275 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ ℂ ∧ 𝑣 ∈ ℂ)) → (𝑢 + 𝑣) ∈ ℂ) |
6 | dvadd.f | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
7 | dvaddxx.g | . . . . . 6 ⊢ (𝜑 → 𝐺:𝑋⟶ℂ) | |
8 | dvadd.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
9 | 1, 8 | ssexd 4068 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ V) |
10 | inidm 3285 | . . . . . 6 ⊢ (𝑋 ∩ 𝑋) = 𝑋 | |
11 | 5, 6, 7, 9, 9, 10 | off 5994 | . . . . 5 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺):𝑋⟶ℂ) |
12 | elpm2r 6560 | . . . . 5 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ ((𝐹 ∘𝑓 + 𝐺):𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → (𝐹 ∘𝑓 + 𝐺) ∈ (ℂ ↑pm 𝑆)) | |
13 | 3, 1, 11, 8, 12 | syl22anc 1217 | . . . 4 ⊢ (𝜑 → (𝐹 ∘𝑓 + 𝐺) ∈ (ℂ ↑pm 𝑆)) |
14 | dvfgg 12826 | . . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ (𝐹 ∘𝑓 + 𝐺) ∈ (ℂ ↑pm 𝑆)) → (𝑆 D (𝐹 ∘𝑓 + 𝐺)):dom (𝑆 D (𝐹 ∘𝑓 + 𝐺))⟶ℂ) | |
15 | 1, 13, 14 | syl2anc 408 | . . 3 ⊢ (𝜑 → (𝑆 D (𝐹 ∘𝑓 + 𝐺)):dom (𝑆 D (𝐹 ∘𝑓 + 𝐺))⟶ℂ) |
16 | 15 | ffund 5276 | . 2 ⊢ (𝜑 → Fun (𝑆 D (𝐹 ∘𝑓 + 𝐺))) |
17 | recnprss 12825 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
18 | 1, 17 | syl 14 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
19 | dvadd.df | . . . 4 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐹)) | |
20 | elpm2r 6560 | . . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
21 | 3, 1, 6, 8, 20 | syl22anc 1217 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
22 | dvfgg 12826 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
23 | 1, 21, 22 | syl2anc 408 | . . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
24 | ffun 5275 | . . . . 5 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) | |
25 | funfvbrb 5533 | . . . . 5 ⊢ (Fun (𝑆 D 𝐹) → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))) | |
26 | 23, 24, 25 | 3syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ dom (𝑆 D 𝐹) ↔ 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶))) |
27 | 19, 26 | mpbid 146 | . . 3 ⊢ (𝜑 → 𝐶(𝑆 D 𝐹)((𝑆 D 𝐹)‘𝐶)) |
28 | dvadd.dg | . . . 4 ⊢ (𝜑 → 𝐶 ∈ dom (𝑆 D 𝐺)) | |
29 | elpm2r 6560 | . . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐺:𝑋⟶ℂ ∧ 𝑋 ⊆ 𝑆)) → 𝐺 ∈ (ℂ ↑pm 𝑆)) | |
30 | 3, 1, 7, 8, 29 | syl22anc 1217 | . . . . . 6 ⊢ (𝜑 → 𝐺 ∈ (ℂ ↑pm 𝑆)) |
31 | dvfgg 12826 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐺 ∈ (ℂ ↑pm 𝑆)) → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) | |
32 | 1, 30, 31 | syl2anc 408 | . . . . 5 ⊢ (𝜑 → (𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ) |
33 | ffun 5275 | . . . . 5 ⊢ ((𝑆 D 𝐺):dom (𝑆 D 𝐺)⟶ℂ → Fun (𝑆 D 𝐺)) | |
34 | funfvbrb 5533 | . . . . 5 ⊢ (Fun (𝑆 D 𝐺) → (𝐶 ∈ dom (𝑆 D 𝐺) ↔ 𝐶(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝐶))) | |
35 | 32, 33, 34 | 3syl 17 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ dom (𝑆 D 𝐺) ↔ 𝐶(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝐶))) |
36 | 28, 35 | mpbid 146 | . . 3 ⊢ (𝜑 → 𝐶(𝑆 D 𝐺)((𝑆 D 𝐺)‘𝐶)) |
37 | eqid 2139 | . . 3 ⊢ (MetOpen‘(abs ∘ − )) = (MetOpen‘(abs ∘ − )) | |
38 | 6, 8, 7, 18, 27, 36, 37 | dvaddxxbr 12834 | . 2 ⊢ (𝜑 → 𝐶(𝑆 D (𝐹 ∘𝑓 + 𝐺))(((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶))) |
39 | funbrfv 5460 | . 2 ⊢ (Fun (𝑆 D (𝐹 ∘𝑓 + 𝐺)) → (𝐶(𝑆 D (𝐹 ∘𝑓 + 𝐺))(((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)) → ((𝑆 D (𝐹 ∘𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶)))) | |
40 | 16, 38, 39 | sylc 62 | 1 ⊢ (𝜑 → ((𝑆 D (𝐹 ∘𝑓 + 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘𝐶) + ((𝑆 D 𝐺)‘𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⊆ wss 3071 {cpr 3528 class class class wbr 3929 dom cdm 4539 ∘ ccom 4543 Fun wfun 5117 ⟶wf 5119 ‘cfv 5123 (class class class)co 5774 ∘𝑓 cof 5980 ↑pm cpm 6543 ℂcc 7618 ℝcr 7619 + caddc 7623 − cmin 7933 abscabs 10769 MetOpencmopn 12154 D cdv 12793 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738 ax-arch 7739 ax-caucvg 7740 ax-addf 7742 |
This theorem depends on definitions: df-bi 116 df-stab 816 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-of 5982 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-map 6544 df-pm 6545 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-2 8779 df-3 8780 df-4 8781 df-n0 8978 df-z 9055 df-uz 9327 df-q 9412 df-rp 9442 df-xneg 9559 df-xadd 9560 df-seqfrec 10219 df-exp 10293 df-cj 10614 df-re 10615 df-im 10616 df-rsqrt 10770 df-abs 10771 df-rest 12122 df-topgen 12141 df-psmet 12156 df-xmet 12157 df-met 12158 df-bl 12159 df-mopn 12160 df-top 12165 df-topon 12178 df-bases 12210 df-ntr 12265 df-cn 12357 df-cnp 12358 df-tx 12422 df-limced 12794 df-dvap 12795 |
This theorem is referenced by: dviaddf 12838 |
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