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Mirrors > Home > MPE Home > Th. List > dvco | Structured version Visualization version GIF version |
Description: The chain rule for derivatives at a point. For the (more general) relation version, see dvcobr 24546. (Contributed by Mario Carneiro, 9-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
Ref | Expression |
---|---|
dvco.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvco.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
dvco.g | ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) |
dvco.y | ⊢ (𝜑 → 𝑌 ⊆ 𝑇) |
dvco.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvco.t | ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) |
dvco.df | ⊢ (𝜑 → (𝐺‘𝐶) ∈ dom (𝑆 D 𝐹)) |
dvco.dg | ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺)) |
Ref | Expression |
---|---|
dvco | ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘(𝐺‘𝐶)) · ((𝑇 D 𝐺)‘𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvco.t | . . 3 ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) | |
2 | dvfg 24507 | . . 3 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ) | |
3 | ffun 6520 | . . 3 ⊢ ((𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ → Fun (𝑇 D (𝐹 ∘ 𝐺))) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ (𝜑 → Fun (𝑇 D (𝐹 ∘ 𝐺))) |
5 | dvco.f | . . 3 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
6 | dvco.x | . . 3 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
7 | dvco.g | . . 3 ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) | |
8 | dvco.y | . . 3 ⊢ (𝜑 → 𝑌 ⊆ 𝑇) | |
9 | dvco.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
10 | recnprss 24505 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
12 | recnprss 24505 | . . . 4 ⊢ (𝑇 ∈ {ℝ, ℂ} → 𝑇 ⊆ ℂ) | |
13 | 1, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ ℂ) |
14 | fvexd 6688 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹)‘(𝐺‘𝐶)) ∈ V) | |
15 | fvexd 6688 | . . 3 ⊢ (𝜑 → ((𝑇 D 𝐺)‘𝐶) ∈ V) | |
16 | dvco.df | . . . 4 ⊢ (𝜑 → (𝐺‘𝐶) ∈ dom (𝑆 D 𝐹)) | |
17 | dvfg 24507 | . . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
18 | ffun 6520 | . . . . 5 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) | |
19 | funfvbrb 6824 | . . . . 5 ⊢ (Fun (𝑆 D 𝐹) → ((𝐺‘𝐶) ∈ dom (𝑆 D 𝐹) ↔ (𝐺‘𝐶)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝐶)))) | |
20 | 9, 17, 18, 19 | 4syl 19 | . . . 4 ⊢ (𝜑 → ((𝐺‘𝐶) ∈ dom (𝑆 D 𝐹) ↔ (𝐺‘𝐶)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝐶)))) |
21 | 16, 20 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝐺‘𝐶)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝐶))) |
22 | dvco.dg | . . . 4 ⊢ (𝜑 → 𝐶 ∈ dom (𝑇 D 𝐺)) | |
23 | dvfg 24507 | . . . . 5 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ) | |
24 | ffun 6520 | . . . . 5 ⊢ ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ → Fun (𝑇 D 𝐺)) | |
25 | funfvbrb 6824 | . . . . 5 ⊢ (Fun (𝑇 D 𝐺) → (𝐶 ∈ dom (𝑇 D 𝐺) ↔ 𝐶(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝐶))) | |
26 | 1, 23, 24, 25 | 4syl 19 | . . . 4 ⊢ (𝜑 → (𝐶 ∈ dom (𝑇 D 𝐺) ↔ 𝐶(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝐶))) |
27 | 22, 26 | mpbid 234 | . . 3 ⊢ (𝜑 → 𝐶(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝐶)) |
28 | eqid 2824 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
29 | 5, 6, 7, 8, 11, 13, 14, 15, 21, 27, 28 | dvcobr 24546 | . 2 ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝐶)) · ((𝑇 D 𝐺)‘𝐶))) |
30 | funbrfv 6719 | . 2 ⊢ (Fun (𝑇 D (𝐹 ∘ 𝐺)) → (𝐶(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝐶)) · ((𝑇 D 𝐺)‘𝐶)) → ((𝑇 D (𝐹 ∘ 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘(𝐺‘𝐶)) · ((𝑇 D 𝐺)‘𝐶)))) | |
31 | 4, 29, 30 | sylc 65 | 1 ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺))‘𝐶) = (((𝑆 D 𝐹)‘(𝐺‘𝐶)) · ((𝑇 D 𝐺)‘𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 Vcvv 3497 ⊆ wss 3939 {cpr 4572 class class class wbr 5069 dom cdm 5558 ∘ ccom 5562 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 ℝcr 10539 · cmul 10545 TopOpenctopn 16698 ℂfldccnfld 20548 D cdv 24464 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-pm 8412 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-fi 8878 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-pt 16721 df-prds 16724 df-xrs 16778 df-qtop 16783 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-met 20542 df-bl 20543 df-mopn 20544 df-fbas 20545 df-fg 20546 df-cnfld 20549 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-cld 21630 df-ntr 21631 df-cls 21632 df-nei 21709 df-lp 21747 df-perf 21748 df-cn 21838 df-cnp 21839 df-haus 21926 df-tx 22173 df-hmeo 22366 df-fil 22457 df-fm 22549 df-flim 22550 df-flf 22551 df-xms 22933 df-ms 22934 df-tms 22935 df-cncf 23489 df-limc 24467 df-dv 24468 |
This theorem is referenced by: dvcof 24548 |
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