Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 24545. (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 24506 | . . 3 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ) | |
3 | ffun 6519 | . . 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 24504 | . . . 4 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
12 | recnprss 24504 | . . . 4 ⊢ (𝑇 ∈ {ℝ, ℂ} → 𝑇 ⊆ ℂ) | |
13 | 1, 12 | syl 17 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ ℂ) |
14 | fvexd 6687 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹)‘(𝐺‘𝐶)) ∈ V) | |
15 | fvexd 6687 | . . 3 ⊢ (𝜑 → ((𝑇 D 𝐺)‘𝐶) ∈ V) | |
16 | dvco.df | . . . 4 ⊢ (𝜑 → (𝐺‘𝐶) ∈ dom (𝑆 D 𝐹)) | |
17 | dvfg 24506 | . . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
18 | ffun 6519 | . . . . 5 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) | |
19 | funfvbrb 6823 | . . . . 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 24506 | . . . . 5 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ) | |
24 | ffun 6519 | . . . . 5 ⊢ ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ → Fun (𝑇 D 𝐺)) | |
25 | funfvbrb 6823 | . . . . 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 2823 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
29 | 5, 6, 7, 8, 11, 13, 14, 15, 21, 27, 28 | dvcobr 24545 | . 2 ⊢ (𝜑 → 𝐶(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝐶)) · ((𝑇 D 𝐺)‘𝐶))) |
30 | funbrfv 6718 | . 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 1537 ∈ wcel 2114 Vcvv 3496 ⊆ wss 3938 {cpr 4571 class class class wbr 5068 dom cdm 5557 ∘ ccom 5561 Fun wfun 6351 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 · cmul 10544 TopOpenctopn 16697 ℂfldccnfld 20547 D cdv 24463 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 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 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-mulg 18227 df-cntz 18449 df-cmn 18910 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-limc 24466 df-dv 24467 |
This theorem is referenced by: dvcof 24547 |
Copyright terms: Public domain | W3C validator |