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Mirrors > Home > MPE Home > Th. List > dvcof | Structured version Visualization version GIF version |
Description: The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.) |
Ref | Expression |
---|---|
dvcof.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
dvcof.t | ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) |
dvcof.f | ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) |
dvcof.g | ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) |
dvcof.df | ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) |
dvcof.dg | ⊢ (𝜑 → dom (𝑇 D 𝐺) = 𝑌) |
Ref | Expression |
---|---|
dvcof | ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (((𝑆 D 𝐹) ∘ 𝐺) ∘f · (𝑇 D 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcof.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ) | |
2 | 1 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋⟶ℂ) |
3 | dvcof.df | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) | |
4 | dvbsss 24654 | . . . . . 6 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 | |
5 | 3, 4 | eqsstrrdi 3932 | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
6 | 5 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑋 ⊆ 𝑆) |
7 | dvcof.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) | |
8 | 7 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐺:𝑌⟶𝑋) |
9 | dvcof.dg | . . . . . 6 ⊢ (𝜑 → dom (𝑇 D 𝐺) = 𝑌) | |
10 | dvbsss 24654 | . . . . . 6 ⊢ dom (𝑇 D 𝐺) ⊆ 𝑇 | |
11 | 9, 10 | eqsstrrdi 3932 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇) |
12 | 11 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑌 ⊆ 𝑇) |
13 | dvcof.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
14 | 13 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ {ℝ, ℂ}) |
15 | dvcof.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) | |
16 | 15 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑇 ∈ {ℝ, ℂ}) |
17 | 7 | ffvelrnda 6861 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ 𝑋) |
18 | 3 | adantr 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → dom (𝑆 D 𝐹) = 𝑋) |
19 | 17, 18 | eleqtrrd 2836 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ dom (𝑆 D 𝐹)) |
20 | 9 | eleq2d 2818 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥 ∈ 𝑌)) |
21 | 20 | biimpar 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑇 D 𝐺)) |
22 | 2, 6, 8, 12, 14, 16, 19, 21 | dvco 24699 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥) = (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥))) |
23 | 22 | mpteq2dva 5125 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))) |
24 | dvfg 24658 | . . . . 5 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ) | |
25 | 15, 24 | syl 17 | . . . 4 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ) |
26 | recnprss 24656 | . . . . . . . 8 ⊢ (𝑇 ∈ {ℝ, ℂ} → 𝑇 ⊆ ℂ) | |
27 | 15, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ℂ) |
28 | fco 6528 | . . . . . . . 8 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ) | |
29 | 1, 7, 28 | syl2anc 587 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ) |
30 | 27, 29, 11 | dvbss 24653 | . . . . . 6 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) ⊆ 𝑌) |
31 | recnprss 24656 | . . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
32 | 14, 31 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ⊆ ℂ) |
33 | 16, 26 | syl 17 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑇 ⊆ ℂ) |
34 | fvexd 6689 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑆 D 𝐹)‘(𝐺‘𝑥)) ∈ V) | |
35 | fvexd 6689 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D 𝐺)‘𝑥) ∈ V) | |
36 | dvfg 24658 | . . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
37 | ffun 6507 | . . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) | |
38 | funfvbrb 6828 | . . . . . . . . . 10 ⊢ (Fun (𝑆 D 𝐹) → ((𝐺‘𝑥) ∈ dom (𝑆 D 𝐹) ↔ (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥)))) | |
39 | 14, 36, 37, 38 | 4syl 19 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝐺‘𝑥) ∈ dom (𝑆 D 𝐹) ↔ (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥)))) |
40 | 19, 39 | mpbid 235 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥))) |
41 | dvfg 24658 | . . . . . . . . . 10 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ) | |
42 | ffun 6507 | . . . . . . . . . 10 ⊢ ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ → Fun (𝑇 D 𝐺)) | |
43 | funfvbrb 6828 | . . . . . . . . . 10 ⊢ (Fun (𝑇 D 𝐺) → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥))) | |
44 | 16, 41, 42, 43 | 4syl 19 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥))) |
45 | 21, 44 | mpbid 235 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥)) |
46 | eqid 2738 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
47 | 2, 6, 8, 12, 32, 33, 34, 35, 40, 45, 46 | dvcobr 24698 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥))) |
48 | reldv 24622 | . . . . . . . 8 ⊢ Rel (𝑇 D (𝐹 ∘ 𝐺)) | |
49 | 48 | releldmi 5791 | . . . . . . 7 ⊢ (𝑥(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)) → 𝑥 ∈ dom (𝑇 D (𝐹 ∘ 𝐺))) |
50 | 47, 49 | syl 17 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑇 D (𝐹 ∘ 𝐺))) |
51 | 30, 50 | eqelssd 3898 | . . . . 5 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) = 𝑌) |
52 | 51 | feq2d 6490 | . . . 4 ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ ↔ (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ)) |
53 | 25, 52 | mpbid 235 | . . 3 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ) |
54 | 53 | feqmptd 6737 | . 2 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥))) |
55 | 15, 11 | ssexd 5192 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
56 | 7 | feqmptd 6737 | . . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑌 ↦ (𝐺‘𝑥))) |
57 | 13, 36 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) |
58 | 3 | feq2d 6490 | . . . . . 6 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
59 | 57, 58 | mpbid 235 | . . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
60 | 59 | feqmptd 6737 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑦))) |
61 | fveq2 6674 | . . . 4 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝑆 D 𝐹)‘𝑦) = ((𝑆 D 𝐹)‘(𝐺‘𝑥))) | |
62 | 17, 56, 60, 61 | fmptco 6901 | . . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ∘ 𝐺) = (𝑥 ∈ 𝑌 ↦ ((𝑆 D 𝐹)‘(𝐺‘𝑥)))) |
63 | 15, 41 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ) |
64 | 9 | feq2d 6490 | . . . . 5 ⊢ (𝜑 → ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ ↔ (𝑇 D 𝐺):𝑌⟶ℂ)) |
65 | 63, 64 | mpbid 235 | . . . 4 ⊢ (𝜑 → (𝑇 D 𝐺):𝑌⟶ℂ) |
66 | 65 | feqmptd 6737 | . . 3 ⊢ (𝜑 → (𝑇 D 𝐺) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D 𝐺)‘𝑥))) |
67 | 55, 34, 35, 62, 66 | offval2 7444 | . 2 ⊢ (𝜑 → (((𝑆 D 𝐹) ∘ 𝐺) ∘f · (𝑇 D 𝐺)) = (𝑥 ∈ 𝑌 ↦ (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))) |
68 | 23, 54, 67 | 3eqtr4d 2783 | 1 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (((𝑆 D 𝐹) ∘ 𝐺) ∘f · (𝑇 D 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ⊆ wss 3843 {cpr 4518 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5525 ∘ ccom 5529 Fun wfun 6333 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 ∘f cof 7423 ℂcc 10613 ℝcr 10614 · cmul 10620 TopOpenctopn 16798 ℂfldccnfld 20217 D cdv 24615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 ax-addf 10694 ax-mulf 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-2o 8132 df-er 8320 df-map 8439 df-pm 8440 df-ixp 8508 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-fi 8948 df-sup 8979 df-inf 8980 df-oi 9047 df-card 9441 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-dec 12180 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-icc 12828 df-fz 12982 df-fzo 13125 df-seq 13461 df-exp 13522 df-hash 13783 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-starv 16683 df-sca 16684 df-vsca 16685 df-ip 16686 df-tset 16687 df-ple 16688 df-ds 16690 df-unif 16691 df-hom 16692 df-cco 16693 df-rest 16799 df-topn 16800 df-0g 16818 df-gsum 16819 df-topgen 16820 df-pt 16821 df-prds 16824 df-xrs 16878 df-qtop 16883 df-imas 16884 df-xps 16886 df-mre 16960 df-mrc 16961 df-acs 16963 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-submnd 18073 df-mulg 18343 df-cntz 18565 df-cmn 19026 df-psmet 20209 df-xmet 20210 df-met 20211 df-bl 20212 df-mopn 20213 df-fbas 20214 df-fg 20215 df-cnfld 20218 df-top 21645 df-topon 21662 df-topsp 21684 df-bases 21697 df-cld 21770 df-ntr 21771 df-cls 21772 df-nei 21849 df-lp 21887 df-perf 21888 df-cn 21978 df-cnp 21979 df-haus 22066 df-tx 22313 df-hmeo 22506 df-fil 22597 df-fm 22689 df-flim 22690 df-flf 22691 df-xms 23073 df-ms 23074 df-tms 23075 df-cncf 23630 df-limc 24618 df-dv 24619 |
This theorem is referenced by: dvmptco 24724 dvsinax 42996 dvcosax 43009 |
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