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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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋⟶ℂ) |
3 | dvcof.df | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋) | |
4 | dvbsss 24500 | . . . . . 6 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 | |
5 | 3, 4 | eqsstrrdi 4022 | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑋 ⊆ 𝑆) |
7 | dvcof.g | . . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶𝑋) | |
8 | 7 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐺:𝑌⟶𝑋) |
9 | dvcof.dg | . . . . . 6 ⊢ (𝜑 → dom (𝑇 D 𝐺) = 𝑌) | |
10 | dvbsss 24500 | . . . . . 6 ⊢ dom (𝑇 D 𝐺) ⊆ 𝑇 | |
11 | 9, 10 | eqsstrrdi 4022 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇) |
12 | 11 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑌 ⊆ 𝑇) |
13 | dvcof.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ {ℝ, ℂ}) |
15 | dvcof.t | . . . . 5 ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ}) | |
16 | 15 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑇 ∈ {ℝ, ℂ}) |
17 | 7 | ffvelrnda 6851 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ 𝑋) |
18 | 3 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → dom (𝑆 D 𝐹) = 𝑋) |
19 | 17, 18 | eleqtrrd 2916 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ dom (𝑆 D 𝐹)) |
20 | 9 | eleq2d 2898 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥 ∈ 𝑌)) |
21 | 20 | biimpar 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑇 D 𝐺)) |
22 | 2, 6, 8, 12, 14, 16, 19, 21 | dvco 24544 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥) = (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥))) |
23 | 22 | mpteq2dva 5161 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))) |
24 | dvfg 24504 | . . . . 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 24502 | . . . . . . . 8 ⊢ (𝑇 ∈ {ℝ, ℂ} → 𝑇 ⊆ ℂ) | |
27 | 15, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ℂ) |
28 | fco 6531 | . . . . . . . 8 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ) | |
29 | 1, 7, 28 | syl2anc 586 | . . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ) |
30 | 27, 29, 11 | dvbss 24499 | . . . . . 6 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) ⊆ 𝑌) |
31 | recnprss 24502 | . . . . . . . . 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 6685 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑆 D 𝐹)‘(𝐺‘𝑥)) ∈ V) | |
35 | fvexd 6685 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D 𝐺)‘𝑥) ∈ V) | |
36 | dvfg 24504 | . . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ) | |
37 | ffun 6517 | . . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹)) | |
38 | funfvbrb 6821 | . . . . . . . . . 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 234 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥))) |
41 | dvfg 24504 | . . . . . . . . . 10 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ) | |
42 | ffun 6517 | . . . . . . . . . 10 ⊢ ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ → Fun (𝑇 D 𝐺)) | |
43 | funfvbrb 6821 | . . . . . . . . . 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 234 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥)) |
46 | eqid 2821 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
47 | 2, 6, 8, 12, 32, 33, 34, 35, 40, 45, 46 | dvcobr 24543 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥))) |
48 | reldv 24468 | . . . . . . . 8 ⊢ Rel (𝑇 D (𝐹 ∘ 𝐺)) | |
49 | 48 | releldmi 5818 | . . . . . . 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 3988 | . . . . 5 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) = 𝑌) |
52 | 51 | feq2d 6500 | . . . 4 ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ ↔ (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ)) |
53 | 25, 52 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ) |
54 | 53 | feqmptd 6733 | . 2 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥))) |
55 | 15, 11 | ssexd 5228 | . . 3 ⊢ (𝜑 → 𝑌 ∈ V) |
56 | 7 | feqmptd 6733 | . . . 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 6500 | . . . . . 6 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ)) |
59 | 57, 58 | mpbid 234 | . . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ) |
60 | 59 | feqmptd 6733 | . . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑦))) |
61 | fveq2 6670 | . . . 4 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝑆 D 𝐹)‘𝑦) = ((𝑆 D 𝐹)‘(𝐺‘𝑥))) | |
62 | 17, 56, 60, 61 | fmptco 6891 | . . 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 6500 | . . . . 5 ⊢ (𝜑 → ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ ↔ (𝑇 D 𝐺):𝑌⟶ℂ)) |
65 | 63, 64 | mpbid 234 | . . . 4 ⊢ (𝜑 → (𝑇 D 𝐺):𝑌⟶ℂ) |
66 | 65 | feqmptd 6733 | . . 3 ⊢ (𝜑 → (𝑇 D 𝐺) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D 𝐺)‘𝑥))) |
67 | 55, 34, 35, 62, 66 | offval2 7426 | . 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 2866 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ∘ ccom 5559 Fun wfun 6349 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∘f cof 7407 ℂcc 10535 ℝcr 10536 · cmul 10542 TopOpenctopn 16695 ℂfldccnfld 20545 D cdv 24461 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465 |
This theorem is referenced by: dvmptco 24569 dvsinax 42217 dvcosax 42231 |
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