dvcof - Metamath Proof Explorer

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Theorem dvcof 26116

Description: The chain rule for everywhere-differentiable functions. (Contributed by Mario Carneiro, 10-Aug-2014.) (Revised by Mario Carneiro, 10-Feb-2015.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
dvcof	⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (((𝑆 D 𝐹) ∘ 𝐺) ∘_f · (𝑇 D 𝐺)))

Proof of Theorem dvcof Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvcof.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝑋⟶ℂ)
2	1	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋⟶ℂ)
3		dvcof.df	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
4		dvbsss 26070	. . . . . 6 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
5	3, 4	eqsstrrdi 3982	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
6	5	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑋 ⊆ 𝑆)
7		dvcof.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶𝑋)
8	7	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐺:𝑌⟶𝑋)
9		dvcof.dg	. . . . . 6 ⊢ (𝜑 → dom (𝑇 D 𝐺) = 𝑌)
10		dvbsss 26070	. . . . . 6 ⊢ dom (𝑇 D 𝐺) ⊆ 𝑇
11	9, 10	eqsstrrdi 3982	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	11	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑌 ⊆ 𝑇)
13		dvcof.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
14	13	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ {ℝ, ℂ})
15		dvcof.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ})
16	15	adantr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑇 ∈ {ℝ, ℂ})
17	7	ffvelcdmda 7079	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ 𝑋)
18	3	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → dom (𝑆 D 𝐹) = 𝑋)
19	17, 18	eleqtrrd 2866	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ dom (𝑆 D 𝐹))
20	9	eleq2d 2849	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥 ∈ 𝑌))
21	20	biimpar 482	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑇 D 𝐺))
22	2, 6, 8, 12, 14, 16, 19, 21	dvco 26115	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥) = (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))
23	22	mpteq2dva 5204	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥))))
24		dvfg 26074	. . . . 5 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ)
25	15, 24	syl 18	. . . 4 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ)
26		recnprss 26072	. . . . . . . 8 ⊢ (𝑇 ∈ {ℝ, ℂ} → 𝑇 ⊆ ℂ)
27	15, 26	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ℂ)
28		fco 6730	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
29	1, 7, 28	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
30	27, 29, 11	dvbss 26069	. . . . . 6 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) ⊆ 𝑌)
31		recnprss 26072	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
32	14, 31	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ⊆ ℂ)
33	16, 26	syl 18	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑇 ⊆ ℂ)
34		dvfg 26074	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
35		ffun 6708	. . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
36		funfvbrb 7046	. . . . . . . . . 10 ⊢ (Fun (𝑆 D 𝐹) → ((𝐺‘𝑥) ∈ dom (𝑆 D 𝐹) ↔ (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥))))
37	14, 34, 35, 36	4syl 20	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝐺‘𝑥) ∈ dom (𝑆 D 𝐹) ↔ (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥))))
38	19, 37	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥)))
39		dvfg 26074	. . . . . . . . . 10 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ)
40		ffun 6708	. . . . . . . . . 10 ⊢ ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ → Fun (𝑇 D 𝐺))
41		funfvbrb 7046	. . . . . . . . . 10 ⊢ (Fun (𝑇 D 𝐺) → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥)))
42	16, 39, 40, 41	4syl 20	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥)))
43	21, 42	mpbid 235	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥))
44		eqid 2763	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	2, 6, 8, 12, 32, 33, 38, 43, 44	dvcobr 26114	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))
46		reldv 26038	. . . . . . . 8 ⊢ Rel (𝑇 D (𝐹 ∘ 𝐺))
47	46	releldmi 5938	. . . . . . 7 ⊢ (𝑥(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)) → 𝑥 ∈ dom (𝑇 D (𝐹 ∘ 𝐺)))
48	45, 47	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑇 D (𝐹 ∘ 𝐺)))
49	30, 48	eqelssd 3958	. . . . 5 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) = 𝑌)
50	49	feq2d 6689	. . . 4 ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ ↔ (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ))
51	25, 50	mpbid 235	. . 3 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ)
52	51	feqmptd 6949	. 2 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥)))
53	15, 11	ssexd 5295	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
54		fvexd 6896	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑆 D 𝐹)‘(𝐺‘𝑥)) ∈ V)
55		fvexd 6896	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D 𝐺)‘𝑥) ∈ V)
56	7	feqmptd 6949	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑌 ↦ (𝐺‘𝑥)))
57	13, 34	syl 18	. . . . . 6 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
58	3	feq2d 6689	. . . . . 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 6949	. . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑦)))
61		fveq2 6881	. . . 4 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝑆 D 𝐹)‘𝑦) = ((𝑆 D 𝐹)‘(𝐺‘𝑥)))
62	17, 56, 60, 61	fmptco 7125	. . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ∘ 𝐺) = (𝑥 ∈ 𝑌 ↦ ((𝑆 D 𝐹)‘(𝐺‘𝑥))))
63	15, 39	syl 18	. . . . 5 ⊢ (𝜑 → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ)
64	9	feq2d 6689	. . . . 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 6949	. . 3 ⊢ (𝜑 → (𝑇 D 𝐺) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D 𝐺)‘𝑥)))
67	53, 54, 55, 62, 66	offval2 7694	. 2 ⊢ (𝜑 → (((𝑆 D 𝐹) ∘ 𝐺) ∘_f · (𝑇 D 𝐺)) = (𝑥 ∈ 𝑌 ↦ (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥))))
68	23, 52, 67	3eqtr4d 2808	1 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (((𝑆 D 𝐹) ∘ 𝐺) ∘_f · (𝑇 D 𝐺)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 Vcvv 3455 ⊆ wss 3905 {cpr 4591 class class class wbr 5109 ↦ cmpt 5192 dom cdm 5661 ∘ ccom 5665 Fun wfun 6530 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ∘_f cof 7672 ℂcc 11102 ℝcr 11103 · cmul 11109 TopOpenctopn 17478 ℂ_fldccnfld 21531 D cdv 26031

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-icc 13383 df-fz 13540 df-fzo 13688 df-seq 14043 df-exp 14103 df-hash 14372 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-hom 17338 df-cco 17339 df-rest 17479 df-topn 17480 df-0g 17498 df-gsum 17499 df-topgen 17500 df-pt 17501 df-prds 17504 df-xrs 17560 df-qtop 17565 df-imas 17566 df-xps 17568 df-mre 17642 df-mrc 17643 df-acs 17645 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-mulg 19138 df-cntz 19391 df-cmn 19856 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-fbas 21528 df-fg 21529 df-cnfld 21532 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cld 23185 df-ntr 23186 df-cls 23187 df-nei 23264 df-lp 23302 df-perf 23303 df-cn 23393 df-cnp 23394 df-haus 23481 df-tx 23728 df-hmeo 23921 df-fil 24012 df-fm 24104 df-flim 24105 df-flf 24106 df-xms 24486 df-ms 24487 df-tms 24488 df-cncf 25046 df-limc 26034 df-dv 26035

This theorem is used by: dvmptco 26140 dvsinax 46655 dvcosax 46668

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