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Theorem dvcof 25934

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 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋⟶ℂ)
3		dvcof.df	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
4		dvbsss 25888	. . . . . 6 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
5	3, 4	eqsstrrdi 3960	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
6	5	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑋 ⊆ 𝑆)
7		dvcof.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶𝑋)
8	7	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐺:𝑌⟶𝑋)
9		dvcof.dg	. . . . . 6 ⊢ (𝜑 → dom (𝑇 D 𝐺) = 𝑌)
10		dvbsss 25888	. . . . . 6 ⊢ dom (𝑇 D 𝐺) ⊆ 𝑇
11	9, 10	eqsstrrdi 3960	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	11	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑌 ⊆ 𝑇)
13		dvcof.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
14	13	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ {ℝ, ℂ})
15		dvcof.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ})
16	15	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑇 ∈ {ℝ, ℂ})
17	7	ffvelcdmda 7026	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ 𝑋)
18	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → dom (𝑆 D 𝐹) = 𝑋)
19	17, 18	eleqtrrd 2842	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ dom (𝑆 D 𝐹))
20	9	eleq2d 2825	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥 ∈ 𝑌))
21	20	biimpar 478	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑇 D 𝐺))
22	2, 6, 8, 12, 14, 16, 19, 21	dvco 25933	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥) = (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))
23	22	mpteq2dva 5166	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥))))
24		dvfg 25892	. . . . 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 25890	. . . . . . . 8 ⊢ (𝑇 ∈ {ℝ, ℂ} → 𝑇 ⊆ ℂ)
27	15, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ℂ)
28		fco 6680	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
29	1, 7, 28	syl2anc 590	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
30	27, 29, 11	dvbss 25887	. . . . . 6 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) ⊆ 𝑌)
31		recnprss 25890	. . . . . . . . 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		dvfg 25892	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
35		ffun 6659	. . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
36		funfvbrb 6993	. . . . . . . . . 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 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝐺‘𝑥) ∈ dom (𝑆 D 𝐹) ↔ (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥))))
38	19, 37	mpbid 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥)))
39		dvfg 25892	. . . . . . . . . 10 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ)
40		ffun 6659	. . . . . . . . . 10 ⊢ ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ → Fun (𝑇 D 𝐺))
41		funfvbrb 6993	. . . . . . . . . 10 ⊢ (Fun (𝑇 D 𝐺) → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥)))
42	16, 39, 40, 41	4syl 19	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥)))
43	21, 42	mpbid 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥))
44		eqid 2739	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	2, 6, 8, 12, 32, 33, 38, 43, 44	dvcobr 25932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))
46		reldv 25856	. . . . . . . 8 ⊢ Rel (𝑇 D (𝐹 ∘ 𝐺))
47	46	releldmi 5891	. . . . . . 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 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑇 D (𝐹 ∘ 𝐺)))
49	30, 48	eqelssd 3936	. . . . 5 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) = 𝑌)
50	49	feq2d 6640	. . . 4 ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ ↔ (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ))
51	25, 50	mpbid 233	. . 3 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ)
52	51	feqmptd 6896	. 2 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥)))
53	15, 11	ssexd 5253	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
54		fvexd 6843	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑆 D 𝐹)‘(𝐺‘𝑥)) ∈ V)
55		fvexd 6843	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D 𝐺)‘𝑥) ∈ V)
56	7	feqmptd 6896	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝑌 ↦ (𝐺‘𝑥)))
57	13, 34	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
58	3	feq2d 6640	. . . . . 6 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ))
59	57, 58	mpbid 233	. . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ)
60	59	feqmptd 6896	. . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑦)))
61		fveq2 6828	. . . 4 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝑆 D 𝐹)‘𝑦) = ((𝑆 D 𝐹)‘(𝐺‘𝑥)))
62	17, 56, 60, 61	fmptco 7072	. . 3 ⊢ (𝜑 → ((𝑆 D 𝐹) ∘ 𝐺) = (𝑥 ∈ 𝑌 ↦ ((𝑆 D 𝐹)‘(𝐺‘𝑥))))
63	15, 39	syl 17	. . . . 5 ⊢ (𝜑 → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ)
64	9	feq2d 6640	. . . . 5 ⊢ (𝜑 → ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ ↔ (𝑇 D 𝐺):𝑌⟶ℂ))
65	63, 64	mpbid 233	. . . 4 ⊢ (𝜑 → (𝑇 D 𝐺):𝑌⟶ℂ)
66	65	feqmptd 6896	. . 3 ⊢ (𝜑 → (𝑇 D 𝐺) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D 𝐺)‘𝑥)))
67	53, 54, 55, 62, 66	offval2 7641	. 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 2784	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 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 Vcvv 3431 ⊆ wss 3883 {cpr 4558 class class class wbr 5073 ↦ cmpt 5154 dom cdm 5619 ∘ ccom 5623 Fun wfun 6480 ⟶wf 6482 ‘cfv 6486 (class class class)co 7357 ∘_f cof 7619 ℂcc 11028 ℝcr 11029 · cmul 11035 TopOpenctopn 17376 ℂ_fldccnfld 21348 D cdv 25849

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 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5200 ax-sep 5219 ax-nul 5229 ax-pow 5295 ax-pr 5363 ax-un 7679 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108 ax-addf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4263 df-if 4456 df-pw 4532 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4840 df-int 4879 df-iun 4924 df-iin 4925 df-br 5074 df-opab 5136 df-mpt 5155 df-tr 5181 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7314 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7621 df-om 7808 df-1st 7932 df-2nd 7933 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9855 df-pnf 11173 df-mnf 11174 df-xr 11175 df-ltxr 11176 df-le 11177 df-sub 11371 df-neg 11372 df-div 11800 df-nn 12167 df-2 12236 df-3 12237 df-4 12238 df-5 12239 df-6 12240 df-7 12241 df-8 12242 df-9 12243 df-n0 12430 df-z 12517 df-dec 12637 df-uz 12781 df-q 12891 df-rp 12935 df-xneg 13055 df-xadd 13056 df-xmul 13057 df-icc 13297 df-fz 13454 df-fzo 13601 df-seq 13956 df-exp 14016 df-hash 14285 df-cj 15053 df-re 15054 df-im 15055 df-sqrt 15189 df-abs 15190 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17172 df-ress 17193 df-plusg 17225 df-mulr 17226 df-starv 17227 df-sca 17228 df-vsca 17229 df-ip 17230 df-tset 17231 df-ple 17232 df-ds 17234 df-unif 17235 df-hom 17236 df-cco 17237 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-xrs 17458 df-qtop 17463 df-imas 17464 df-xps 17466 df-mre 17540 df-mrc 17541 df-acs 17543 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18744 df-mulg 19036 df-cntz 19284 df-cmn 19749 df-psmet 21340 df-xmet 21341 df-met 21342 df-bl 21343 df-mopn 21344 df-fbas 21345 df-fg 21346 df-cnfld 21349 df-top 22878 df-topon 22895 df-topsp 22917 df-bases 22930 df-cld 23003 df-ntr 23004 df-cls 23005 df-nei 23082 df-lp 23120 df-perf 23121 df-cn 23211 df-cnp 23212 df-haus 23299 df-tx 23546 df-hmeo 23739 df-fil 23830 df-fm 23922 df-flim 23923 df-flf 23924 df-xms 24304 df-ms 24305 df-tms 24306 df-cncf 24864 df-limc 25852 df-dv 25853

This theorem is referenced by: dvmptco 25958 dvsinax 46364 dvcosax 46377

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