dvcof - Metamath Proof Explorer

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Theorem dvcof 25923

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 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋⟶ℂ)
3		dvcof.df	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
4		dvbsss 25874	. . . . . 6 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
5	3, 4	eqsstrrdi 3981	. . . . 5 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
6	5	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑋 ⊆ 𝑆)
7		dvcof.g	. . . . 5 ⊢ (𝜑 → 𝐺:𝑌⟶𝑋)
8	7	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐺:𝑌⟶𝑋)
9		dvcof.dg	. . . . . 6 ⊢ (𝜑 → dom (𝑇 D 𝐺) = 𝑌)
10		dvbsss 25874	. . . . . 6 ⊢ dom (𝑇 D 𝐺) ⊆ 𝑇
11	9, 10	eqsstrrdi 3981	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑇)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑌 ⊆ 𝑇)
13		dvcof.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
14	13	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ {ℝ, ℂ})
15		dvcof.t	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ})
16	15	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑇 ∈ {ℝ, ℂ})
17	7	ffvelcdmda 7038	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ 𝑋)
18	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → dom (𝑆 D 𝐹) = 𝑋)
19	17, 18	eleqtrrd 2840	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥) ∈ dom (𝑆 D 𝐹))
20	9	eleq2d 2823	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ dom (𝑇 D 𝐺) ↔ 𝑥 ∈ 𝑌))
21	20	biimpar 477	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑇 D 𝐺))
22	2, 6, 8, 12, 14, 16, 19, 21	dvco 25922	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥) = (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))
23	22	mpteq2dva 5193	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥))))
24		dvfg 25878	. . . . 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 25876	. . . . . . . 8 ⊢ (𝑇 ∈ {ℝ, ℂ} → 𝑇 ⊆ ℂ)
27	15, 26	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ⊆ ℂ)
28		fco 6694	. . . . . . . 8 ⊢ ((𝐹:𝑋⟶ℂ ∧ 𝐺:𝑌⟶𝑋) → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
29	1, 7, 28	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝐹 ∘ 𝐺):𝑌⟶ℂ)
30	27, 29, 11	dvbss 25873	. . . . . 6 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) ⊆ 𝑌)
31		recnprss 25876	. . . . . . . . 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 25878	. . . . . . . . . 10 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ)
35		ffun 6673	. . . . . . . . . 10 ⊢ ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ → Fun (𝑆 D 𝐹))
36		funfvbrb 7005	. . . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐺‘𝑥)(𝑆 D 𝐹)((𝑆 D 𝐹)‘(𝐺‘𝑥)))
39		dvfg 25878	. . . . . . . . . 10 ⊢ (𝑇 ∈ {ℝ, ℂ} → (𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ)
40		ffun 6673	. . . . . . . . . 10 ⊢ ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ → Fun (𝑇 D 𝐺))
41		funfvbrb 7005	. . . . . . . . . 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 232	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D 𝐺)((𝑇 D 𝐺)‘𝑥))
44		eqid 2737	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
45	2, 6, 8, 12, 32, 33, 38, 43, 44	dvcobr 25920	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑇 D (𝐹 ∘ 𝐺))(((𝑆 D 𝐹)‘(𝐺‘𝑥)) · ((𝑇 D 𝐺)‘𝑥)))
46		reldv 25842	. . . . . . . 8 ⊢ Rel (𝑇 D (𝐹 ∘ 𝐺))
47	46	releldmi 5905	. . . . . . 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 3957	. . . . 5 ⊢ (𝜑 → dom (𝑇 D (𝐹 ∘ 𝐺)) = 𝑌)
50	49	feq2d 6654	. . . 4 ⊢ (𝜑 → ((𝑇 D (𝐹 ∘ 𝐺)):dom (𝑇 D (𝐹 ∘ 𝐺))⟶ℂ ↔ (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ))
51	25, 50	mpbid 232	. . 3 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)):𝑌⟶ℂ)
52	51	feqmptd 6910	. 2 ⊢ (𝜑 → (𝑇 D (𝐹 ∘ 𝐺)) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D (𝐹 ∘ 𝐺))‘𝑥)))
53	15, 11	ssexd 5271	. . 3 ⊢ (𝜑 → 𝑌 ∈ V)
54		fvexd 6857	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑆 D 𝐹)‘(𝐺‘𝑥)) ∈ V)
55		fvexd 6857	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑇 D 𝐺)‘𝑥) ∈ V)
56	7	feqmptd 6910	. . . 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 6654	. . . . . 6 ⊢ (𝜑 → ((𝑆 D 𝐹):dom (𝑆 D 𝐹)⟶ℂ ↔ (𝑆 D 𝐹):𝑋⟶ℂ))
59	57, 58	mpbid 232	. . . . 5 ⊢ (𝜑 → (𝑆 D 𝐹):𝑋⟶ℂ)
60	59	feqmptd 6910	. . . 4 ⊢ (𝜑 → (𝑆 D 𝐹) = (𝑦 ∈ 𝑋 ↦ ((𝑆 D 𝐹)‘𝑦)))
61		fveq2 6842	. . . 4 ⊢ (𝑦 = (𝐺‘𝑥) → ((𝑆 D 𝐹)‘𝑦) = ((𝑆 D 𝐹)‘(𝐺‘𝑥)))
62	17, 56, 60, 61	fmptco 7084	. . 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 6654	. . . . 5 ⊢ (𝜑 → ((𝑇 D 𝐺):dom (𝑇 D 𝐺)⟶ℂ ↔ (𝑇 D 𝐺):𝑌⟶ℂ))
65	63, 64	mpbid 232	. . . 4 ⊢ (𝜑 → (𝑇 D 𝐺):𝑌⟶ℂ)
66	65	feqmptd 6910	. . 3 ⊢ (𝜑 → (𝑇 D 𝐺) = (𝑥 ∈ 𝑌 ↦ ((𝑇 D 𝐺)‘𝑥)))
67	53, 54, 55, 62, 66	offval2 7652	. 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 2782	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3442 ⊆ wss 3903 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5632 ∘ ccom 5636 Fun wfun 6494 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ∘_f cof 7630 ℂcc 11036 ℝcr 11037 · cmul 11043 TopOpenctopn 17353 ℂ_fldccnfld 21324 D cdv 25835

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-mulg 19013 df-cntz 19261 df-cmn 19726 df-psmet 21316 df-xmet 21317 df-met 21318 df-bl 21319 df-mopn 21320 df-fbas 21321 df-fg 21322 df-cnfld 21325 df-top 22853 df-topon 22870 df-topsp 22892 df-bases 22905 df-cld 22978 df-ntr 22979 df-cls 22980 df-nei 23057 df-lp 23095 df-perf 23096 df-cn 23186 df-cnp 23187 df-haus 23274 df-tx 23521 df-hmeo 23714 df-fil 23805 df-fm 23897 df-flim 23898 df-flf 23899 df-xms 24279 df-ms 24280 df-tms 24281 df-cncf 24842 df-limc 25838 df-dv 25839

This theorem is referenced by: dvmptco 25947 dvsinax 46275 dvcosax 46288

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