dvmptco - Metamath Proof Explorer

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Theorem dvmptco 24569

Description: Function-builder for derivative, chain rule. (Contributed by Mario Carneiro, 1-Sep-2014.)

**Hypotheses**
Ref	Expression
dvmptco.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvmptco.t	⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ})
dvmptco.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
dvmptco.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
dvmptco.c	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ ℂ)
dvmptco.d	⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ 𝑊)
dvmptco.da	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
dvmptco.dc	⊢ (𝜑 → (𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑦 ∈ 𝑌 ↦ 𝐷))
dvmptco.e	⊢ (𝑦 = 𝐴 → 𝐶 = 𝐸)
dvmptco.f	⊢ (𝑦 = 𝐴 → 𝐷 = 𝐹)

**Assertion**
Ref	Expression
dvmptco	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐸)) = (𝑥 ∈ 𝑋 ↦ (𝐹 · 𝐵)))

Distinct variable groups: 𝑦,𝐴 𝑥,𝐶 𝑥,𝐷 𝑦,𝐸 𝑦,𝐹 𝑦,𝑇 𝑥,𝑉 𝑥,𝑦,𝜑 𝑦,𝑊 𝑥,𝑋 𝑥,𝑌,𝑦 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥,𝑦) 𝐶(𝑦) 𝐷(𝑦) 𝑆(𝑥,𝑦) 𝑇(𝑥) 𝐸(𝑥) 𝐹(𝑥) 𝑉(𝑦) 𝑊(𝑥) 𝑋(𝑦)

**Proof of Theorem dvmptco**
Step	Hyp	Ref	Expression
1		dvmptco.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ {ℝ, ℂ})
2		dvmptco.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
3		dvmptco.c	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐶 ∈ ℂ)
4	3	fmpttd 6879	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐶):𝑌⟶ℂ)
5		dvmptco.a	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑌)
6	5	fmpttd 6879	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌)
7		dvmptco.dc	. . . . 5 ⊢ (𝜑 → (𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) = (𝑦 ∈ 𝑌 ↦ 𝐷))
8	7	dmeqd 5774	. . . 4 ⊢ (𝜑 → dom (𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) = dom (𝑦 ∈ 𝑌 ↦ 𝐷))
9		dvmptco.d	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ 𝑊)
10	9	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ 𝑌 𝐷 ∈ 𝑊)
11		dmmptg 6096	. . . . 5 ⊢ (∀𝑦 ∈ 𝑌 𝐷 ∈ 𝑊 → dom (𝑦 ∈ 𝑌 ↦ 𝐷) = 𝑌)
12	10, 11	syl 17	. . . 4 ⊢ (𝜑 → dom (𝑦 ∈ 𝑌 ↦ 𝐷) = 𝑌)
13	8, 12	eqtrd 2856	. . 3 ⊢ (𝜑 → dom (𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) = 𝑌)
14		dvmptco.da	. . . . 5 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
15	14	dmeqd 5774	. . . 4 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
16		dvmptco.b	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
17	16	ralrimiva 3182	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
18		dmmptg 6096	. . . . 5 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
19	17, 18	syl 17	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
20	15, 19	eqtrd 2856	. . 3 ⊢ (𝜑 → dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
21	1, 2, 4, 6, 13, 20	dvcof 24545	. 2 ⊢ (𝜑 → (𝑆 D ((𝑦 ∈ 𝑌 ↦ 𝐶) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (((𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_f · (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))))
22		eqidd 2822	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴))
23		eqidd 2822	. . . 4 ⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 𝐶) = (𝑦 ∈ 𝑌 ↦ 𝐶))
24		dvmptco.e	. . . 4 ⊢ (𝑦 = 𝐴 → 𝐶 = 𝐸)
25	5, 22, 23, 24	fmptco 6891	. . 3 ⊢ (𝜑 → ((𝑦 ∈ 𝑌 ↦ 𝐶) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐸))
26	25	oveq2d 7172	. 2 ⊢ (𝜑 → (𝑆 D ((𝑦 ∈ 𝑌 ↦ 𝐶) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴))) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐸)))
27		ovex 7189	. . . . 5 ⊢ (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ V
28	27	dmex 7616	. . . 4 ⊢ dom (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ∈ V
29	20, 28	eqeltrrdi 2922	. . 3 ⊢ (𝜑 → 𝑋 ∈ V)
30	1, 3, 9, 7	dvmptcl 24556	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 𝐷 ∈ ℂ)
31	7, 30	fmpt3d 6880	. . . . . 6 ⊢ (𝜑 → (𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)):𝑌⟶ℂ)
32		fco 6531	. . . . . 6 ⊢ (((𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)):𝑌⟶ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶𝑌) → ((𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶ℂ)
33	31, 6, 32	syl2anc 586	. . . . 5 ⊢ (𝜑 → ((𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶ℂ)
34		dvmptco.f	. . . . . . 7 ⊢ (𝑦 = 𝐴 → 𝐷 = 𝐹)
35	5, 22, 7, 34	fmptco 6891	. . . . . 6 ⊢ (𝜑 → ((𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐹))
36	35	feq1d 6499	. . . . 5 ⊢ (𝜑 → (((𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)):𝑋⟶ℂ ↔ (𝑥 ∈ 𝑋 ↦ 𝐹):𝑋⟶ℂ))
37	33, 36	mpbid 234	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐹):𝑋⟶ℂ)
38	37	fvmptelrn 6877	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐹 ∈ ℂ)
39	29, 38, 16, 35, 14	offval2 7426	. 2 ⊢ (𝜑 → (((𝑇 D (𝑦 ∈ 𝑌 ↦ 𝐶)) ∘ (𝑥 ∈ 𝑋 ↦ 𝐴)) ∘_f · (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))) = (𝑥 ∈ 𝑋 ↦ (𝐹 · 𝐵)))
40	21, 26, 39	3eqtr3d 2864	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐸)) = (𝑥 ∈ 𝑋 ↦ (𝐹 · 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 Vcvv 3494 {cpr 4569 ↦ cmpt 5146 dom cdm 5555 ∘ ccom 5559 ⟶wf 6351 (class class class)co 7156 ∘_f cof 7407 ℂcc 10535 ℝcr 10536 · cmul 10542 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: dvrecg 24570 dvexp3 24575 dvsincos 24578 dvlipcn 24591 lhop2 24612 itgsubstlem 24645 dvtaylp 24958 taylthlem2 24962 pige3ALT 25105 advlogexp 25238 logtayl 25243 dvcxp1 25321 dvcxp2 25322 dvcncxp1 25324 loglesqrt 25339 dvatan 25513 lgamgulmlem2 25607 logdivsum 26109 log2sumbnd 26120 itgexpif 31877 dvtan 34957 dvasin 34993 areacirclem1 34997 expgrowthi 40685 expgrowth 40687 binomcxplemdvbinom 40705 dvsinexp 42215 dvxpaek 42245 fourierdlem28 42440 fourierdlem39 42451 fourierdlem56 42467 fourierdlem60 42471 fourierdlem61 42472 etransclem46 42585

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