dvmptres3 - Metamath Proof Explorer

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Theorem dvmptres3 24553

Description: Function-builder for derivative: restrict a derivative to a subset. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Hypotheses**
Ref	Expression
dvmptres3.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
dvmptres3.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvmptres3.x	⊢ (𝜑 → 𝑋 ∈ 𝐽)
dvmptres3.y	⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌)
dvmptres3.a	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
dvmptres3.b	⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
dvmptres3.d	⊢ (𝜑 → (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))

**Assertion**
Ref	Expression
dvmptres3	⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))

Distinct variable groups: 𝜑,𝑥 𝑥,𝑋 𝑥,𝑌 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥) 𝑆(𝑥) 𝐽(𝑥) 𝑉(𝑥)

**Proof of Theorem dvmptres3**
Step	Hyp	Ref	Expression
1		dvmptres3.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		dvmptres3.a	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
3	2	fmpttd 6879	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ)
4		dvmptres3.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
5		dvmptres3.d	. . . . 5 ⊢ (𝜑 → (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 𝐵))
6	5	dmeqd 5774	. . . 4 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
7		eqid 2821	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
8		dvmptres3.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
9	7, 8	dmmptd 6493	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
10	6, 9	eqtrd 2856	. . 3 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
11		dvmptres3.j	. . . 4 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
12	11	dvres3a 24512	. . 3 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) ∧ (𝑋 ∈ 𝐽 ∧ dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)) → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
13	1, 3, 4, 10, 12	syl22anc 836	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
14		rescom 5879	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋)
15		resres 5866	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
16	14, 15	eqtri 2844	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
17		dvmptres3.y	. . . . . 6 ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌)
18	17	reseq2d 5853	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
19	16, 18	syl5eq 2868	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
20		ffn 6514	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ → (𝑥 ∈ 𝑋 ↦ 𝐴) Fn 𝑋)
21		fnresdm 6466	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) Fn 𝑋 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴))
22	3, 20, 21	3syl 18	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴))
23	22	reseq1d 5852	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆))
24		inss2 4206	. . . . . 6 ⊢ (𝑆 ∩ 𝑋) ⊆ 𝑋
25	17, 24	eqsstrrdi 4022	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
26	25	resmptd 5908	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴))
27	19, 23, 26	3eqtr3d 2864	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐴))
28	27	oveq2d 7172	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))
29		rescom 5879	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋)
30		resres 5866	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
31	29, 30	eqtri 2844	. . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
32	17	reseq2d 5853	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
33	31, 32	syl5eq 2868	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
34	8	ralrimiva 3182	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
35	7	fnmpt 6488	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ 𝐵) Fn 𝑋)
36		fnresdm 6466	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) Fn 𝑋 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐵))
37	34, 35, 36	3syl 18	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐵))
38	37, 5	eqtr4d 2859	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)))
39	38	reseq1d 5852	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
40	25	resmptd 5908	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐵))
41	33, 39, 40	3eqtr3d 2864	. 2 ⊢ (𝜑 → ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐵))
42	13, 28, 41	3eqtr3d 2864	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∩ cin 3935 {cpr 4569 ↦ cmpt 5146 dom cdm 5555 ↾ cres 5557 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 TopOpenctopn 16695 ℂ_fldccnfld 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

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-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-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 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-seq 13371 df-exp 13431 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-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-topgen 16717 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-cnp 21836 df-haus 21923 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-limc 24464 df-dv 24465

This theorem is referenced by: dvmptid 24554 dvmptc 24555 taylthlem1 24961 taylthlem2 24962 pige3ALT 25105 dvcxp1 25321 dvreasin 34995 dvreacos 34996 areacirclem1 34997

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