dvmptres3 - Metamath Proof Explorer

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Theorem dvmptres3 26331

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 7147	. . 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 5899	. . . 4 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
7		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
8		dvmptres3.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
9	7, 8	dmmptd 6714	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
10	6, 9	eqtrd 2798	. . 3 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
11		dvmptres3.j	. . . 4 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
12	11	dvres3a 26289	. . 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 852	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
14		rescom 6009	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋)
15		resres 5999	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
16	14, 15	eqtri 2786	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
17		dvmptres3.y	. . . . . 6 ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌)
18	17	reseq2d 5986	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
19	16, 18	eqtrid 2810	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
20		ffn 6739	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ → (𝑥 ∈ 𝑋 ↦ 𝐴) Fn 𝑋)
21		fnresdm 6688	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴) Fn 𝑋 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴))
22	3, 20, 21	3syl 19	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴))
23	22	reseq1d 5985	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆))
24		inss2 4137	. . . . . 6 ⊢ (𝑆 ∩ 𝑋) ⊆ 𝑋
25	17, 24	eqsstrrdi 3958	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
26	25	resmptd 6050	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴))
27	19, 23, 26	3eqtr3d 2806	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐴))
28	27	oveq2d 7467	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))
29		rescom 6009	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋)
30		resres 5999	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
31	29, 30	eqtri 2786	. . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
32	17	reseq2d 5986	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
33	31, 32	eqtrid 2810	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
34	8	ralrimiva 3158	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
35	7	fnmpt 6709	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ 𝐵) Fn 𝑋)
36		fnresdm 6688	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐵) Fn 𝑋 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐵))
37	34, 35, 36	3syl 19	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐵))
38	37, 5	eqtr4d 2801	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)))
39	38	reseq1d 5985	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
40	25	resmptd 6050	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐵))
41	33, 39, 40	3eqtr3d 2806	. 2 ⊢ (𝜑 → ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐵))
42	13, 28, 41	3eqtr3d 2806	1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)) = (𝑥 ∈ 𝑌 ↦ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ∩ cin 4092 {cpr 4590 ↦ cmpt 5192 dom cdm 5658 ↾ cres 5660 Fn wfn 6561 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 TopOpenctopn 17642 ℂ_fldccnfld 21729 D cdv 26238

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-om 7912 df-1st 8035 df-2nd 8036 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-er 8755 df-map 8887 df-pm 8888 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fi 9440 df-sup 9471 df-inf 9472 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-icc 13523 df-fz 13680 df-seq 14185 df-exp 14245 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-struct 17369 df-slot 17404 df-ndx 17416 df-base 17432 df-plusg 17486 df-mulr 17487 df-starv 17488 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-euf 17499 df-rest 17643 df-topn 17644 df-topgen 17664 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cnp 23601 df-haus 23688 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-limc 26241 df-dv 26242

This theorem is used by: dvmptid 26332 dvmptc 26333 taylthlem1 26757 taylthlem2 26758 pige3ALT 26905 dvcxp1 27125 dvreasin 38892 dvreacos 38893 areacirclem1 38894 readvrec2 43678 readvcot 43681

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