dvmptres3 - Metamath Proof Explorer

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Theorem dvmptres3 26269

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 7113	. . 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 5887	. . . 4 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = dom (𝑥 ∈ 𝑋 ↦ 𝐵))
7		eqid 2761	. . . . 5 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐵) = (𝑥 ∈ 𝑋 ↦ 𝐵)
8		dvmptres3.b	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐵 ∈ 𝑉)
9	7, 8	dmmptd 6682	. . . 4 ⊢ (𝜑 → dom (𝑥 ∈ 𝑋 ↦ 𝐵) = 𝑋)
10	6, 9	eqtrd 2796	. . 3 ⊢ (𝜑 → dom (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) = 𝑋)
11		dvmptres3.j	. . . 4 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
12	11	dvres3a 26227	. . 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 5993	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋)
15		resres 5983	. . . . . 6 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
16	14, 15	eqtri 2784	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋))
17		dvmptres3.y	. . . . . 6 ⊢ (𝜑 → (𝑆 ∩ 𝑋) = 𝑌)
18	17	reseq2d 5970	. . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
19	16, 18	eqtrid 2808	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))
20		ffn 6707	. . . . . 6 ⊢ ((𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ → (𝑥 ∈ 𝑋 ↦ 𝐴) Fn 𝑋)
21		fnresdm 6656	. . . . . 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 5969	. . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆))
24		inss2 4183	. . . . . 6 ⊢ (𝑆 ∩ 𝑋) ⊆ 𝑋
25	17, 24	eqsstrrdi 3976	. . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑋)
26	25	resmptd 6032	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴))
27	19, 23, 26	3eqtr3d 2804	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐴))
28	27	oveq2d 7434	. 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑆)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴)))
29		rescom 5993	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋)
30		resres 5983	. . . . 5 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑆) ↾ 𝑋) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
31	29, 30	eqtri 2784	. . . 4 ⊢ (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋))
32	17	reseq2d 5970	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ (𝑆 ∩ 𝑋)) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
33	31, 32	eqtrid 2808	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌))
34	8	ralrimiva 3155	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉)
35	7	fnmpt 6677	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝑋 ↦ 𝐵) Fn 𝑋)
36		fnresdm 6656	. . . . . 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 2799	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) = (ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)))
39	38	reseq1d 5969	. . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑋) ↾ 𝑆) = ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆))
40	25	resmptd 6032	. . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐵) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐵))
41	33, 39, 40	3eqtr3d 2804	. 2 ⊢ (𝜑 → ((ℂ D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ 𝑆) = (𝑥 ∈ 𝑌 ↦ 𝐵))
42	13, 28, 41	3eqtr3d 2804	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 2145 ∀wral 3077 ∩ cin 3898 {cpr 4586 ↦ cmpt 5186 dom cdm 5651 ↾ cres 5653 Fn wfn 6532 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ℂcc 11191 ℝcr 11192 TopOpenctopn 17585 ℂ_fldccnfld 21671 D cdv 26176

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271

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 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-om 7876 df-1st 7999 df-2nd 8000 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-er 8710 df-map 8842 df-pm 8843 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fi 9396 df-sup 9427 df-inf 9428 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-icc 13476 df-fz 13633 df-seq 14138 df-exp 14198 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-struct 17318 df-slot 17353 df-ndx 17365 df-base 17381 df-plusg 17434 df-mulr 17435 df-starv 17436 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-rest 17586 df-topn 17587 df-topgen 17607 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cnp 23539 df-haus 23626 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-xms 24632 df-ms 24633 df-limc 26179 df-dv 26180

This theorem is used by: dvmptid 26270 dvmptc 26271 taylthlem1 26693 taylthlem2 26694 pige3ALT 26841 dvcxp1 27061 dvreasin 38604 dvreacos 38605 areacirclem1 38606 readvrec2 43392 readvcot 43395

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