dvconstss - Intuitionistic Logic Explorer

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Theorem dvconstss 15380

Description: Derivative of a constant function defined on an open set. (Contributed by Jim Kingdon, 6-Oct-2025.)

**Hypotheses**
Ref	Expression
dvconstss.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvconstss.j	⊢ 𝐽 = (𝐾 ↾_t 𝑆)
dvconstss.k	⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
dvconstss.x	⊢ (𝜑 → 𝑋 ∈ 𝐽)
dvconstss.a	⊢ (𝜑 → 𝐴 ∈ ℂ)

**Assertion**
Ref	Expression
dvconstss	⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))

Proof of Theorem dvconstss Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvconstss.s	. 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		dvconstss.j	. 2 ⊢ 𝐽 = (𝐾 ↾_t 𝑆)
3		dvconstss.k	. 2 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
4		dvconstss.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ)
5		fconst6g 5526	. . 3 ⊢ (𝐴 ∈ ℂ → (𝑋 × {𝐴}):𝑋⟶ℂ)
6	4, 5	syl 14	. 2 ⊢ (𝜑 → (𝑋 × {𝐴}):𝑋⟶ℂ)
7		dvconstss.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐽)
8		simpr2 1028	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ 𝑋)
9		fvconst2g 5857	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑧) = 𝐴)
10	4, 8, 9	syl2an2r 597	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((𝑋 × {𝐴})‘𝑧) = 𝐴)
11		simpr1 1027	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ 𝑋)
12		fvconst2g 5857	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
13	4, 11, 12	syl2an2r 597	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
14	10, 13	oveq12d 6025	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) = (𝐴 − 𝐴))
15	4	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝐴 ∈ ℂ)
16	15	subidd 8453	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝐴 − 𝐴) = 0)
17	14, 16	eqtrd 2262	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) = 0)
18	17	oveq1d 6022	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = (0 / (𝑧 − 𝑥)))
19		restsspw 13290	. . . . . . . . . . 11 ⊢ (𝐾 ↾_t 𝑆) ⊆ 𝒫 𝑆
20	2, 19	eqsstri 3256	. . . . . . . . . 10 ⊢ 𝐽 ⊆ 𝒫 𝑆
21	20, 7	sselid 3222	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
22	21	elpwid 3660	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
23		recnprss 15369	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
24	1, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
25	22, 24	sstrd 3234	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑋 ⊆ ℂ)
27	26, 8	sseldd 3225	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ ℂ)
28	26, 11	sseldd 3225	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ ℂ)
29	27, 28	subcld 8465	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) ∈ ℂ)
30		simpr3 1029	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 # 𝑥)
31	27, 28, 30	subap0d 8799	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) # 0)
32	29, 31	div0apd 8942	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (0 / (𝑧 − 𝑥)) = 0)
33	18, 32	eqtrd 2262	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = 0)
34		0cn 8146	. 2 ⊢ 0 ∈ ℂ
35	1, 2, 3, 6, 7, 33, 34	dvidsslem 15375	1 ⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1002 = wceq 1395 ∈ wcel 2200 ⊆ wss 3197 𝒫 cpw 3649 {csn 3666 {cpr 3667 class class class wbr 4083 × cxp 4717 ∘ ccom 4723 ⟶wf 5314 ‘cfv 5318 (class class class)co 6007 ℂcc 8005 ℝcr 8006 0cc0 8007 − cmin 8325 # cap 8736 / cdiv 8827 abscabs 11516 ↾_t crest 13280 MetOpencmopn 14513 D cdv 15337

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-mulrcl 8106 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-0lt1 8113 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-precex 8117 ax-cnre 8118 ax-pre-ltirr 8119 ax-pre-ltwlin 8120 ax-pre-lttrn 8121 ax-pre-apti 8122 ax-pre-ltadd 8123 ax-pre-mulgt0 8124 ax-pre-mulext 8125 ax-arch 8126 ax-caucvg 8127

This theorem depends on definitions: df-bi 117 df-stab 836 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-isom 5327 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-1st 6292 df-2nd 6293 df-recs 6457 df-frec 6543 df-map 6805 df-pm 6806 df-sup 7159 df-inf 7160 df-pnf 8191 df-mnf 8192 df-xr 8193 df-ltxr 8194 df-le 8195 df-sub 8327 df-neg 8328 df-reap 8730 df-ap 8737 df-div 8828 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-n0 9378 df-z 9455 df-uz 9731 df-q 9823 df-rp 9858 df-xneg 9976 df-xadd 9977 df-seqfrec 10678 df-exp 10769 df-cj 11361 df-re 11362 df-im 11363 df-rsqrt 11517 df-abs 11518 df-rest 13282 df-topgen 13301 df-psmet 14515 df-xmet 14516 df-met 14517 df-bl 14518 df-mopn 14519 df-top 14680 df-topon 14693 df-bases 14725 df-ntr 14778 df-cn 14870 df-cnp 14871 df-cncf 15253 df-limced 15338 df-dvap 15339

This theorem is referenced by: dvmptfsum 15407

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