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Theorem dvconstss 15612

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 5568	. . 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 1031	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ 𝑋)
9		fvconst2g 5900	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑧) = 𝐴)
10	4, 8, 9	syl2an2r 599	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((𝑋 × {𝐴})‘𝑧) = 𝐴)
11		simpr1 1030	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ 𝑋)
12		fvconst2g 5900	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
13	4, 11, 12	syl2an2r 599	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
14	10, 13	oveq12d 6070	. . . . 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 8577	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝐴 − 𝐴) = 0)
17	14, 16	eqtrd 2267	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) = 0)
18	17	oveq1d 6067	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = (0 / (𝑧 − 𝑥)))
19		restsspw 13483	. . . . . . . . . . 11 ⊢ (𝐾 ↾_t 𝑆) ⊆ 𝒫 𝑆
20	2, 19	eqsstri 3272	. . . . . . . . . 10 ⊢ 𝐽 ⊆ 𝒫 𝑆
21	20, 7	sselid 3238	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
22	21	elpwid 3682	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
23		recnprss 15601	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
24	1, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
25	22, 24	sstrd 3250	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑋 ⊆ ℂ)
27	26, 8	sseldd 3241	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ ℂ)
28	26, 11	sseldd 3241	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ ℂ)
29	27, 28	subcld 8589	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) ∈ ℂ)
30		simpr3 1032	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 # 𝑥)
31	27, 28, 30	subap0d 8923	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) # 0)
32	29, 31	div0apd 9066	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (0 / (𝑧 − 𝑥)) = 0)
33	18, 32	eqtrd 2267	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = 0)
34		0cn 8271	. 2 ⊢ 0 ∈ ℂ
35	1, 2, 3, 6, 7, 33, 34	dvidsslem 15607	1 ⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 1005 = wceq 1398 ∈ wcel 2205 ⊆ wss 3213 𝒫 cpw 3671 {csn 3691 {cpr 3692 class class class wbr 4111 × cxp 4749 ∘ ccom 4755 ⟶wf 5350 ‘cfv 5354 (class class class)co 6052 ℂcc 8130 ℝcr 8131 0cc0 8132 − cmin 8449 # cap 8860 / cdiv 8951 abscabs 11690 ↾_t crest 13473 MetOpencmopn 14738 D cdv 15569

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-coll 4227 ax-sep 4230 ax-nul 4238 ax-pow 4289 ax-pr 4324 ax-un 4556 ax-setind 4661 ax-iinf 4712 ax-cnex 8223 ax-resscn 8224 ax-1cn 8225 ax-1re 8226 ax-icn 8227 ax-addcl 8228 ax-addrcl 8229 ax-mulcl 8230 ax-mulrcl 8231 ax-addcom 8232 ax-mulcom 8233 ax-addass 8234 ax-mulass 8235 ax-distr 8236 ax-i2m1 8237 ax-0lt1 8238 ax-1rid 8239 ax-0id 8240 ax-rnegex 8241 ax-precex 8242 ax-cnre 8243 ax-pre-ltirr 8244 ax-pre-ltwlin 8245 ax-pre-lttrn 8246 ax-pre-apti 8247 ax-pre-ltadd 8248 ax-pre-mulgt0 8249 ax-pre-mulext 8250 ax-arch 8251 ax-caucvg 8252

This theorem depends on definitions: df-bi 117 df-stab 839 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-reu 2529 df-rmo 2530 df-rab 2531 df-v 2817 df-sbc 3045 df-csb 3141 df-dif 3215 df-un 3217 df-in 3219 df-ss 3226 df-nul 3511 df-if 3623 df-pw 3673 df-sn 3697 df-pr 3698 df-op 3700 df-uni 3917 df-int 3952 df-iun 3995 df-br 4112 df-opab 4174 df-mpt 4175 df-tr 4211 df-id 4416 df-po 4419 df-iso 4420 df-iord 4489 df-on 4491 df-ilim 4492 df-suc 4494 df-iom 4715 df-xp 4757 df-rel 4758 df-cnv 4759 df-co 4760 df-dm 4761 df-rn 4762 df-res 4763 df-ima 4764 df-iota 5314 df-fun 5356 df-fn 5357 df-f 5358 df-f1 5359 df-fo 5360 df-f1o 5361 df-fv 5362 df-isom 5363 df-riota 6005 df-ov 6055 df-oprab 6056 df-mpo 6057 df-1st 6336 df-2nd 6337 df-recs 6538 df-frec 6624 df-map 6886 df-pm 6887 df-sup 7277 df-inf 7278 df-pnf 8315 df-mnf 8316 df-xr 8317 df-ltxr 8318 df-le 8319 df-sub 8451 df-neg 8452 df-reap 8854 df-ap 8861 df-div 8952 df-inn 9243 df-2 9301 df-3 9302 df-4 9303 df-n0 9502 df-z 9583 df-uz 9860 df-q 9958 df-rp 9993 df-xneg 10111 df-xadd 10112 df-seqfrec 10817 df-exp 10908 df-cj 11535 df-re 11536 df-im 11537 df-rsqrt 11691 df-abs 11692 df-rest 13475 df-topgen 13494 df-psmet 14740 df-xmet 14741 df-met 14742 df-bl 14743 df-mopn 14744 df-top 14912 df-topon 14925 df-bases 14957 df-ntr 15010 df-cn 15102 df-cnp 15103 df-cncf 15485 df-limced 15570 df-dvap 15571

This theorem is referenced by: dvmptfsum 15639

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