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Theorem dvconstss 15941

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 5594	. . 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 1035	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ 𝑋)
9		fvconst2g 5932	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑧) = 𝐴)
10	4, 8, 9	syl2an2r 603	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((𝑋 × {𝐴})‘𝑧) = 𝐴)
11		simpr1 1034	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ 𝑋)
12		fvconst2g 5932	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
13	4, 11, 12	syl2an2r 603	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
14	10, 13	oveq12d 6106	. . . . 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 8632	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝐴 − 𝐴) = 0)
17	14, 16	eqtrd 2271	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) = 0)
18	17	oveq1d 6103	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = (0 / (𝑧 − 𝑥)))
19		restsspw 13661	. . . . . . . . . . 11 ⊢ (𝐾 ↾_t 𝑆) ⊆ 𝒫 𝑆
20	2, 19	eqsstri 3280	. . . . . . . . . 10 ⊢ 𝐽 ⊆ 𝒫 𝑆
21	20, 7	sselid 3246	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
22	21	elpwid 3700	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
23		recnprss 15930	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
24	1, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
25	22, 24	sstrd 3258	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑋 ⊆ ℂ)
27	26, 8	sseldd 3249	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ ℂ)
28	26, 11	sseldd 3249	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ ℂ)
29	27, 28	subcld 8644	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) ∈ ℂ)
30		simpr3 1036	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 # 𝑥)
31	27, 28, 30	subap0d 8980	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) # 0)
32	29, 31	div0apd 9125	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (0 / (𝑧 − 𝑥)) = 0)
33	18, 32	eqtrd 2271	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = 0)
34		0cn 8324	. 2 ⊢ 0 ∈ ℂ
35	1, 2, 3, 6, 7, 33, 34	dvidsslem 15936	1 ⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ∧ wa 104 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ⊆ wss 3220 𝒫 cpw 3688 {csn 3709 {cpr 3710 class class class wbr 4131 × cxp 4774 ∘ ccom 4780 ⟶wf 5376 ‘cfv 5380 (class class class)co 6088 ℂcc 8183 ℝcr 8184 0cc0 8185 − cmin 8504 # cap 8917 / cdiv 9010 abscabs 11784 ↾_t crest 13651 MetOpencmopn 14986 D cdv 15898

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

This proof depends on definitions: df-bi 117 df-stab 843 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-isom 5389 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6581 df-frec 6667 df-map 6929 df-pm 6930 df-sup 7330 df-inf 7331 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9654 df-uz 9937 df-q 10035 df-rp 10071 df-xneg 10190 df-xadd 10191 df-seqfrec 10905 df-exp 10996 df-cj 11628 df-re 11629 df-im 11630 df-rsqrt 11785 df-abs 11786 df-rest 13653 df-topgen 13672 df-psmet 14988 df-xmet 14989 df-met 14990 df-bl 14991 df-mopn 14992 df-top 15241 df-topon 15254 df-bases 15286 df-ntr 15339 df-cn 15431 df-cnp 15432 df-cncf 15814 df-limced 15899 df-dvap 15900

This theorem is used by: dvmptfsum 15968

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