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Theorem dvconstss 14877

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 5453	. . 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 1006	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ 𝑋)
9		fvconst2g 5773	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑧 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑧) = 𝐴)
10	4, 8, 9	syl2an2r 595	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((𝑋 × {𝐴})‘𝑧) = 𝐴)
11		simpr1 1005	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ 𝑋)
12		fvconst2g 5773	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ 𝑋) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
13	4, 11, 12	syl2an2r 595	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((𝑋 × {𝐴})‘𝑥) = 𝐴)
14	10, 13	oveq12d 5937	. . . . 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 8320	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝐴 − 𝐴) = 0)
17	14, 16	eqtrd 2226	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) = 0)
18	17	oveq1d 5934	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = (0 / (𝑧 − 𝑥)))
19		restsspw 12863	. . . . . . . . . . 11 ⊢ (𝐾 ↾_t 𝑆) ⊆ 𝒫 𝑆
20	2, 19	eqsstri 3212	. . . . . . . . . 10 ⊢ 𝐽 ⊆ 𝒫 𝑆
21	20, 7	sselid 3178	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
22	21	elpwid 3613	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
23		recnprss 14866	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
24	1, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
25	22, 24	sstrd 3190	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑋 ⊆ ℂ)
27	26, 8	sseldd 3181	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ ℂ)
28	26, 11	sseldd 3181	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ ℂ)
29	27, 28	subcld 8332	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) ∈ ℂ)
30		simpr3 1007	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 # 𝑥)
31	27, 28, 30	subap0d 8665	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) # 0)
32	29, 31	div0apd 8808	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (0 / (𝑧 − 𝑥)) = 0)
33	18, 32	eqtrd 2226	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = 0)
34		0cn 8013	. 2 ⊢ 0 ∈ ℂ
35	1, 2, 3, 6, 7, 33, 34	dvidsslem 14872	1 ⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 980 = wceq 1364 ∈ wcel 2164 ⊆ wss 3154 𝒫 cpw 3602 {csn 3619 {cpr 3620 class class class wbr 4030 × cxp 4658 ∘ ccom 4664 ⟶wf 5251 ‘cfv 5255 (class class class)co 5919 ℂcc 7872 ℝcr 7873 0cc0 7874 − cmin 8192 # cap 8602 / cdiv 8693 abscabs 11144 ↾_t crest 12853 MetOpencmopn 14040 D cdv 14834

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4145 ax-sep 4148 ax-nul 4156 ax-pow 4204 ax-pr 4239 ax-un 4465 ax-setind 4570 ax-iinf 4621 ax-cnex 7965 ax-resscn 7966 ax-1cn 7967 ax-1re 7968 ax-icn 7969 ax-addcl 7970 ax-addrcl 7971 ax-mulcl 7972 ax-mulrcl 7973 ax-addcom 7974 ax-mulcom 7975 ax-addass 7976 ax-mulass 7977 ax-distr 7978 ax-i2m1 7979 ax-0lt1 7980 ax-1rid 7981 ax-0id 7982 ax-rnegex 7983 ax-precex 7984 ax-cnre 7985 ax-pre-ltirr 7986 ax-pre-ltwlin 7987 ax-pre-lttrn 7988 ax-pre-apti 7989 ax-pre-ltadd 7990 ax-pre-mulgt0 7991 ax-pre-mulext 7992 ax-arch 7993 ax-caucvg 7994

This theorem depends on definitions: df-bi 117 df-stab 832 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2987 df-csb 3082 df-dif 3156 df-un 3158 df-in 3160 df-ss 3167 df-nul 3448 df-if 3559 df-pw 3604 df-sn 3625 df-pr 3626 df-op 3628 df-uni 3837 df-int 3872 df-iun 3915 df-br 4031 df-opab 4092 df-mpt 4093 df-tr 4129 df-id 4325 df-po 4328 df-iso 4329 df-iord 4398 df-on 4400 df-ilim 4401 df-suc 4403 df-iom 4624 df-xp 4666 df-rel 4667 df-cnv 4668 df-co 4669 df-dm 4670 df-rn 4671 df-res 4672 df-ima 4673 df-iota 5216 df-fun 5257 df-fn 5258 df-f 5259 df-f1 5260 df-fo 5261 df-f1o 5262 df-fv 5263 df-isom 5264 df-riota 5874 df-ov 5922 df-oprab 5923 df-mpo 5924 df-1st 6195 df-2nd 6196 df-recs 6360 df-frec 6446 df-map 6706 df-pm 6707 df-sup 7045 df-inf 7046 df-pnf 8058 df-mnf 8059 df-xr 8060 df-ltxr 8061 df-le 8062 df-sub 8194 df-neg 8195 df-reap 8596 df-ap 8603 df-div 8694 df-inn 8985 df-2 9043 df-3 9044 df-4 9045 df-n0 9244 df-z 9321 df-uz 9596 df-q 9688 df-rp 9723 df-xneg 9841 df-xadd 9842 df-seqfrec 10522 df-exp 10613 df-cj 10989 df-re 10990 df-im 10991 df-rsqrt 11145 df-abs 11146 df-rest 12855 df-topgen 12874 df-psmet 14042 df-xmet 14043 df-met 14044 df-bl 14045 df-mopn 14046 df-top 14177 df-topon 14190 df-bases 14222 df-ntr 14275 df-cn 14367 df-cnp 14368 df-cncf 14750 df-limced 14835 df-dvap 14836

This theorem is referenced by: dvmptfsum 14904

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