dvconstss - Intuitionistic Logic Explorer

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Theorem dvconstss 15337

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 5500	. . 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 1009	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ 𝑋)
9		fvconst2g 5826	. . . . . . 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 1008	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ 𝑋)
12		fvconst2g 5826	. . . . . . 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 5992	. . . . 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 8413	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝐴 − 𝐴) = 0)
17	14, 16	eqtrd 2242	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) = 0)
18	17	oveq1d 5989	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = (0 / (𝑧 − 𝑥)))
19		restsspw 13248	. . . . . . . . . . 11 ⊢ (𝐾 ↾_t 𝑆) ⊆ 𝒫 𝑆
20	2, 19	eqsstri 3236	. . . . . . . . . 10 ⊢ 𝐽 ⊆ 𝒫 𝑆
21	20, 7	sselid 3202	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
22	21	elpwid 3640	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
23		recnprss 15326	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
24	1, 23	syl 14	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
25	22, 24	sstrd 3214	. . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
26	25	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑋 ⊆ ℂ)
27	26, 8	sseldd 3205	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 ∈ ℂ)
28	26, 11	sseldd 3205	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑥 ∈ ℂ)
29	27, 28	subcld 8425	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) ∈ ℂ)
30		simpr3 1010	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → 𝑧 # 𝑥)
31	27, 28, 30	subap0d 8759	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (𝑧 − 𝑥) # 0)
32	29, 31	div0apd 8902	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → (0 / (𝑧 − 𝑥)) = 0)
33	18, 32	eqtrd 2242	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑧 ∈ 𝑋 ∧ 𝑧 # 𝑥)) → ((((𝑋 × {𝐴})‘𝑧) − ((𝑋 × {𝐴})‘𝑥)) / (𝑧 − 𝑥)) = 0)
34		0cn 8106	. 2 ⊢ 0 ∈ ℂ
35	1, 2, 3, 6, 7, 33, 34	dvidsslem 15332	1 ⊢ (𝜑 → (𝑆 D (𝑋 × {𝐴})) = (𝑋 × {0}))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 983 = wceq 1375 ∈ wcel 2180 ⊆ wss 3177 𝒫 cpw 3629 {csn 3646 {cpr 3647 class class class wbr 4062 × cxp 4694 ∘ ccom 4700 ⟶wf 5290 ‘cfv 5294 (class class class)co 5974 ℂcc 7965 ℝcr 7966 0cc0 7967 − cmin 8285 # cap 8696 / cdiv 8787 abscabs 11474 ↾_t crest 13238 MetOpencmopn 14470 D cdv 15294

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 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-coll 4178 ax-sep 4181 ax-nul 4189 ax-pow 4237 ax-pr 4272 ax-un 4501 ax-setind 4606 ax-iinf 4657 ax-cnex 8058 ax-resscn 8059 ax-1cn 8060 ax-1re 8061 ax-icn 8062 ax-addcl 8063 ax-addrcl 8064 ax-mulcl 8065 ax-mulrcl 8066 ax-addcom 8067 ax-mulcom 8068 ax-addass 8069 ax-mulass 8070 ax-distr 8071 ax-i2m1 8072 ax-0lt1 8073 ax-1rid 8074 ax-0id 8075 ax-rnegex 8076 ax-precex 8077 ax-cnre 8078 ax-pre-ltirr 8079 ax-pre-ltwlin 8080 ax-pre-lttrn 8081 ax-pre-apti 8082 ax-pre-ltadd 8083 ax-pre-mulgt0 8084 ax-pre-mulext 8085 ax-arch 8086 ax-caucvg 8087

This theorem depends on definitions: df-bi 117 df-stab 835 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-eu 2060 df-mo 2061 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-ral 2493 df-rex 2494 df-reu 2495 df-rmo 2496 df-rab 2497 df-v 2781 df-sbc 3009 df-csb 3105 df-dif 3179 df-un 3181 df-in 3183 df-ss 3190 df-nul 3472 df-if 3583 df-pw 3631 df-sn 3652 df-pr 3653 df-op 3655 df-uni 3868 df-int 3903 df-iun 3946 df-br 4063 df-opab 4125 df-mpt 4126 df-tr 4162 df-id 4361 df-po 4364 df-iso 4365 df-iord 4434 df-on 4436 df-ilim 4437 df-suc 4439 df-iom 4660 df-xp 4702 df-rel 4703 df-cnv 4704 df-co 4705 df-dm 4706 df-rn 4707 df-res 4708 df-ima 4709 df-iota 5254 df-fun 5296 df-fn 5297 df-f 5298 df-f1 5299 df-fo 5300 df-f1o 5301 df-fv 5302 df-isom 5303 df-riota 5927 df-ov 5977 df-oprab 5978 df-mpo 5979 df-1st 6256 df-2nd 6257 df-recs 6421 df-frec 6507 df-map 6767 df-pm 6768 df-sup 7119 df-inf 7120 df-pnf 8151 df-mnf 8152 df-xr 8153 df-ltxr 8154 df-le 8155 df-sub 8287 df-neg 8288 df-reap 8690 df-ap 8697 df-div 8788 df-inn 9079 df-2 9137 df-3 9138 df-4 9139 df-n0 9338 df-z 9415 df-uz 9691 df-q 9783 df-rp 9818 df-xneg 9936 df-xadd 9937 df-seqfrec 10637 df-exp 10728 df-cj 11319 df-re 11320 df-im 11321 df-rsqrt 11475 df-abs 11476 df-rest 13240 df-topgen 13259 df-psmet 14472 df-xmet 14473 df-met 14474 df-bl 14475 df-mopn 14476 df-top 14637 df-topon 14650 df-bases 14682 df-ntr 14735 df-cn 14827 df-cnp 14828 df-cncf 15210 df-limced 15295 df-dvap 15296

This theorem is referenced by: dvmptfsum 15364

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