eldvap - Intuitionistic Logic Explorer

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Theorem eldvap 14918

Description: The differentiable predicate. A function 𝐹 is differentiable at 𝐵 with derivative 𝐶 iff 𝐹 is defined in a neighborhood of 𝐵 and the difference quotient has limit 𝐶 at 𝐵. (Contributed by Mario Carneiro, 7-Aug-2014.) (Revised by Jim Kingdon, 27-Jun-2023.)

**Hypotheses**
Ref	Expression
dvval.t	⊢ 𝑇 = (𝐾 ↾_t 𝑆)
dvval.k	⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
eldvap.g	⊢ 𝐺 = (𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝐵} ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))
eldv.s	⊢ (𝜑 → 𝑆 ⊆ ℂ)
eldv.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
eldv.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)

**Assertion**
Ref	Expression
eldvap	⊢ (𝜑 → (𝐵(𝑆 D 𝐹)𝐶 ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 lim_ℂ 𝐵))))

Distinct variable groups: 𝑤,𝐴,𝑧 𝑤,𝐹,𝑧 𝑤,𝑆,𝑧 𝑧,𝐵,𝑤 Allowed substitution hints: 𝜑(𝑧,𝑤) 𝐶(𝑧,𝑤) 𝑇(𝑧,𝑤) 𝐺(𝑧,𝑤) 𝐾(𝑧,𝑤)

Proof of Theorem eldvap Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldv.s	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
2		eldv.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3		eldv.a	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
4		dvval.t	. . . . . 6 ⊢ 𝑇 = (𝐾 ↾_t 𝑆)
5		dvval.k	. . . . . 6 ⊢ 𝐾 = (MetOpen‘(abs ∘ − ))
6	4, 5	dvfvalap 14917	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆) → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝑇)‘𝐴) × ℂ)))
7	1, 2, 3, 6	syl3anc 1249	. . . 4 ⊢ (𝜑 → ((𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)) ∧ (𝑆 D 𝐹) ⊆ (((int‘𝑇)‘𝐴) × ℂ)))
8	7	simpld 112	. . 3 ⊢ (𝜑 → (𝑆 D 𝐹) = ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)))
9	8	eleq2d 2266	. 2 ⊢ (𝜑 → (⟨𝐵, 𝐶⟩ ∈ (𝑆 D 𝐹) ↔ ⟨𝐵, 𝐶⟩ ∈ ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥))))
10		df-br 4034	. . 3 ⊢ (𝐵(𝑆 D 𝐹)𝐶 ↔ ⟨𝐵, 𝐶⟩ ∈ (𝑆 D 𝐹))
11	10	bicomi 132	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ (𝑆 D 𝐹) ↔ 𝐵(𝑆 D 𝐹)𝐶)
12		breq2 4037	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑤 # 𝑥 ↔ 𝑤 # 𝐵))
13	12	rabbidv 2752	. . . . . 6 ⊢ (𝑥 = 𝐵 → {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} = {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝐵})
14		fveq2 5558	. . . . . . . 8 ⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵))
15	14	oveq2d 5938	. . . . . . 7 ⊢ (𝑥 = 𝐵 → ((𝐹‘𝑧) − (𝐹‘𝑥)) = ((𝐹‘𝑧) − (𝐹‘𝐵)))
16		oveq2 5930	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑧 − 𝑥) = (𝑧 − 𝐵))
17	15, 16	oveq12d 5940	. . . . . 6 ⊢ (𝑥 = 𝐵 → (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥)) = (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))
18	13, 17	mpteq12dv 4115	. . . . 5 ⊢ (𝑥 = 𝐵 → (𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = (𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝐵} ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵))))
19		eldvap.g	. . . . 5 ⊢ 𝐺 = (𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝐵} ↦ (((𝐹‘𝑧) − (𝐹‘𝐵)) / (𝑧 − 𝐵)))
20	18, 19	eqtr4di 2247	. . . 4 ⊢ (𝑥 = 𝐵 → (𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) = 𝐺)
21		id 19	. . . 4 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
22	20, 21	oveq12d 5940	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥) = (𝐺 lim_ℂ 𝐵))
23	22	opeliunxp2 4806	. 2 ⊢ (⟨𝐵, 𝐶⟩ ∈ ∪ 𝑥 ∈ ((int‘𝑇)‘𝐴)({𝑥} × ((𝑧 ∈ {𝑤 ∈ 𝐴 ∣ 𝑤 # 𝑥} ↦ (((𝐹‘𝑧) − (𝐹‘𝑥)) / (𝑧 − 𝑥))) lim_ℂ 𝑥)) ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 lim_ℂ 𝐵)))
24	9, 11, 23	3bitr3g 222	1 ⊢ (𝜑 → (𝐵(𝑆 D 𝐹)𝐶 ↔ (𝐵 ∈ ((int‘𝑇)‘𝐴) ∧ 𝐶 ∈ (𝐺 lim_ℂ 𝐵))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2167 {crab 2479 ⊆ wss 3157 {csn 3622 ⟨cop 3625 ∪ ciun 3916 class class class wbr 4033 ↦ cmpt 4094 × cxp 4661 ∘ ccom 4667 ⟶wf 5254 ‘cfv 5258 (class class class)co 5922 ℂcc 7877 − cmin 8197 # cap 8608 / cdiv 8699 abscabs 11162 ↾_t crest 12910 MetOpencmopn 14097 intcnt 14329 lim_ℂ climc 14890 D cdv 14891

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997 ax-arch 7998 ax-caucvg 7999

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-po 4331 df-iso 4332 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-isom 5267 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-map 6709 df-pm 6710 df-sup 7050 df-inf 7051 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-2 9049 df-3 9050 df-4 9051 df-n0 9250 df-z 9327 df-uz 9602 df-q 9694 df-rp 9729 df-xneg 9847 df-xadd 9848 df-seqfrec 10540 df-exp 10631 df-cj 11007 df-re 11008 df-im 11009 df-rsqrt 11163 df-abs 11164 df-rest 12912 df-topgen 12931 df-psmet 14099 df-xmet 14100 df-met 14101 df-bl 14102 df-mopn 14103 df-top 14234 df-topon 14247 df-bases 14279 df-ntr 14332 df-limced 14892 df-dvap 14893

This theorem is referenced by: dvcl 14919 dvfgg 14924 dvidlemap 14927 dvidrelem 14928 dvidsslem 14929 dvcnp2cntop 14935 dvaddxxbr 14937 dvmulxxbr 14938 dvcoapbr 14943 dvcjbr 14944 dvrecap 14949 dveflem 14962

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