dvun - Mathbox for Steven Nguyen

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Theorem dvun 42389

Description: Condition for the union of the derivatives of two disjoint functions to be equal to the derivative of the union of the two functions. If 𝐴 and 𝐵 are open sets, this condition (dvun.n) is satisfied by isopn3i 23090. (Contributed by SN, 30-Sep-2025.)

**Hypotheses**
Ref	Expression
dvun.j	⊢ 𝐽 = (𝐾 ↾_t 𝑆)
dvun.k	⊢ 𝐾 = (TopOpen‘ℂ_fld)
dvun.s	⊢ (𝜑 → 𝑆 ⊆ ℂ)
dvun.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
dvun.g	⊢ (𝜑 → 𝐺:𝐵⟶ℂ)
dvun.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)
dvun.b	⊢ (𝜑 → 𝐵 ⊆ 𝑆)
dvun.d	⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
dvun.n	⊢ (𝜑 → (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵)) = ((int‘𝐽)‘(𝐴 ∪ 𝐵)))

**Assertion**
Ref	Expression
dvun	⊢ (𝜑 → ((𝑆 D 𝐹) ∪ (𝑆 D 𝐺)) = (𝑆 D (𝐹 ∪ 𝐺)))

**Proof of Theorem dvun**
Step	Hyp	Ref	Expression
1		resundi 6011	. . 3 ⊢ ((𝑆 D (𝐹 ∪ 𝐺)) ↾ (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵))) = (((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)) ∪ ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)))
2		dvun.n	. . . 4 ⊢ (𝜑 → (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵)) = ((int‘𝐽)‘(𝐴 ∪ 𝐵)))
3	2	reseq2d 5997	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))))
4	1, 3	eqtr3id 2791	. 2 ⊢ (𝜑 → (((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)) ∪ ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))))
5		dvun.s	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
6		dvun.f	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
7		dvun.g	. . . . . 6 ⊢ (𝜑 → 𝐺:𝐵⟶ℂ)
8		dvun.d	. . . . . 6 ⊢ (𝜑 → (𝐴 ∩ 𝐵) = ∅)
9	6, 7, 8	fun2d 6772	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶ℂ)
10		dvun.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
11		dvun.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑆)
12	10, 11	unssd 4192	. . . . 5 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝑆)
13		dvun.k	. . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂ_fld)
14		dvun.j	. . . . . 6 ⊢ 𝐽 = (𝐾 ↾_t 𝑆)
15	13, 14	dvres 25946	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶ℂ) ∧ ((𝐴 ∪ 𝐵) ⊆ 𝑆 ∧ 𝐴 ⊆ 𝑆)) → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐴)) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)))
16	5, 9, 12, 10, 15	syl22anc 839	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐴)) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)))
17	6	ffnd 6737	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
18	7	ffnd 6737	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐵)
19		fnunres1 6680	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)
20	17, 18, 8, 19	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)
21	20	oveq2d 7447	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐴)) = (𝑆 D 𝐹))
22	16, 21	eqtr3d 2779	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)) = (𝑆 D 𝐹))
23	13, 14	dvres 25946	. . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶ℂ) ∧ ((𝐴 ∪ 𝐵) ⊆ 𝑆 ∧ 𝐵 ⊆ 𝑆)) → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐵)) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)))
24	5, 9, 12, 11, 23	syl22anc 839	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐵)) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)))
25		fnunres2 6681	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
26	17, 18, 8, 25	syl3anc 1373	. . . . 5 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
27	26	oveq2d 7447	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐵)) = (𝑆 D 𝐺))
28	24, 27	eqtr3d 2779	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)) = (𝑆 D 𝐺))
29	22, 28	uneq12d 4169	. 2 ⊢ (𝜑 → (((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)) ∪ ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵))) = ((𝑆 D 𝐹) ∪ (𝑆 D 𝐺)))
30	13, 14	dvres 25946	. . . 4 ⊢ (((𝑆 ⊆ ℂ ∧ (𝐹 ∪ 𝐺):(𝐴 ∪ 𝐵)⟶ℂ) ∧ ((𝐴 ∪ 𝐵) ⊆ 𝑆 ∧ (𝐴 ∪ 𝐵) ⊆ 𝑆)) → (𝑆 D ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))))
31	5, 9, 12, 12, 30	syl22anc 839	. . 3 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))))
32	9	ffnd 6737	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
33		fnresdm 6687	. . . . 5 ⊢ ((𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵) → ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵)) = (𝐹 ∪ 𝐺))
34	32, 33	syl 17	. . . 4 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵)) = (𝐹 ∪ 𝐺))
35	34	oveq2d 7447	. . 3 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵))) = (𝑆 D (𝐹 ∪ 𝐺)))
36	31, 35	eqtr3d 2779	. 2 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))) = (𝑆 D (𝐹 ∪ 𝐺)))
37	4, 29, 36	3eqtr3d 2785	1 ⊢ (𝜑 → ((𝑆 D 𝐹) ∪ (𝑆 D 𝐺)) = (𝑆 D (𝐹 ∪ 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1540 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ↾ cres 5687 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ↾_t crest 17465 TopOpenctopn 17466 ℂ_fldccnfld 21364 intcnt 23025 D cdv 25898

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fi 9451 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-fz 13548 df-seq 14043 df-exp 14103 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17467 df-topn 17468 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-cnp 23236 df-xms 24330 df-ms 24331 df-limc 25901 df-dv 25902

This theorem is referenced by: redvmptabs 42390

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