dvun - Mathbox for Steven Nguyen

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Theorem dvun 42843

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 23072. (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 5952	. . 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 5938	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))))
4	1, 3	eqtr3id 2789	. 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 6698	. . . . 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 4128	. . . . 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 25903	. . . . 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 844	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐴)) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)))
17	6	ffnd 6663	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
18	7	ffnd 6663	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐵)
19		fnunres1 6604	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)
20	17, 18, 8, 19	syl3anc 1379	. . . . 5 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹)
21	20	oveq2d 7379	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐴)) = (𝑆 D 𝐹))
22	16, 21	eqtr3d 2777	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)) = (𝑆 D 𝐹))
23	13, 14	dvres 25903	. . . . 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 844	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐵)) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)))
25		fnunres2 6605	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
26	17, 18, 8, 25	syl3anc 1379	. . . . 5 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺)
27	26	oveq2d 7379	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐵)) = (𝑆 D 𝐺))
28	24, 27	eqtr3d 2777	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)) = (𝑆 D 𝐺))
29	22, 28	uneq12d 4106	. 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 25903	. . . 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 844	. . 3 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))))
32	9	ffnd 6663	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
33		fnresdm 6611	. . . . 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 7379	. . 3 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵))) = (𝑆 D (𝐹 ∪ 𝐺)))
36	31, 35	eqtr3d 2777	. 2 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))) = (𝑆 D (𝐹 ∪ 𝐺)))
37	4, 29, 36	3eqtr3d 2783	1 ⊢ (𝜑 → ((𝑆 D 𝐹) ∪ (𝑆 D 𝐺)) = (𝑆 D (𝐹 ∪ 𝐺)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1547 ∪ cun 3888 ∩ cin 3889 ⊆ wss 3890 ∅c0 4268 ↾ cres 5627 Fn wfn 6487 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℂcc 11034 ↾_t crest 17381 TopOpenctopn 17382 ℂ_fldccnfld 21354 intcnt 23007 D cdv 25855

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-om 7814 df-1st 7938 df-2nd 7939 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-pm 8773 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fi 9321 df-sup 9352 df-inf 9353 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-fz 13460 df-seq 13962 df-exp 14022 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-struct 17115 df-slot 17150 df-ndx 17162 df-base 17178 df-plusg 17231 df-mulr 17232 df-starv 17233 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-rest 17383 df-topn 17384 df-topgen 17404 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-ntr 23010 df-cls 23011 df-cnp 23218 df-xms 24310 df-ms 24311 df-limc 25858 df-dv 25859

This theorem is referenced by: redvmptabs 42844

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