dvun - Mathbox for Steven Nguyen

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Theorem dvun 42332

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 22985. (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 5948	. . 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 5934	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))))
4	1, 3	eqtr3id 2778	. 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 6692	. . . . 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 4145	. . . . 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 25828	. . . . 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 838	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐴)) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)))
17	6	ffnd 6657	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
18	7	ffnd 6657	. . . . . 6 ⊢ (𝜑 → 𝐺 Fn 𝐵)
19		fnunres1 6598	. . . . . 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 7369	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐴)) = (𝑆 D 𝐹))
22	16, 21	eqtr3d 2766	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)) = (𝑆 D 𝐹))
23	13, 14	dvres 25828	. . . . 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 838	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐵)) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)))
25		fnunres2 6599	. . . . . 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 7369	. . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐵)) = (𝑆 D 𝐺))
28	24, 27	eqtr3d 2766	. . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)) = (𝑆 D 𝐺))
29	22, 28	uneq12d 4122	. 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 25828	. . . 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 838	. . 3 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))))
32	9	ffnd 6657	. . . . 5 ⊢ (𝜑 → (𝐹 ∪ 𝐺) Fn (𝐴 ∪ 𝐵))
33		fnresdm 6605	. . . . 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 7369	. . 3 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵))) = (𝑆 D (𝐹 ∪ 𝐺)))
36	31, 35	eqtr3d 2766	. 2 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))) = (𝑆 D (𝐹 ∪ 𝐺)))
37	4, 29, 36	3eqtr3d 2772	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 3903 ∩ cin 3904 ⊆ wss 3905 ∅c0 4286 ↾ cres 5625 Fn wfn 6481 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 ℂcc 11026 ↾_t crest 17342 TopOpenctopn 17343 ℂ_fldccnfld 21279 intcnt 22920 D cdv 25780

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8632 df-map 8762 df-pm 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fi 9320 df-sup 9351 df-inf 9352 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12610 df-uz 12754 df-q 12868 df-rp 12912 df-xneg 13032 df-xadd 13033 df-xmul 13034 df-fz 13429 df-seq 13927 df-exp 13987 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17139 df-plusg 17192 df-mulr 17193 df-starv 17194 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-rest 17344 df-topn 17345 df-topgen 17365 df-psmet 21271 df-xmet 21272 df-met 21273 df-bl 21274 df-mopn 21275 df-cnfld 21280 df-top 22797 df-topon 22814 df-topsp 22836 df-bases 22849 df-cld 22922 df-ntr 22923 df-cls 22924 df-cnp 23131 df-xms 24224 df-ms 24225 df-limc 25783 df-dv 25784

This theorem is referenced by: redvmptabs 42333

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