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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvun | Structured version Visualization version GIF version |
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 23105. (Contributed by SN, 30-Sep-2025.) |
Ref | Expression |
---|---|
dvun.j | ⊢ 𝐽 = (𝐾 ↾t 𝑆) |
dvun.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
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))) |
Ref | Expression |
---|---|
dvun | ⊢ (𝜑 → ((𝑆 D 𝐹) ∪ (𝑆 D 𝐺)) = (𝑆 D (𝐹 ∪ 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resundi 6013 | . . 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 5999 | . . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ (((int‘𝐽)‘𝐴) ∪ ((int‘𝐽)‘𝐵))) = ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵)))) |
4 | 1, 3 | eqtr3id 2788 | . 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 4201 | . . . . 5 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ⊆ 𝑆) |
13 | dvun.k | . . . . . 6 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
14 | dvun.j | . . . . . 6 ⊢ 𝐽 = (𝐾 ↾t 𝑆) | |
15 | 13, 14 | dvres 25960 | . . . . 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 1370 | . . . . 5 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) ↾ 𝐴) = 𝐹) |
21 | 20 | oveq2d 7446 | . . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐴)) = (𝑆 D 𝐹)) |
22 | 16, 21 | eqtr3d 2776 | . . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐴)) = (𝑆 D 𝐹)) |
23 | 13, 14 | dvres 25960 | . . . . 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 1370 | . . . . 5 ⊢ (𝜑 → ((𝐹 ∪ 𝐺) ↾ 𝐵) = 𝐺) |
27 | 26 | oveq2d 7446 | . . . 4 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ 𝐵)) = (𝑆 D 𝐺)) |
28 | 24, 27 | eqtr3d 2776 | . . 3 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘𝐵)) = (𝑆 D 𝐺)) |
29 | 22, 28 | uneq12d 4178 | . 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 25960 | . . . 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 7446 | . . 3 ⊢ (𝜑 → (𝑆 D ((𝐹 ∪ 𝐺) ↾ (𝐴 ∪ 𝐵))) = (𝑆 D (𝐹 ∪ 𝐺))) |
36 | 31, 35 | eqtr3d 2776 | . 2 ⊢ (𝜑 → ((𝑆 D (𝐹 ∪ 𝐺)) ↾ ((int‘𝐽)‘(𝐴 ∪ 𝐵))) = (𝑆 D (𝐹 ∪ 𝐺))) |
37 | 4, 29, 36 | 3eqtr3d 2782 | 1 ⊢ (𝜑 → ((𝑆 D 𝐹) ∪ (𝑆 D 𝐺)) = (𝑆 D (𝐹 ∪ 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∪ cun 3960 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 ↾ cres 5690 Fn wfn 6557 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ↾t crest 17466 TopOpenctopn 17467 ℂfldccnfld 21381 intcnt 23040 D cdv 25912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fi 9448 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17468 df-topn 17469 df-topgen 17489 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-cnp 23251 df-xms 24345 df-ms 24346 df-limc 25915 df-dv 25916 |
This theorem is referenced by: redvmptabs 42368 |
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