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Mirrors > Home > MPE Home > Th. List > dvmptntr | Structured version Visualization version GIF version |
Description: Function-builder for derivative: expand the function from an open set to its closure. (Contributed by Mario Carneiro, 1-Sep-2014.) (Revised by Mario Carneiro, 11-Feb-2015.) |
Ref | Expression |
---|---|
dvmptntr.s | ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
dvmptntr.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
dvmptntr.a | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) |
dvmptntr.j | ⊢ 𝐽 = (𝐾 ↾t 𝑆) |
dvmptntr.k | ⊢ 𝐾 = (TopOpen‘ℂfld) |
dvmptntr.i | ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌) |
Ref | Expression |
---|---|
dvmptntr | ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvmptntr.j | . . . . . . . . 9 ⊢ 𝐽 = (𝐾 ↾t 𝑆) | |
2 | dvmptntr.k | . . . . . . . . . . 11 ⊢ 𝐾 = (TopOpen‘ℂfld) | |
3 | 2 | cnfldtopon 22396 | . . . . . . . . . 10 ⊢ 𝐾 ∈ (TopOn‘ℂ) |
4 | dvmptntr.s | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ) | |
5 | resttopon 20775 | . . . . . . . . . 10 ⊢ ((𝐾 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
6 | 3, 4, 5 | sylancr 694 | . . . . . . . . 9 ⊢ (𝜑 → (𝐾 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
7 | 1, 6 | syl5eqel 2692 | . . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑆)) |
8 | topontop 20541 | . . . . . . . 8 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝐽 ∈ Top) | |
9 | 7, 8 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐽 ∈ Top) |
10 | dvmptntr.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
11 | toponuni 20542 | . . . . . . . . 9 ⊢ (𝐽 ∈ (TopOn‘𝑆) → 𝑆 = ∪ 𝐽) | |
12 | 7, 11 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑆 = ∪ 𝐽) |
13 | 10, 12 | sseqtrd 3604 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ ∪ 𝐽) |
14 | eqid 2610 | . . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
15 | 14 | ntridm 20682 | . . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋)) |
16 | 9, 13, 15 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑋)) |
17 | dvmptntr.i | . . . . . . 7 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = 𝑌) | |
18 | 17 | fveq2d 6107 | . . . . . 6 ⊢ (𝜑 → ((int‘𝐽)‘((int‘𝐽)‘𝑋)) = ((int‘𝐽)‘𝑌)) |
19 | 16, 18 | eqtr3d 2646 | . . . . 5 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) = ((int‘𝐽)‘𝑌)) |
20 | 19 | reseq2d 5317 | . . . 4 ⊢ (𝜑 → ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌))) |
21 | dvmptntr.a | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ) | |
22 | eqid 2610 | . . . . . 6 ⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) | |
23 | 21, 22 | fmptd 6292 | . . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) |
24 | 2, 1 | dvres 23481 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) ∧ (𝑋 ⊆ 𝑆 ∧ 𝑋 ⊆ 𝑆)) → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋))) |
25 | 4, 23, 10, 10, 24 | syl22anc 1319 | . . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑋))) |
26 | 14 | ntrss2 20671 | . . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑋) ⊆ 𝑋) |
27 | 9, 13, 26 | syl2anc 691 | . . . . . . 7 ⊢ (𝜑 → ((int‘𝐽)‘𝑋) ⊆ 𝑋) |
28 | 17, 27 | eqsstr3d 3603 | . . . . . 6 ⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
29 | 28, 10 | sstrd 3578 | . . . . 5 ⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
30 | 2, 1 | dvres 23481 | . . . . 5 ⊢ (((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶ℂ) ∧ (𝑋 ⊆ 𝑆 ∧ 𝑌 ⊆ 𝑆)) → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌))) |
31 | 4, 23, 10, 29, 30 | syl22anc 1319 | . . . 4 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = ((𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) ↾ ((int‘𝐽)‘𝑌))) |
32 | 20, 25, 31 | 3eqtr4d 2654 | . . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌))) |
33 | ssid 3587 | . . . . 5 ⊢ 𝑋 ⊆ 𝑋 | |
34 | resmpt 5369 | . . . . 5 ⊢ (𝑋 ⊆ 𝑋 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴)) | |
35 | 33, 34 | mp1i 13 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋) = (𝑥 ∈ 𝑋 ↦ 𝐴)) |
36 | 35 | oveq2d 6565 | . . 3 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑋)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))) |
37 | 32, 36 | eqtr3d 2646 | . 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴))) |
38 | 28 | resmptd 5371 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌) = (𝑥 ∈ 𝑌 ↦ 𝐴)) |
39 | 38 | oveq2d 6565 | . 2 ⊢ (𝜑 → (𝑆 D ((𝑥 ∈ 𝑋 ↦ 𝐴) ↾ 𝑌)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴))) |
40 | 37, 39 | eqtr3d 2646 | 1 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑆 D (𝑥 ∈ 𝑌 ↦ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ∪ cuni 4372 ↦ cmpt 4643 ↾ cres 5040 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ↾t crest 15904 TopOpenctopn 15905 ℂfldccnfld 19567 Topctop 20517 TopOnctopon 20518 intcnt 20631 D cdv 23433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fi 8200 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-seq 12664 df-exp 12723 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-starv 15783 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-rest 15906 df-topn 15907 df-topgen 15927 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-cnp 20842 df-xms 21935 df-ms 21936 df-limc 23436 df-dv 23437 |
This theorem is referenced by: rolle 23557 cmvth 23558 dvlip 23560 dvlipcn 23561 dvle 23574 dvfsumabs 23590 ftc2 23611 itgparts 23614 itgsubstlem 23615 lgamgulmlem2 24556 ftc2nc 32664 areacirc 32675 itgsin0pilem1 38841 itgsbtaddcnst 38874 |
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