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Mirrors > Home > MPE Home > Th. List > ftc1cn | Structured version Visualization version GIF version |
Description: Strengthen the assumptions of ftc1 24332 to when the function 𝐹 is continuous on the entire interval (𝐴, 𝐵); in this case we can calculate D 𝐺 exactly. (Contributed by Mario Carneiro, 1-Sep-2014.) |
Ref | Expression |
---|---|
ftc1cn.g | ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) |
ftc1cn.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ftc1cn.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ftc1cn.le | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
ftc1cn.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) |
ftc1cn.i | ⊢ (𝜑 → 𝐹 ∈ 𝐿1) |
Ref | Expression |
---|---|
ftc1cn | ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvf 24198 | . . . . 5 ⊢ (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ | |
2 | 1 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ) |
3 | 2 | ffund 6342 | . . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺)) |
4 | ax-resscn 10384 | . . . . . . 7 ⊢ ℝ ⊆ ℂ | |
5 | 4 | a1i 11 | . . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ) |
6 | ftc1cn.g | . . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) | |
7 | ftc1cn.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
8 | ftc1cn.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
9 | ftc1cn.le | . . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
10 | ssidd 3876 | . . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
11 | ioossre 12607 | . . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ) |
13 | ftc1cn.i | . . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐿1) | |
14 | ftc1cn.f | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ)) | |
15 | cncff 23194 | . . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ) | |
16 | 14, 15 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ) |
17 | 6, 7, 8, 9, 10, 12, 13, 16 | ftc1lem2 24326 | . . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
18 | iccssre 12627 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
19 | 7, 8, 18 | syl2anc 576 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
20 | eqid 2772 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
21 | 20 | tgioo2 23104 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
22 | 5, 17, 19, 21, 20 | dvbssntr 24191 | . . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) |
23 | iccntr 23122 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) | |
24 | 7, 8, 23 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
25 | 22, 24 | sseqtrd 3893 | . . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵)) |
26 | 7 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ) |
27 | 8 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ) |
28 | 9 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵) |
29 | ssidd 3876 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
30 | 11 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ℝ) |
31 | 13 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ 𝐿1) |
32 | simpr 477 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴(,)𝐵)) | |
33 | 11, 4 | sstri 3863 | . . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℂ |
34 | ssid 3875 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
35 | eqid 2772 | . . . . . . . . . . 11 ⊢ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) | |
36 | 20 | cnfldtopon 23084 | . . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
37 | 36 | toponrestid 21223 | . . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
38 | 20, 35, 37 | cncfcn 23210 | . . . . . . . . . 10 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))) |
39 | 33, 34, 38 | mp2an 679 | . . . . . . . . 9 ⊢ ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)) |
40 | 14, 39 | syl6eleq 2870 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))) |
41 | 40 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))) |
42 | 33 | a1i 11 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ) |
43 | resttopon 21463 | . . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))) | |
44 | 36, 42, 43 | sylancr 578 | . . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))) |
45 | toponuni 21216 | . . . . . . . . . 10 ⊢ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)) → (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))) | |
46 | 44, 45 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))) |
47 | 46 | eleq2d 2845 | . . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))) |
48 | 47 | biimpa 469 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))) |
49 | eqid 2772 | . . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) | |
50 | 49 | cncnpi 21580 | . . . . . . 7 ⊢ ((𝐹 ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)) ∧ 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦)) |
51 | 41, 48, 50 | syl2anc 576 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ ((((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) CnP (TopOpen‘ℂfld))‘𝑦)) |
52 | 6, 26, 27, 28, 29, 30, 31, 32, 51, 21, 35, 20 | ftc1 24332 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦(ℝ D 𝐺)(𝐹‘𝑦)) |
53 | vex 3412 | . . . . . 6 ⊢ 𝑦 ∈ V | |
54 | fvex 6506 | . . . . . 6 ⊢ (𝐹‘𝑦) ∈ V | |
55 | 53, 54 | breldm 5620 | . . . . 5 ⊢ (𝑦(ℝ D 𝐺)(𝐹‘𝑦) → 𝑦 ∈ dom (ℝ D 𝐺)) |
56 | 52, 55 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ dom (ℝ D 𝐺)) |
57 | 25, 56 | eqelssd 3874 | . . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵)) |
58 | df-fn 6185 | . . 3 ⊢ ((ℝ D 𝐺) Fn (𝐴(,)𝐵) ↔ (Fun (ℝ D 𝐺) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵))) | |
59 | 3, 57, 58 | sylanbrc 575 | . 2 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵)) |
60 | 16 | ffnd 6339 | . 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵)) |
61 | 3 | adantr 473 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺)) |
62 | funbrfv 6540 | . . 3 ⊢ (Fun (ℝ D 𝐺) → (𝑦(ℝ D 𝐺)(𝐹‘𝑦) → ((ℝ D 𝐺)‘𝑦) = (𝐹‘𝑦))) | |
63 | 61, 52, 62 | sylc 65 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑦) = (𝐹‘𝑦)) |
64 | 59, 60, 63 | eqfnfvd 6624 | 1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2048 ⊆ wss 3825 ∪ cuni 4706 class class class wbr 4923 ↦ cmpt 5002 dom cdm 5400 ran crn 5401 Fun wfun 6176 Fn wfn 6177 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 ℂcc 10325 ℝcr 10326 ≤ cle 10467 (,)cioo 12547 [,]cicc 12550 ↾t crest 16540 TopOpenctopn 16541 topGenctg 16557 ℂfldccnfld 20237 TopOnctopon 21212 intcnt 21319 Cn ccn 21526 CnP ccnp 21527 –cn→ccncf 23177 𝐿1cibl 23911 ∫citg 23912 D cdv 24154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-inf2 8890 ax-cc 9647 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 ax-pre-sup 10405 ax-addf 10406 ax-mulf 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-symdif 4101 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-iin 4789 df-disj 4892 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-se 5360 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-isom 6191 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-of 7221 df-ofr 7222 df-om 7391 df-1st 7494 df-2nd 7495 df-supp 7627 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-1o 7897 df-2o 7898 df-oadd 7901 df-omul 7902 df-er 8081 df-map 8200 df-pm 8201 df-ixp 8252 df-en 8299 df-dom 8300 df-sdom 8301 df-fin 8302 df-fsupp 8621 df-fi 8662 df-sup 8693 df-inf 8694 df-oi 8761 df-dju 9116 df-card 9154 df-acn 9157 df-cda 9380 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-div 11091 df-nn 11432 df-2 11496 df-3 11497 df-4 11498 df-5 11499 df-6 11500 df-7 11501 df-8 11502 df-9 11503 df-n0 11701 df-z 11787 df-dec 11905 df-uz 12052 df-q 12156 df-rp 12198 df-xneg 12317 df-xadd 12318 df-xmul 12319 df-ioo 12551 df-ioc 12552 df-ico 12553 df-icc 12554 df-fz 12702 df-fzo 12843 df-fl 12970 df-mod 13046 df-seq 13178 df-exp 13238 df-hash 13499 df-cj 14309 df-re 14310 df-im 14311 df-sqrt 14445 df-abs 14446 df-clim 14696 df-rlim 14697 df-sum 14894 df-struct 16331 df-ndx 16332 df-slot 16333 df-base 16335 df-sets 16336 df-ress 16337 df-plusg 16424 df-mulr 16425 df-starv 16426 df-sca 16427 df-vsca 16428 df-ip 16429 df-tset 16430 df-ple 16431 df-ds 16433 df-unif 16434 df-hom 16435 df-cco 16436 df-rest 16542 df-topn 16543 df-0g 16561 df-gsum 16562 df-topgen 16563 df-pt 16564 df-prds 16567 df-xrs 16621 df-qtop 16626 df-imas 16627 df-xps 16629 df-mre 16705 df-mrc 16706 df-acs 16708 df-mgm 17700 df-sgrp 17742 df-mnd 17753 df-submnd 17794 df-mulg 18002 df-cntz 18208 df-cmn 18658 df-psmet 20229 df-xmet 20230 df-met 20231 df-bl 20232 df-mopn 20233 df-fbas 20234 df-fg 20235 df-cnfld 20238 df-top 21196 df-topon 21213 df-topsp 21235 df-bases 21248 df-cld 21321 df-ntr 21322 df-cls 21323 df-nei 21400 df-lp 21438 df-perf 21439 df-cn 21529 df-cnp 21530 df-haus 21617 df-cmp 21689 df-tx 21864 df-hmeo 22057 df-fil 22148 df-fm 22240 df-flim 22241 df-flf 22242 df-xms 22623 df-ms 22624 df-tms 22625 df-cncf 23179 df-ovol 23758 df-vol 23759 df-mbf 23913 df-itg1 23914 df-itg2 23915 df-ibl 23916 df-itg 23917 df-0p 23964 df-limc 24157 df-dv 24158 |
This theorem is referenced by: ftc2 24334 itgsubstlem 24338 |
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