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Mirrors > Home > MPE Home > Th. List > ftc1cn | Structured version Visualization version GIF version |
Description: Strengthen the assumptions of ftc1 25488 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 25353 | . . . . 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 6708 | . . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺)) |
4 | ax-resscn 11149 | . . . . . . 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 4001 | . . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
11 | ioossre 13367 | . . . . . . . 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 24338 | . . . . . . . 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 25482 | . . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
18 | iccssre 13388 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
19 | 7, 8, 18 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
20 | eqid 2731 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
21 | 20 | tgioo2 24248 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
22 | 5, 17, 19, 21, 20 | dvbssntr 25346 | . . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) |
23 | iccntr 24266 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) | |
24 | 7, 8, 23 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
25 | 22, 24 | sseqtrd 4018 | . . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵)) |
26 | 7 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ) |
27 | 8 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ) |
28 | 9 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵) |
29 | ssidd 4001 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
30 | 11 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ℝ) |
31 | 13 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ 𝐿1) |
32 | simpr 485 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴(,)𝐵)) | |
33 | 11, 4 | sstri 3987 | . . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℂ |
34 | ssid 4000 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
35 | eqid 2731 | . . . . . . . . . . 11 ⊢ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) | |
36 | 20 | cnfldtopon 24228 | . . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
37 | 36 | toponrestid 22352 | . . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
38 | 20, 35, 37 | cncfcn 24355 | . . . . . . . . . 10 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))) |
39 | 33, 34, 38 | mp2an 690 | . . . . . . . . 9 ⊢ ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)) |
40 | 14, 39 | eleqtrdi 2842 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))) |
41 | 40 | adantr 481 | . . . . . . 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 22594 | . . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))) | |
44 | 36, 42, 43 | sylancr 587 | . . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))) |
45 | toponuni 22345 | . . . . . . . . . 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 2818 | . . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))) |
48 | 47 | biimpa 477 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))) |
49 | eqid 2731 | . . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) | |
50 | 49 | cncnpi 22711 | . . . . . . 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 584 | . . . . . 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 25488 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦(ℝ D 𝐺)(𝐹‘𝑦)) |
53 | vex 3477 | . . . . . 6 ⊢ 𝑦 ∈ V | |
54 | fvex 6891 | . . . . . 6 ⊢ (𝐹‘𝑦) ∈ V | |
55 | 53, 54 | breldm 5900 | . . . . 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 3999 | . . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵)) |
58 | df-fn 6535 | . . 3 ⊢ ((ℝ D 𝐺) Fn (𝐴(,)𝐵) ↔ (Fun (ℝ D 𝐺) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵))) | |
59 | 3, 57, 58 | sylanbrc 583 | . 2 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵)) |
60 | 16 | ffnd 6705 | . 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵)) |
61 | 3 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺)) |
62 | funbrfv 6929 | . . 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 7021 | 1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3944 ∪ cuni 4901 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 ran crn 5670 Fun wfun 6526 Fn wfn 6527 ⟶wf 6528 ‘cfv 6532 (class class class)co 7393 ℂcc 11090 ℝcr 11091 ≤ cle 11231 (,)cioo 13306 [,]cicc 13309 ↾t crest 17348 TopOpenctopn 17349 topGenctg 17365 ℂfldccnfld 20878 TopOnctopon 22341 intcnt 22450 Cn ccn 22657 CnP ccnp 22658 –cn→ccncf 24321 𝐿1cibl 25063 ∫citg 25064 D cdv 25309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-inf2 9618 ax-cc 10412 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-pre-sup 11170 ax-addf 11171 ax-mulf 11172 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-symdif 4238 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-disj 5107 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-isom 6541 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-ofr 7654 df-om 7839 df-1st 7957 df-2nd 7958 df-supp 8129 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-2o 8449 df-oadd 8452 df-omul 8453 df-er 8686 df-map 8805 df-pm 8806 df-ixp 8875 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-fsupp 9345 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9487 df-dju 9878 df-card 9916 df-acn 9919 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-div 11854 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-7 12262 df-8 12263 df-9 12264 df-n0 12455 df-z 12541 df-dec 12660 df-uz 12805 df-q 12915 df-rp 12957 df-xneg 13074 df-xadd 13075 df-xmul 13076 df-ioo 13310 df-ioc 13311 df-ico 13312 df-icc 13313 df-fz 13467 df-fzo 13610 df-fl 13739 df-mod 13817 df-seq 13949 df-exp 14010 df-hash 14273 df-cj 15028 df-re 15029 df-im 15030 df-sqrt 15164 df-abs 15165 df-clim 15414 df-rlim 15415 df-sum 15615 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17350 df-topn 17351 df-0g 17369 df-gsum 17370 df-topgen 17371 df-pt 17372 df-prds 17375 df-xrs 17430 df-qtop 17435 df-imas 17436 df-xps 17438 df-mre 17512 df-mrc 17513 df-acs 17515 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-mulg 18923 df-cntz 19147 df-cmn 19614 df-psmet 20870 df-xmet 20871 df-met 20872 df-bl 20873 df-mopn 20874 df-fbas 20875 df-fg 20876 df-cnfld 20879 df-top 22325 df-topon 22342 df-topsp 22364 df-bases 22378 df-cld 22452 df-ntr 22453 df-cls 22454 df-nei 22531 df-lp 22569 df-perf 22570 df-cn 22660 df-cnp 22661 df-haus 22748 df-cmp 22820 df-tx 22995 df-hmeo 23188 df-fil 23279 df-fm 23371 df-flim 23372 df-flf 23373 df-xms 23755 df-ms 23756 df-tms 23757 df-cncf 24323 df-ovol 24910 df-vol 24911 df-mbf 25065 df-itg1 25066 df-itg2 25067 df-ibl 25068 df-itg 25069 df-0p 25116 df-limc 25312 df-dv 25313 |
This theorem is referenced by: ftc2 25490 itgsubstlem 25494 |
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