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Mirrors > Home > MPE Home > Th. List > ftc1cn | Structured version Visualization version GIF version |
Description: Strengthen the assumptions of ftc1 24633 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 24499 | . . . . 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 6513 | . . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺)) |
4 | ax-resscn 10588 | . . . . . . 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 3990 | . . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
11 | ioossre 12792 | . . . . . . . 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 23495 | . . . . . . . 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 24627 | . . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
18 | iccssre 12812 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
19 | 7, 8, 18 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
20 | eqid 2821 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
21 | 20 | tgioo2 23405 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) |
22 | 5, 17, 19, 21, 20 | dvbssntr 24492 | . . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) |
23 | iccntr 23423 | . . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) | |
24 | 7, 8, 23 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵)) |
25 | 22, 24 | sseqtrd 4007 | . . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵)) |
26 | 7 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ) |
27 | 8 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ) |
28 | 9 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵) |
29 | ssidd 3990 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
30 | 11 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ℝ) |
31 | 13 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ 𝐿1) |
32 | simpr 487 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴(,)𝐵)) | |
33 | 11, 4 | sstri 3976 | . . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℂ |
34 | ssid 3989 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
35 | eqid 2821 | . . . . . . . . . . 11 ⊢ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) | |
36 | 20 | cnfldtopon 23385 | . . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
37 | 36 | toponrestid 21523 | . . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
38 | 20, 35, 37 | cncfcn 23511 | . . . . . . . . . 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 2923 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))) |
41 | 40 | adantr 483 | . . . . . . 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 21763 | . . . . . . . . . . 11 ⊢ (((TopOpen‘ℂfld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))) | |
44 | 36, 42, 43 | sylancr 589 | . . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵))) |
45 | toponuni 21516 | . . . . . . . . . 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 2898 | . . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))) |
48 | 47 | biimpa 479 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))) |
49 | eqid 2821 | . . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) | |
50 | 49 | cncnpi 21880 | . . . . . . 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 586 | . . . . . 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 24633 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦(ℝ D 𝐺)(𝐹‘𝑦)) |
53 | vex 3498 | . . . . . 6 ⊢ 𝑦 ∈ V | |
54 | fvex 6678 | . . . . . 6 ⊢ (𝐹‘𝑦) ∈ V | |
55 | 53, 54 | breldm 5772 | . . . . 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 3988 | . . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵)) |
58 | df-fn 6353 | . . 3 ⊢ ((ℝ D 𝐺) Fn (𝐴(,)𝐵) ↔ (Fun (ℝ D 𝐺) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵))) | |
59 | 3, 57, 58 | sylanbrc 585 | . 2 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵)) |
60 | 16 | ffnd 6510 | . 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵)) |
61 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺)) |
62 | funbrfv 6711 | . . 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 6800 | 1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 ∪ cuni 4832 class class class wbr 5059 ↦ cmpt 5139 dom cdm 5550 ran crn 5551 Fun wfun 6344 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 ℝcr 10530 ≤ cle 10670 (,)cioo 12732 [,]cicc 12735 ↾t crest 16688 TopOpenctopn 16689 topGenctg 16705 ℂfldccnfld 20539 TopOnctopon 21512 intcnt 21619 Cn ccn 21826 CnP ccnp 21827 –cn→ccncf 23478 𝐿1cibl 24212 ∫citg 24213 D cdv 24455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cc 9851 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-symdif 4219 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-disj 5025 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-ofr 7404 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-dju 9324 df-card 9362 df-acn 9365 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-mulg 18219 df-cntz 18441 df-cmn 18902 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-ovol 24059 df-vol 24060 df-mbf 24214 df-itg1 24215 df-itg2 24216 df-ibl 24217 df-itg 24218 df-0p 24265 df-limc 24458 df-dv 24459 |
This theorem is referenced by: ftc2 24635 itgsubstlem 24639 |
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