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| Mirrors > Home > MPE Home > Th. List > ftc1cn | Structured version Visualization version GIF version | ||
| Description: Strengthen the assumptions of ftc1 26083 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 25942 | . . . . 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 6740 | . . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺)) |
| 4 | ax-resscn 11212 | . . . . . . 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 4007 | . . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
| 11 | ioossre 13448 | . . . . . . . 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 24919 | . . . . . . . 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 26077 | . . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
| 18 | iccssre 13469 | . . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
| 19 | 7, 8, 18 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
| 20 | tgioo4 24826 | . . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂfld) ↾t ℝ) | |
| 21 | eqid 2737 | . . . . . 6 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
| 22 | 5, 17, 19, 20, 21 | dvbssntr 25935 | . . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵))) |
| 23 | iccntr 24843 | . . . . . 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 4020 | . . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵)) |
| 26 | 7 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ) |
| 27 | 8 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ) |
| 28 | 9 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵) |
| 29 | ssidd 4007 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵)) | |
| 30 | 11 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ℝ) |
| 31 | 13 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ 𝐿1) |
| 32 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴(,)𝐵)) | |
| 33 | 11, 4 | sstri 3993 | . . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℂ |
| 34 | ssid 4006 | . . . . . . . . . 10 ⊢ ℂ ⊆ ℂ | |
| 35 | eqid 2737 | . . . . . . . . . . 11 ⊢ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) | |
| 36 | 21 | cnfldtopon 24803 | . . . . . . . . . . . 12 ⊢ (TopOpen‘ℂfld) ∈ (TopOn‘ℂ) |
| 37 | 36 | toponrestid 22927 | . . . . . . . . . . 11 ⊢ (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ) |
| 38 | 21, 35, 37 | cncfcn 24936 | . . . . . . . . . 10 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))) |
| 39 | 33, 34, 38 | mp2an 692 | . . . . . . . . 9 ⊢ ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld)) |
| 40 | 14, 39 | eleqtrdi 2851 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) Cn (TopOpen‘ℂfld))) |
| 41 | 40 | adantr 480 | . . . . . . 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 23169 | . . . . . . . . . . 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 22920 | . . . . . . . . . 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 2827 | . . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)))) |
| 48 | 47 | biimpa 476 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵))) |
| 49 | eqid 2737 | . . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂfld) ↾t (𝐴(,)𝐵)) | |
| 50 | 49 | cncnpi 23286 | . . . . . . 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, 20, 35, 21 | ftc1 26083 | . . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦(ℝ D 𝐺)(𝐹‘𝑦)) |
| 53 | vex 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 54 | fvex 6919 | . . . . . 6 ⊢ (𝐹‘𝑦) ∈ V | |
| 55 | 53, 54 | breldm 5919 | . . . . 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 4005 | . . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵)) |
| 58 | df-fn 6564 | . . 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 6737 | . 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵)) |
| 61 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺)) |
| 62 | funbrfv 6957 | . . 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 7054 | 1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ran crn 5686 Fun wfun 6555 Fn wfn 6556 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 ≤ cle 11296 (,)cioo 13387 [,]cicc 13390 ↾t crest 17465 TopOpenctopn 17466 topGenctg 17482 ℂfldccnfld 21364 TopOnctopon 22916 intcnt 23025 Cn ccn 23232 CnP ccnp 23233 –cn→ccncf 24902 𝐿1cibl 25652 ∫citg 25653 D cdv 25898 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cc 10475 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 ax-addf 11234 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-symdif 4253 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-disj 5111 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-er 8745 df-map 8868 df-pm 8869 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-fi 9451 df-sup 9482 df-inf 9483 df-oi 9550 df-dju 9941 df-card 9979 df-acn 9982 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-ioo 13391 df-ioc 13392 df-ico 13393 df-icc 13394 df-fz 13548 df-fzo 13695 df-fl 13832 df-mod 13910 df-seq 14043 df-exp 14103 df-hash 14370 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-clim 15524 df-rlim 15525 df-sum 15723 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17467 df-topn 17468 df-0g 17486 df-gsum 17487 df-topgen 17488 df-pt 17489 df-prds 17492 df-xrs 17547 df-qtop 17552 df-imas 17553 df-xps 17555 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-psmet 21356 df-xmet 21357 df-met 21358 df-bl 21359 df-mopn 21360 df-fbas 21361 df-fg 21362 df-cnfld 21365 df-top 22900 df-topon 22917 df-topsp 22939 df-bases 22953 df-cld 23027 df-ntr 23028 df-cls 23029 df-nei 23106 df-lp 23144 df-perf 23145 df-cn 23235 df-cnp 23236 df-haus 23323 df-cmp 23395 df-tx 23570 df-hmeo 23763 df-fil 23854 df-fm 23946 df-flim 23947 df-flf 23948 df-xms 24330 df-ms 24331 df-tms 24332 df-cncf 24904 df-ovol 25499 df-vol 25500 df-mbf 25654 df-itg1 25655 df-itg2 25656 df-ibl 25657 df-itg 25658 df-0p 25705 df-limc 25901 df-dv 25902 |
| This theorem is referenced by: ftc2 26085 itgsubstlem 26089 |
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