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Mirrors > Home > MPE Home > Th. List > ftc1 | Structured version Visualization version GIF version |
Description: The Fundamental Theorem of Calculus, part one. The function formed by varying the right endpoint of an integral is differentiable at 𝐶 with derivative 𝐹(𝐶) if the original function is continuous at 𝐶. This is part of Metamath 100 proof #15. (Contributed by Mario Carneiro, 1-Sep-2014.) |
Ref | Expression |
---|---|
ftc1.g | ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) |
ftc1.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ftc1.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ftc1.le | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
ftc1.s | ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) |
ftc1.d | ⊢ (𝜑 → 𝐷 ⊆ ℝ) |
ftc1.i | ⊢ (𝜑 → 𝐹 ∈ 𝐿1) |
ftc1.c | ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) |
ftc1.f | ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶)) |
ftc1.j | ⊢ 𝐽 = (𝐿 ↾t ℝ) |
ftc1.k | ⊢ 𝐾 = (𝐿 ↾t 𝐷) |
ftc1.l | ⊢ 𝐿 = (TopOpen‘ℂfld) |
Ref | Expression |
---|---|
ftc1 | ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ftc1.j | . . . . . . 7 ⊢ 𝐽 = (𝐿 ↾t ℝ) | |
2 | ftc1.l | . . . . . . . 8 ⊢ 𝐿 = (TopOpen‘ℂfld) | |
3 | 2 | tgioo2 23338 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾t ℝ) |
4 | 1, 3 | eqtr4i 2844 | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) |
5 | retop 23297 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
6 | 4, 5 | eqeltri 2906 | . . . . 5 ⊢ 𝐽 ∈ Top |
7 | 6 | a1i 11 | . . . 4 ⊢ (𝜑 → 𝐽 ∈ Top) |
8 | ftc1.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
9 | ftc1.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
10 | iccssre 12806 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
11 | 8, 9, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
12 | iooretop 23301 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
13 | 12, 4 | eleqtrri 2909 | . . . . 5 ⊢ (𝐴(,)𝐵) ∈ 𝐽 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ 𝐽) |
15 | ioossicc 12810 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
17 | uniretop 23298 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
18 | 4 | unieqi 4839 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
19 | 17, 18 | eqtr4i 2844 | . . . . 5 ⊢ ℝ = ∪ 𝐽 |
20 | 19 | ssntr 21594 | . . . 4 ⊢ (((𝐽 ∈ Top ∧ (𝐴[,]𝐵) ⊆ ℝ) ∧ ((𝐴(,)𝐵) ∈ 𝐽 ∧ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))) → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵))) |
21 | 7, 11, 14, 16, 20 | syl22anc 834 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵))) |
22 | ftc1.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
23 | 21, 22 | sseldd 3965 | . 2 ⊢ (𝜑 → 𝐶 ∈ ((int‘𝐽)‘(𝐴[,]𝐵))) |
24 | ftc1.g | . . 3 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡) | |
25 | ftc1.le | . . 3 ⊢ (𝜑 → 𝐴 ≤ 𝐵) | |
26 | ftc1.s | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ 𝐷) | |
27 | ftc1.d | . . 3 ⊢ (𝜑 → 𝐷 ⊆ ℝ) | |
28 | ftc1.i | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐿1) | |
29 | ftc1.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐾 CnP 𝐿)‘𝐶)) | |
30 | ftc1.k | . . 3 ⊢ 𝐾 = (𝐿 ↾t 𝐷) | |
31 | eqid 2818 | . . 3 ⊢ (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) = (𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) | |
32 | 24, 8, 9, 25, 26, 27, 28, 22, 29, 1, 30, 2, 31 | ftc1lem6 24565 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
33 | ax-resscn 10582 | . . . 4 ⊢ ℝ ⊆ ℂ | |
34 | 33 | a1i 11 | . . 3 ⊢ (𝜑 → ℝ ⊆ ℂ) |
35 | 24, 8, 9, 25, 26, 27, 28, 22, 29, 1, 30, 2 | ftc1lem3 24562 | . . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
36 | 24, 8, 9, 25, 26, 27, 28, 35 | ftc1lem2 24560 | . . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
37 | 1, 2, 31, 34, 36, 11 | eldv 24423 | . 2 ⊢ (𝜑 → (𝐶(ℝ D 𝐺)(𝐹‘𝐶) ↔ (𝐶 ∈ ((int‘𝐽)‘(𝐴[,]𝐵)) ∧ (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)))) |
38 | 23, 32, 37 | mpbir2and 709 | 1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 ∪ cuni 4830 class class class wbr 5057 ↦ cmpt 5137 ran crn 5549 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 ≤ cle 10664 − cmin 10858 / cdiv 11285 (,)cioo 12726 [,]cicc 12729 ↾t crest 16682 TopOpenctopn 16683 topGenctg 16699 ℂfldccnfld 20473 Topctop 21429 intcnt 21553 CnP ccnp 21761 𝐿1cibl 24145 ∫citg 24146 limℂ climc 24387 D cdv 24388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-symdif 4216 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-disj 5023 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-ofr 7399 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-dju 9318 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ioc 12731 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-mod 13226 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-ntr 21556 df-cn 21763 df-cnp 21764 df-cmp 21923 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-cncf 23413 df-ovol 23992 df-vol 23993 df-mbf 24147 df-itg1 24148 df-itg2 24149 df-ibl 24150 df-itg 24151 df-0p 24198 df-limc 24391 df-dv 24392 |
This theorem is referenced by: ftc1cn 24567 |
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