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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 22606 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾t ℝ) |
4 | 1, 3 | eqtr4i 2647 | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) |
5 | retop 22565 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
6 | 4, 5 | eqeltri 2697 | . . . . 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 12255 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
11 | 8, 9, 10 | syl2anc 693 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
12 | iooretop 22569 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
13 | 12, 4 | eleqtrri 2700 | . . . . 5 ⊢ (𝐴(,)𝐵) ∈ 𝐽 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ 𝐽) |
15 | ioossicc 12259 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
17 | uniretop 22566 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
18 | 4 | unieqi 4445 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
19 | 17, 18 | eqtr4i 2647 | . . . . 5 ⊢ ℝ = ∪ 𝐽 |
20 | 19 | ssntr 20862 | . . . 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 1327 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵))) |
22 | ftc1.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
23 | 21, 22 | sseldd 3604 | . 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 2622 | . . 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 23804 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
33 | ax-resscn 9993 | . . . 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 23801 | . . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
36 | 24, 8, 9, 25, 26, 27, 28, 35 | ftc1lem2 23799 | . . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
37 | 1, 2, 31, 34, 36, 11 | eldv 23662 | . 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 957 | 1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∖ cdif 3571 ⊆ wss 3574 {csn 4177 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 ran crn 5115 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 ≤ cle 10075 − cmin 10266 / cdiv 10684 (,)cioo 12175 [,]cicc 12178 ↾t crest 16081 TopOpenctopn 16082 topGenctg 16098 ℂfldccnfld 19746 Topctop 20698 intcnt 20821 CnP ccnp 21029 𝐿1cibl 23386 ∫citg 23387 limℂ climc 23626 D cdv 23627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cc 9257 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-ofr 6898 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-omul 7565 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-acn 8768 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-ntr 20824 df-cn 21031 df-cnp 21032 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-ovol 23233 df-vol 23234 df-mbf 23388 df-itg1 23389 df-itg2 23390 df-ibl 23391 df-itg 23392 df-0p 23437 df-limc 23630 df-dv 23631 |
This theorem is referenced by: ftc1cn 23806 |
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