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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 22414 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾t ℝ) |
4 | 1, 3 | eqtr4i 2635 | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) |
5 | retop 22375 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
6 | 4, 5 | eqeltri 2684 | . . . . 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 12126 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
11 | 8, 9, 10 | syl2anc 691 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
12 | iooretop 22379 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
13 | 12, 4 | eleqtrri 2687 | . . . . 5 ⊢ (𝐴(,)𝐵) ∈ 𝐽 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ 𝐽) |
15 | ioossicc 12130 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
17 | uniretop 22376 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
18 | 4 | unieqi 4381 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
19 | 17, 18 | eqtr4i 2635 | . . . . 5 ⊢ ℝ = ∪ 𝐽 |
20 | 19 | ssntr 20672 | . . . 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 1319 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵))) |
22 | ftc1.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
23 | 21, 22 | sseldd 3569 | . 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 2610 | . . 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 23608 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
33 | ax-resscn 9872 | . . . 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 23605 | . . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
36 | 24, 8, 9, 25, 26, 27, 28, 35 | ftc1lem2 23603 | . . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
37 | 1, 2, 31, 34, 36, 11 | eldv 23468 | . 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 959 | 1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 ran crn 5039 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 ℝcr 9814 ≤ cle 9954 − cmin 10145 / cdiv 10563 (,)cioo 12046 [,]cicc 12049 ↾t crest 15904 TopOpenctopn 15905 topGenctg 15921 ℂfldccnfld 19567 Topctop 20517 intcnt 20631 CnP ccnp 20839 𝐿1cibl 23192 ∫citg 23193 limℂ climc 23432 D cdv 23433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cc 9140 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-ofr 6796 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-omul 7452 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-acn 8651 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-rlim 14068 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-ntr 20634 df-cn 20841 df-cnp 20842 df-cmp 21000 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-cncf 22489 df-ovol 23040 df-vol 23041 df-mbf 23194 df-itg1 23195 df-itg2 23196 df-ibl 23197 df-itg 23198 df-0p 23243 df-limc 23436 df-dv 23437 |
This theorem is referenced by: ftc1cn 23610 |
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