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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 23405 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾t ℝ) |
4 | 1, 3 | eqtr4i 2847 | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) |
5 | retop 23364 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
6 | 4, 5 | eqeltri 2909 | . . . . 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 12812 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
11 | 8, 9, 10 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
12 | iooretop 23368 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
13 | 12, 4 | eleqtrri 2912 | . . . . 5 ⊢ (𝐴(,)𝐵) ∈ 𝐽 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ 𝐽) |
15 | ioossicc 12816 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
17 | uniretop 23365 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
18 | 4 | unieqi 4841 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
19 | 17, 18 | eqtr4i 2847 | . . . . 5 ⊢ ℝ = ∪ 𝐽 |
20 | 19 | ssntr 21660 | . . . 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 836 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵))) |
22 | ftc1.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
23 | 21, 22 | sseldd 3968 | . 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 2821 | . . 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 24632 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
33 | ax-resscn 10588 | . . . 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 24629 | . . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
36 | 24, 8, 9, 25, 26, 27, 28, 35 | ftc1lem2 24627 | . . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
37 | 1, 2, 31, 34, 36, 11 | eldv 24490 | . 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 711 | 1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∖ cdif 3933 ⊆ wss 3936 {csn 4561 ∪ cuni 4832 class class class wbr 5059 ↦ cmpt 5139 ran crn 5551 ‘cfv 6350 (class class class)co 7150 ℂcc 10529 ℝcr 10530 ≤ cle 10670 − cmin 10864 / cdiv 11291 (,)cioo 12732 [,]cicc 12735 ↾t crest 16688 TopOpenctopn 16689 topGenctg 16705 ℂfldccnfld 20539 Topctop 21495 intcnt 21619 CnP ccnp 21827 𝐿1cibl 24212 ∫citg 24213 limℂ climc 24454 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-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-ntr 21622 df-cn 21829 df-cnp 21830 df-cmp 21989 df-tx 22164 df-hmeo 22357 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: ftc1cn 24634 |
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