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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 24839 | . . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾t ℝ) |
4 | 1, 3 | eqtr4i 2766 | . . . . . 6 ⊢ 𝐽 = (topGen‘ran (,)) |
5 | retop 24798 | . . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top | |
6 | 4, 5 | eqeltri 2835 | . . . . 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 13466 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ) | |
11 | 8, 9, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ) |
12 | iooretop 24802 | . . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,)) | |
13 | 12, 4 | eleqtrri 2838 | . . . . 5 ⊢ (𝐴(,)𝐵) ∈ 𝐽 |
14 | 13 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ 𝐽) |
15 | ioossicc 13470 | . . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵) | |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)) |
17 | uniretop 24799 | . . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
18 | 4 | unieqi 4924 | . . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,)) |
19 | 17, 18 | eqtr4i 2766 | . . . . 5 ⊢ ℝ = ∪ 𝐽 |
20 | 19 | ssntr 23082 | . . . 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 839 | . . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵))) |
22 | ftc1.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵)) | |
23 | 21, 22 | sseldd 3996 | . 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 2735 | . . 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 26097 | . 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) limℂ 𝐶)) |
33 | ax-resscn 11210 | . . . 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 26094 | . . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ) |
36 | 24, 8, 9, 25, 26, 27, 28, 35 | ftc1lem2 26092 | . . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ) |
37 | 1, 2, 31, 34, 36, 11 | eldv 25948 | . 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 713 | 1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 ∪ cuni 4912 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 ≤ cle 11294 − cmin 11490 / cdiv 11918 (,)cioo 13384 [,]cicc 13387 ↾t crest 17467 TopOpenctopn 17468 topGenctg 17484 ℂfldccnfld 21382 Topctop 22915 intcnt 23041 CnP ccnp 23249 𝐿1cibl 25666 ∫citg 25667 limℂ climc 25912 D cdv 25913 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cc 10473 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-disj 5116 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-ofr 7698 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-omul 8510 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-acn 9980 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-hash 14367 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-clim 15521 df-rlim 15522 df-sum 15720 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-ntr 23044 df-cn 23251 df-cnp 23252 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-ovol 25513 df-vol 25514 df-mbf 25668 df-itg1 25669 df-itg2 25670 df-ibl 25671 df-itg 25672 df-0p 25719 df-limc 25916 df-dv 25917 |
This theorem is referenced by: ftc1cn 26099 |
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