ftc1 - Metamath Proof Explorer

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Theorem ftc1 26170

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.)

**Hypotheses**
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‘ℂ_fld)

**Assertion**
Ref	Expression
ftc1	⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶))

Distinct variable groups: 𝑥,𝑡,𝐶 𝑡,𝐷,𝑥 𝑡,𝐴,𝑥 𝑡,𝐵,𝑥 𝜑,𝑡,𝑥 𝑡,𝐹,𝑥 𝑥,𝐿 Allowed substitution hints: 𝐺(𝑥,𝑡) 𝐽(𝑥,𝑡) 𝐾(𝑥,𝑡) 𝐿(𝑡)

Proof of Theorem ftc1 Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ftc1.j	. . . . . . 7 ⊢ 𝐽 = (𝐿 ↾_t ℝ)
2		ftc1.l	. . . . . . . 8 ⊢ 𝐿 = (TopOpen‘ℂ_fld)
3	2	tgioo2 24929	. . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾_t ℝ)
4	1, 3	eqtr4i 2795	. . . . . 6 ⊢ 𝐽 = (topGen‘ran (,))
5		retop 24887	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
6	4, 5	eqeltri 2865	. . . . 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 13456	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
11	8, 9, 10	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
12		iooretop 24891	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
13	12, 4	eleqtrri 2868	. . . . 5 ⊢ (𝐴(,)𝐵) ∈ 𝐽
14	13	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ 𝐽)
15		ioossicc 13460	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
17		uniretop 24888	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
18	4	unieqi 4888	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
19	17, 18	eqtr4i 2795	. . . . 5 ⊢ ℝ = ∪ 𝐽
20	19	ssntr 23184	. . . 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 851	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵)))
22		ftc1.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
23	21, 22	sseldd 3946	. 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 2769	. . 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 26169	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
33		ax-resscn 11157	. . . 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 26166	. . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
36	24, 8, 9, 25, 26, 27, 28, 35	ftc1lem2 26164	. . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
37	1, 2, 31, 34, 36, 11	eldv 26026	. 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))) lim_ℂ 𝐶))))
38	23, 32, 37	mpbir2and 725	1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1567 ∈ wcel 2149 ∖ cdif 3910 ⊆ wss 3913 {csn 4594 ∪ cuni 4876 class class class wbr 5113 ↦ cmpt 5196 ran crn 5663 ‘cfv 6537 (class class class)co 7411 ℂcc 11098 ℝcr 11099 ≤ cle 11244 − cmin 11441 / cdiv 11871 (,)cioo 13372 [,]cicc 13375 ↾_t crest 17473 TopOpenctopn 17474 topGenctg 17490 ℂ_fldccnfld 21491 Topctop 23019 intcnt 23143 CnP ccnp 23351 𝐿¹cibl 25745 ∫citg 25746 lim_ℂ climc 25990 D cdv 25991

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cc 10419 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178 ax-addf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-symdif 4214 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-disj 5081 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-ofr 7676 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8157 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-2o 8454 df-oadd 8457 df-omul 8458 df-er 8694 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9322 df-fi 9371 df-sup 9402 df-inf 9403 df-oi 9472 df-dju 9887 df-card 9925 df-acn 9928 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-ioc 13377 df-ico 13378 df-icc 13379 df-fz 13536 df-fzo 13683 df-fl 13825 df-mod 13903 df-seq 14038 df-exp 14098 df-hash 14367 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15539 df-rlim 15540 df-sum 15738 df-struct 17207 df-sets 17224 df-slot 17242 df-ndx 17254 df-base 17270 df-ress 17291 df-plusg 17323 df-mulr 17324 df-starv 17325 df-sca 17326 df-vsca 17327 df-ip 17328 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-hom 17334 df-cco 17335 df-rest 17475 df-topn 17476 df-0g 17494 df-gsum 17495 df-topgen 17496 df-pt 17497 df-prds 17500 df-xrs 17556 df-qtop 17561 df-imas 17562 df-xps 17564 df-mre 17638 df-mrc 17639 df-acs 17641 df-mgm 18698 df-sgrp 18777 df-mnd 18793 df-submnd 18842 df-mulg 19134 df-cntz 19387 df-cmn 19852 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-ntr 23146 df-cn 23353 df-cnp 23354 df-cmp 23513 df-tx 23688 df-hmeo 23881 df-xms 24446 df-ms 24447 df-tms 24448 df-cncf 25006 df-ovol 25592 df-vol 25593 df-mbf 25747 df-itg1 25748 df-itg2 25749 df-ibl 25750 df-itg 25751 df-0p 25798 df-limc 25994 df-dv 25995

This theorem is referenced by: ftc1cn 26171

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