ftc1 - Metamath Proof Explorer

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Theorem ftc1 25977

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 24719	. . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾_t ℝ)
4	1, 3	eqtr4i 2759	. . . . . 6 ⊢ 𝐽 = (topGen‘ran (,))
5		retop 24677	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
6	4, 5	eqeltri 2829	. . . . 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 13331	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
11	8, 9, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
12		iooretop 24681	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
13	12, 4	eleqtrri 2832	. . . . 5 ⊢ (𝐴(,)𝐵) ∈ 𝐽
14	13	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ 𝐽)
15		ioossicc 13335	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
17		uniretop 24678	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
18	4	unieqi 4870	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
19	17, 18	eqtr4i 2759	. . . . 5 ⊢ ℝ = ∪ 𝐽
20	19	ssntr 22974	. . . 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 838	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵)))
22		ftc1.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
23	21, 22	sseldd 3931	. 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 2733	. . 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 25976	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
33		ax-resscn 11070	. . . 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 25973	. . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
36	24, 8, 9, 25, 26, 27, 28, 35	ftc1lem2 25971	. . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
37	1, 2, 31, 34, 36, 11	eldv 25827	. 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 713	1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2113 ∖ cdif 3895 ⊆ wss 3898 {csn 4575 ∪ cuni 4858 class class class wbr 5093 ↦ cmpt 5174 ran crn 5620 ‘cfv 6486 (class class class)co 7352 ℂcc 11011 ℝcr 11012 ≤ cle 11154 − cmin 11351 / cdiv 11781 (,)cioo 13247 [,]cicc 13250 ↾_t crest 17326 TopOpenctopn 17327 topGenctg 17343 ℂ_fldccnfld 21293 Topctop 22809 intcnt 22933 CnP ccnp 23141 𝐿¹cibl 25546 ∫citg 25547 lim_ℂ climc 25791 D cdv 25792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-inf2 9538 ax-cc 10333 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091 ax-addf 11092

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-symdif 4202 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-disj 5061 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-ofr 7617 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-oadd 8395 df-omul 8396 df-er 8628 df-map 8758 df-pm 8759 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-dju 9801 df-card 9839 df-acn 9842 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-ioc 13252 df-ico 13253 df-icc 13254 df-fz 13410 df-fzo 13557 df-fl 13698 df-mod 13776 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-clim 15397 df-rlim 15398 df-sum 15596 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-ntr 22936 df-cn 23143 df-cnp 23144 df-cmp 23303 df-tx 23478 df-hmeo 23671 df-xms 24236 df-ms 24237 df-tms 24238 df-cncf 24799 df-ovol 25393 df-vol 25394 df-mbf 25548 df-itg1 25549 df-itg2 25550 df-ibl 25551 df-itg 25552 df-0p 25599 df-limc 25795 df-dv 25796

This theorem is referenced by: ftc1cn 25978

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