ftc1 - Metamath Proof Explorer

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Theorem ftc1 25996

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 24738	. . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾_t ℝ)
4	1, 3	eqtr4i 2759	. . . . . 6 ⊢ 𝐽 = (topGen‘ran (,))
5		retop 24696	. . . . . 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 13336	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
11	8, 9, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
12		iooretop 24700	. . . . . 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 13340	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
17		uniretop 24697	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
18	4	unieqi 4872	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
19	17, 18	eqtr4i 2759	. . . . 5 ⊢ ℝ = ∪ 𝐽
20	19	ssntr 22993	. . . 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 25995	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
33		ax-resscn 11074	. . . 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 25992	. . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
36	24, 8, 9, 25, 26, 27, 28, 35	ftc1lem2 25990	. . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
37	1, 2, 31, 34, 36, 11	eldv 25846	. 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 4577 ∪ cuni 4860 class class class wbr 5095 ↦ cmpt 5176 ran crn 5622 ‘cfv 6489 (class class class)co 7355 ℂcc 11015 ℝcr 11016 ≤ cle 11158 − cmin 11355 / cdiv 11785 (,)cioo 13252 [,]cicc 13255 ↾_t crest 17331 TopOpenctopn 17332 topGenctg 17348 ℂ_fldccnfld 21300 Topctop 22828 intcnt 22952 CnP ccnp 23160 𝐿¹cibl 25565 ∫citg 25566 lim_ℂ climc 25810 D cdv 25811

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 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cc 10337 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096

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 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-disj 5063 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-ofr 7620 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-oadd 8398 df-omul 8399 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-dju 9805 df-card 9843 df-acn 9846 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ioc 13257 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-mod 13781 df-seq 13916 df-exp 13976 df-hash 14245 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-clim 15402 df-rlim 15403 df-sum 15601 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-submnd 18700 df-mulg 18989 df-cntz 19237 df-cmn 19702 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-ntr 22955 df-cn 23162 df-cnp 23163 df-cmp 23322 df-tx 23497 df-hmeo 23690 df-xms 24255 df-ms 24256 df-tms 24257 df-cncf 24818 df-ovol 25412 df-vol 25413 df-mbf 25567 df-itg1 25568 df-itg2 25569 df-ibl 25570 df-itg 25571 df-0p 25618 df-limc 25814 df-dv 25815

This theorem is referenced by: ftc1cn 25997

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