ftc1 - Metamath Proof Explorer

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Theorem ftc1 25206

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 23966	. . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾_t ℝ)
4	1, 3	eqtr4i 2769	. . . . . 6 ⊢ 𝐽 = (topGen‘ran (,))
5		retop 23925	. . . . . 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 13161	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
11	8, 9, 10	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
12		iooretop 23929	. . . . . 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 13165	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
17		uniretop 23926	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
18	4	unieqi 4852	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
19	17, 18	eqtr4i 2769	. . . . 5 ⊢ ℝ = ∪ 𝐽
20	19	ssntr 22209	. . . 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 3922	. 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 2738	. . 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 25205	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
33		ax-resscn 10928	. . . 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 25202	. . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
36	24, 8, 9, 25, 26, 27, 28, 35	ftc1lem2 25200	. . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
37	1, 2, 31, 34, 36, 11	eldv 25062	. 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 710	1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 ∖ cdif 3884 ⊆ wss 3887 {csn 4561 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 ran crn 5590 ‘cfv 6433 (class class class)co 7275 ℂcc 10869 ℝcr 10870 ≤ cle 11010 − cmin 11205 / cdiv 11632 (,)cioo 13079 [,]cicc 13082 ↾_t crest 17131 TopOpenctopn 17132 topGenctg 17148 ℂ_fldccnfld 20597 Topctop 22042 intcnt 22168 CnP ccnp 22376 𝐿¹cibl 24781 ∫citg 24782 lim_ℂ climc 25026 D cdv 25027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cc 10191 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-symdif 4176 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-disj 5040 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-ofr 7534 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-oadd 8301 df-omul 8302 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-dju 9659 df-card 9697 df-acn 9700 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ioc 13084 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-fl 13512 df-mod 13590 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-rlim 15198 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-ntr 22171 df-cn 22378 df-cnp 22379 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-ovol 24628 df-vol 24629 df-mbf 24783 df-itg1 24784 df-itg2 24785 df-ibl 24786 df-itg 24787 df-0p 24834 df-limc 25030 df-dv 25031

This theorem is referenced by: ftc1cn 25207

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