ftc1 - Metamath Proof Explorer

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Theorem ftc1 26021

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 24763	. . . . . . 7 ⊢ (topGen‘ran (,)) = (𝐿 ↾_t ℝ)
4	1, 3	eqtr4i 2756	. . . . . 6 ⊢ 𝐽 = (topGen‘ran (,))
5		retop 24722	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
6	4, 5	eqeltri 2821	. . . . 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 13441	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
11	8, 9, 10	syl2anc 582	. . . 4 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
12		iooretop 24726	. . . . . 6 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
13	12, 4	eleqtrri 2824	. . . . 5 ⊢ (𝐴(,)𝐵) ∈ 𝐽
14	13	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ 𝐽)
15		ioossicc 13445	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵)
16	15	a1i 11	. . . 4 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴[,]𝐵))
17		uniretop 24723	. . . . . 6 ⊢ ℝ = ∪ (topGen‘ran (,))
18	4	unieqi 4921	. . . . . 6 ⊢ ∪ 𝐽 = ∪ (topGen‘ran (,))
19	17, 18	eqtr4i 2756	. . . . 5 ⊢ ℝ = ∪ 𝐽
20	19	ssntr 23006	. . . 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 837	. . 3 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ((int‘𝐽)‘(𝐴[,]𝐵)))
22		ftc1.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ (𝐴(,)𝐵))
23	21, 22	sseldd 3977	. 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 2725	. . 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 26020	. 2 ⊢ (𝜑 → (𝐹‘𝐶) ∈ ((𝑧 ∈ ((𝐴[,]𝐵) ∖ {𝐶}) ↦ (((𝐺‘𝑧) − (𝐺‘𝐶)) / (𝑧 − 𝐶))) lim_ℂ 𝐶))
33		ax-resscn 11197	. . . 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 26017	. . . 4 ⊢ (𝜑 → 𝐹:𝐷⟶ℂ)
36	24, 8, 9, 25, 26, 27, 28, 35	ftc1lem2 26015	. . 3 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
37	1, 2, 31, 34, 36, 11	eldv 25871	. 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 711	1 ⊢ (𝜑 → 𝐶(ℝ D 𝐺)(𝐹‘𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 ∪ cuni 4909 class class class wbr 5149 ↦ cmpt 5232 ran crn 5679 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 ≤ cle 11281 − cmin 11476 / cdiv 11903 (,)cioo 13359 [,]cicc 13362 ↾_t crest 17405 TopOpenctopn 17406 topGenctg 17422 ℂ_fldccnfld 21296 Topctop 22839 intcnt 22965 CnP ccnp 23173 𝐿¹cibl 25590 ∫citg 25591 lim_ℂ climc 25835 D cdv 25836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9666 ax-cc 10460 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-symdif 4241 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-ofr 7686 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-dju 9926 df-card 9964 df-acn 9967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ioc 13364 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-clim 15468 df-rlim 15469 df-sum 15669 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-ntr 22968 df-cn 23175 df-cnp 23176 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-ovol 25437 df-vol 25438 df-mbf 25592 df-itg1 25593 df-itg2 25594 df-ibl 25595 df-itg 25596 df-0p 25643 df-limc 25839 df-dv 25840

This theorem is referenced by: ftc1cn 26022

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