ftc1cn - Metamath Proof Explorer

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Theorem ftc1cn 26418

Description: Strengthen the assumptions of ftc1 26417 to when the function 𝐹 is continuous on the entire interval (𝐴, 𝐵); in this case we can calculate D 𝐺 exactly. (Contributed by Mario Carneiro, 1-Sep-2014.)

**Hypotheses**
Ref	Expression
ftc1cn.g	⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
ftc1cn.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
ftc1cn.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
ftc1cn.le	⊢ (𝜑 → 𝐴 ≤ 𝐵)
ftc1cn.f	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
ftc1cn.i	⊢ (𝜑 → 𝐹 ∈ 𝐿¹)

**Assertion**
Ref	Expression
ftc1cn	⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)

Distinct variable groups: 𝑥,𝑡,𝐴 𝑡,𝐵,𝑥 𝑡,𝐹,𝑥 𝜑,𝑡,𝑥 Allowed substitution hints: 𝐺(𝑥, 𝑡)

Proof of Theorem ftc1cn Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvf 26282	. . . . 5 ⊢ (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (ℝ D 𝐺):dom (ℝ D 𝐺)⟶ℂ)
3	2	ffund 6744	. . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺))
4		ax-resscn 11296	. . . . . . 7 ⊢ ℝ ⊆ ℂ
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℝ ⊆ ℂ)
6		ftc1cn.g	. . . . . . 7 ⊢ 𝐺 = (𝑥 ∈ (𝐴[,]𝐵) ↦ ∫(𝐴(,)𝑥)(𝐹‘𝑡) d𝑡)
7		ftc1cn.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
8		ftc1cn.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
9		ftc1cn.le	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
10		ssidd 3936	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
11		ioossre 13578	. . . . . . . 8 ⊢ (𝐴(,)𝐵) ⊆ ℝ
12	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
13		ftc1cn.i	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐿¹)
14		ftc1cn.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
15		cncff 25269	. . . . . . . 8 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → 𝐹:(𝐴(,)𝐵)⟶ℂ)
16	14, 15	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℂ)
17	6, 7, 8, 9, 10, 12, 13, 16	ftc1lem2 26411	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
18		iccssre 13600	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
19	7, 8, 18	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
20		tgioo4 25179	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2763	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5, 17, 19, 20, 21	dvbssntr 26275	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)))
23		iccntr 25196	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
24	7, 8, 23	syl2anc 596	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)) = (𝐴(,)𝐵))
25	22, 24	sseqtrd 3949	. . . 4 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ (𝐴(,)𝐵))
26	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ∈ ℝ)
27	8	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐵 ∈ ℝ)
28	9	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐴 ≤ 𝐵)
29		ssidd 3936	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
30	11	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐴(,)𝐵) ⊆ ℝ)
31	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ 𝐿¹)
32		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ (𝐴(,)𝐵))
33	11, 4	sstri 3922	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℂ
34		ssid 3935	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
35		eqid 2763	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
36	21	cnfldtopon 25156	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
37	36	toponrestid 23294	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
38	21, 35, 37	cncfcn 25286	. . . . . . . . . 10 ⊢ (((𝐴(,)𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
39	33, 34, 38	mp2an 705	. . . . . . . . 9 ⊢ ((𝐴(,)𝐵)–cn→ℂ) = (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld))
40	14, 39	eleqtrdi 2874	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
41	40	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)))
42	33	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℂ)
43		resttopon 23534	. . . . . . . . . . 11 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (𝐴(,)𝐵) ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
44	36, 42, 43	sylancr 599	. . . . . . . . . 10 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)))
45		toponuni 23287	. . . . . . . . . 10 ⊢ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) ∈ (TopOn‘(𝐴(,)𝐵)) → (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
46	44, 45	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (𝐴(,)𝐵) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
47	46	eleq2d 2850	. . . . . . . 8 ⊢ (𝜑 → (𝑦 ∈ (𝐴(,)𝐵) ↔ 𝑦 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))))
48	47	biimpa 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)))
49		eqid 2763	. . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
50	49	cncnpi 23651	. . . . . . 7 ⊢ ((𝐹 ∈ (((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) Cn (TopOpen‘ℂ_fld)) ∧ 𝑦 ∈ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
51	41, 48, 50	syl2anc 596	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐹 ∈ ((((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) CnP (TopOpen‘ℂ_fld))‘𝑦))
52	6, 26, 27, 28, 29, 30, 31, 32, 51, 20, 35, 21	ftc1 26417	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦(ℝ D 𝐺)(𝐹‘𝑦))
53		vex 3458	. . . . . 6 ⊢ 𝑦 ∈ V
54		fvex 6929	. . . . . 6 ⊢ (𝐹‘𝑦) ∈ V
55	53, 54	breldm 5902	. . . . 5 ⊢ (𝑦(ℝ D 𝐺)(𝐹‘𝑦) → 𝑦 ∈ dom (ℝ D 𝐺))
56	52, 55	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ dom (ℝ D 𝐺))
57	25, 56	eqelssd 3934	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
58		df-fn 6569	. . 3 ⊢ ((ℝ D 𝐺) Fn (𝐴(,)𝐵) ↔ (Fun (ℝ D 𝐺) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵)))
59	3, 57, 58	sylanbrc 595	. 2 ⊢ (𝜑 → (ℝ D 𝐺) Fn (𝐴(,)𝐵))
60	16	ffnd 6740	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵))
61	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺))
62		funbrfv 6964	. . 3 ⊢ (Fun (ℝ D 𝐺) → (𝑦(ℝ D 𝐺)(𝐹‘𝑦) → ((ℝ D 𝐺)‘𝑦) = (𝐹‘𝑦)))
63	61, 52, 62	sylc 66	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐺)‘𝑦) = (𝐹‘𝑦))
64	59, 60, 63	eqfnfvd 7063	1 ⊢ (𝜑 → (ℝ D 𝐺) = 𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ∪ cuni 4872 class class class wbr 5108 ↦ cmpt 5192 dom cdm 5658 ran crn 5659 Fun wfun 6560 Fn wfn 6561 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 ≤ cle 11383 (,)cioo 13516 [,]cicc 13519 ↾_t crest 17641 TopOpenctopn 17642 topGenctg 17658 ℂ_fldccnfld 21729 TopOnctopon 23283 intcnt 23390 Cn ccn 23597 CnP ccnp 23598 –cn→ccncf 25252 𝐿¹cibl 25993 ∫citg 25994 D cdv 26238

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cc 10552 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-symdif 4201 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-disj 5076 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-ofr 7726 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-oadd 8518 df-omul 8519 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-dju 10019 df-card 10057 df-acn 10060 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-mod 14050 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-rlim 15700 df-sum 15898 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-ovol 25840 df-vol 25841 df-mbf 25995 df-itg1 25996 df-itg2 25997 df-ibl 25998 df-itg 25999 df-0p 26046 df-limc 26241 df-dv 26242

This theorem is used by: ftc2 26419 itgsubstlem 26423

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