ftc1cn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ftc1cn		Structured version Visualization version GIF version

Theorem ftc1cn 26270

Description: Strengthen the assumptions of ftc1 26269 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 26134	. . . . 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 6707	. . 3 ⊢ (𝜑 → Fun (ℝ D 𝐺))
4		ax-resscn 11181	. . . . . . 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 3954	. . . . . . 7 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ (𝐴(,)𝐵))
11		ioossre 13460	. . . . . . . 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 25121	. . . . . . . 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 26263	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
18		iccssre 13482	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
19	7, 8, 18	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝐴[,]𝐵) ⊆ ℝ)
20		tgioo4 25031	. . . . . 6 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
21		eqid 2760	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
22	5, 17, 19, 20, 21	dvbssntr 26127	. . . . 5 ⊢ (𝜑 → dom (ℝ D 𝐺) ⊆ ((int‘(topGen‘ran (,)))‘(𝐴[,]𝐵)))
23		iccntr 25048	. . . . . 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 3967	. . . 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 3954	. . . . . 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 3940	. . . . . . . . . 10 ⊢ (𝐴(,)𝐵) ⊆ ℂ
34		ssid 3953	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
35		eqid 2760	. . . . . . . . . . 11 ⊢ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
36	21	cnfldtopon 25008	. . . . . . . . . . . 12 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
37	36	toponrestid 23146	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
38	21, 35, 37	cncfcn 25138	. . . . . . . . . 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 2870	. . . . . . . 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 23386	. . . . . . . . . . 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 23139	. . . . . . . . . 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 2846	. . . . . . . 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 2760	. . . . . . . 8 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵)) = ∪ ((TopOpen‘ℂ_fld) ↾_t (𝐴(,)𝐵))
50	49	cncnpi 23503	. . . . . . 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 26269	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦(ℝ D 𝐺)(𝐹‘𝑦))
53		vex 3454	. . . . . 6 ⊢ 𝑦 ∈ V
54		fvex 6891	. . . . . 6 ⊢ (𝐹‘𝑦) ∈ V
55	53, 54	breldm 5892	. . . . 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 3952	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
58		df-fn 6536	. . 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 6703	. 2 ⊢ (𝜑 → 𝐹 Fn (𝐴(,)𝐵))
61	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → Fun (ℝ D 𝐺))
62		funbrfv 6926	. . 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 7025	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 2145 ⊆ wss 3899 ∪ cuni 4867 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ran crn 5656 Fun wfun 6527 Fn wfn 6528 ⟶wf 6529 ‘cfv 6533 (class class class)co 7413 ℂcc 11122 ℝcr 11123 ≤ cle 11268 (,)cioo 13398 [,]cicc 13401 ↾_t crest 17505 TopOpenctopn 17506 topGenctg 17522 ℂ_fldccnfld 21585 TopOnctopon 23135 intcnt 23242 Cn ccn 23449 CnP ccnp 23450 –cn→ccncf 25104 𝐿¹cibl 25845 ∫citg 25846 D cdv 26090

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-inf2 9620 ax-cc 10437 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203

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 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-symdif 4199 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-disj 5071 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-ofr 7679 df-om 7863 df-1st 7986 df-2nd 7987 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-oadd 8459 df-omul 8460 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-fsupp 9332 df-fi 9381 df-sup 9412 df-inf 9413 df-oi 9482 df-dju 9906 df-card 9944 df-acn 9947 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-div 11896 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-q 12998 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-ioo 13402 df-ioc 13403 df-ico 13404 df-icc 13405 df-fz 13562 df-fzo 13710 df-fl 13853 df-mod 13931 df-seq 14066 df-exp 14126 df-hash 14395 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-clim 15575 df-rlim 15576 df-sum 15774 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-starv 17357 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-unif 17365 df-hom 17366 df-cco 17367 df-rest 17507 df-topn 17508 df-0g 17526 df-gsum 17527 df-topgen 17528 df-pt 17529 df-prds 17532 df-xrs 17588 df-qtop 17593 df-imas 17594 df-xps 17596 df-mre 17670 df-mrc 17671 df-acs 17673 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-mulg 19191 df-cntz 19444 df-cmn 19909 df-psmet 21577 df-xmet 21578 df-met 21579 df-bl 21580 df-mopn 21581 df-fbas 21582 df-fg 21583 df-cnfld 21586 df-top 23119 df-topon 23136 df-topsp 23158 df-bases 23171 df-cld 23244 df-ntr 23245 df-cls 23246 df-nei 23323 df-lp 23361 df-perf 23362 df-cn 23452 df-cnp 23453 df-haus 23540 df-cmp 23612 df-tx 23788 df-hmeo 23981 df-fil 24072 df-fm 24164 df-flim 24165 df-flf 24166 df-xms 24546 df-ms 24547 df-tms 24548 df-cncf 25106 df-ovol 25692 df-vol 25693 df-mbf 25847 df-itg1 25848 df-itg2 25849 df-ibl 25850 df-itg 25851 df-0p 25898 df-limc 26093 df-dv 26094

This theorem is used by: ftc2 26271 itgsubstlem 26275

W3C validator