dvcnv - Metamath Proof Explorer

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Theorem dvcnv 25914

Description: A weak version of dvcnvre 25957, valid for both real and complex domains but under the hypothesis that the inverse function is already known to be continuous, and the image set is known to be open. A more advanced proof can show that these conditions are unnecessary. (Contributed by Mario Carneiro, 25-Feb-2015.) (Revised by Mario Carneiro, 8-Sep-2015.)

**Hypotheses**
Ref	Expression
dvcnv.j	⊢ 𝐽 = (TopOpen‘ℂ_fld)
dvcnv.k	⊢ 𝐾 = (𝐽 ↾_t 𝑆)
dvcnv.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvcnv.y	⊢ (𝜑 → 𝑌 ∈ 𝐾)
dvcnv.f	⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
dvcnv.i	⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
dvcnv.d	⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
dvcnv.z	⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))

**Assertion**
Ref	Expression
dvcnv	⊢ (𝜑 → (𝑆 D ^◡𝐹) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥)))))

Distinct variable groups: 𝑥,𝐹 𝑥,𝐽 𝜑,𝑥 𝑥,𝑆 𝑥,𝑌 Allowed substitution hints: 𝐾(𝑥) 𝑋(𝑥)

**Proof of Theorem dvcnv**
Step	Hyp	Ref	Expression
1		dvcnv.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		dvfg 25840	. . . . 5 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 D ^◡𝐹):dom (𝑆 D ^◡𝐹)⟶ℂ)
3	1, 2	syl 17	. . . 4 ⊢ (𝜑 → (𝑆 D ^◡𝐹):dom (𝑆 D ^◡𝐹)⟶ℂ)
4		recnprss 25838	. . . . . . . 8 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
5	1, 4	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
6		dvcnv.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑋–1-1-onto→𝑌)
7		f1ocnv 6794	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
8		f1of 6782	. . . . . . . . 9 ⊢ (^◡𝐹:𝑌–1-1-onto→𝑋 → ^◡𝐹:𝑌⟶𝑋)
9	6, 7, 8	3syl 18	. . . . . . . 8 ⊢ (𝜑 → ^◡𝐹:𝑌⟶𝑋)
10		dvcnv.d	. . . . . . . . . 10 ⊢ (𝜑 → dom (𝑆 D 𝐹) = 𝑋)
11		dvbsss 25836	. . . . . . . . . 10 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
12	10, 11	eqsstrrdi 3989	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
13	12, 5	sstrd 3954	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
14	9, 13	fssd 6687	. . . . . . 7 ⊢ (𝜑 → ^◡𝐹:𝑌⟶ℂ)
15		dvcnv.k	. . . . . . . . 9 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
16		dvcnv.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
17	16	cnfldtopon 24703	. . . . . . . . . 10 ⊢ 𝐽 ∈ (TopOn‘ℂ)
18		resttopon 23081	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
19	17, 5, 18	sylancr 587	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
20	15, 19	eqeltrid 2832	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
21		dvcnv.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
22		toponss 22847	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆)
23	20, 21, 22	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
24	5, 14, 23	dvbss 25835	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D ^◡𝐹) ⊆ 𝑌)
25		f1ocnvfv2 7234	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
26	6, 25	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
27	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ {ℝ, ℂ})
28	21	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑌 ∈ 𝐾)
29	6	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋–1-1-onto→𝑌)
30		dvcnv.i	. . . . . . . . . 10 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
31	30	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ^◡𝐹 ∈ (𝑌–cn→𝑋))
32	10	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → dom (𝑆 D 𝐹) = 𝑋)
33		dvcnv.z	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))
34	33	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ¬ 0 ∈ ran (𝑆 D 𝐹))
35	9	ffvelcdmda 7038	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (^◡𝐹‘𝑥) ∈ 𝑋)
36	16, 15, 27, 28, 29, 31, 32, 34, 35	dvcnvlem 25913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥))(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))))
37	26, 36	eqbrtrrd 5126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))))
38		reldv 25804	. . . . . . . 8 ⊢ Rel (𝑆 D ^◡𝐹)
39	38	releldmi 5901	. . . . . . 7 ⊢ (𝑥(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))) → 𝑥 ∈ dom (𝑆 D ^◡𝐹))
40	37, 39	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥 ∈ dom (𝑆 D ^◡𝐹))
41	24, 40	eqelssd 3965	. . . . 5 ⊢ (𝜑 → dom (𝑆 D ^◡𝐹) = 𝑌)
42	41	feq2d 6654	. . . 4 ⊢ (𝜑 → ((𝑆 D ^◡𝐹):dom (𝑆 D ^◡𝐹)⟶ℂ ↔ (𝑆 D ^◡𝐹):𝑌⟶ℂ))
43	3, 42	mpbid 232	. . 3 ⊢ (𝜑 → (𝑆 D ^◡𝐹):𝑌⟶ℂ)
44	43	feqmptd 6911	. 2 ⊢ (𝜑 → (𝑆 D ^◡𝐹) = (𝑥 ∈ 𝑌 ↦ ((𝑆 D ^◡𝐹)‘𝑥)))
45	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑆 D ^◡𝐹):dom (𝑆 D ^◡𝐹)⟶ℂ)
46	45	ffund 6674	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → Fun (𝑆 D ^◡𝐹))
47		funbrfv 6891	. . . 4 ⊢ (Fun (𝑆 D ^◡𝐹) → (𝑥(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))) → ((𝑆 D ^◡𝐹)‘𝑥) = (1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥)))))
48	46, 37, 47	sylc 65	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ((𝑆 D ^◡𝐹)‘𝑥) = (1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))))
49	48	mpteq2dva 5195	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑆 D ^◡𝐹)‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥)))))
50	44, 49	eqtrd 2764	1 ⊢ (𝜑 → (𝑆 D ^◡𝐹) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥)))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3911 {cpr 4587 class class class wbr 5102 ↦ cmpt 5183 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 Fun wfun 6493 ⟶wf 6495 –1-1-onto→wf1o 6498 ‘cfv 6499 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 / cdiv 11811 ↾_t crest 17359 TopOpenctopn 17360 ℂ_fldccnfld 21296 TopOnctopon 22830 –cn→ccncf 24802 D cdv 25797

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-submnd 18693 df-mulg 18982 df-cntz 19231 df-cmn 19696 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22866 df-cld 22939 df-ntr 22940 df-cls 22941 df-nei 23018 df-lp 23056 df-perf 23057 df-cn 23147 df-cnp 23148 df-haus 23235 df-tx 23482 df-hmeo 23675 df-fil 23766 df-fm 23858 df-flim 23859 df-flf 23860 df-xms 24241 df-ms 24242 df-tms 24243 df-cncf 24804 df-limc 25800 df-dv 25801

This theorem is referenced by: dvcnvre 25957 dvlog 26593

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