dvcnv - Metamath Proof Explorer

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Theorem dvcnv 25888

Description: A weak version of dvcnvre 25931, 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 25814	. . . . 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 25812	. . . . . . . 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 6815	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
8		f1of 6803	. . . . . . . . 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 25810	. . . . . . . . . 10 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
12	10, 11	eqsstrrdi 3995	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
13	12, 5	sstrd 3960	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
14	9, 13	fssd 6708	. . . . . . 7 ⊢ (𝜑 → ^◡𝐹:𝑌⟶ℂ)
15		dvcnv.k	. . . . . . . . 9 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
16		dvcnv.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
17	16	cnfldtopon 24677	. . . . . . . . . 10 ⊢ 𝐽 ∈ (TopOn‘ℂ)
18		resttopon 23055	. . . . . . . . . 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 2833	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
21		dvcnv.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
22		toponss 22821	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆)
23	20, 21, 22	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
24	5, 14, 23	dvbss 25809	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D ^◡𝐹) ⊆ 𝑌)
25		f1ocnvfv2 7255	. . . . . . . . 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 7059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (^◡𝐹‘𝑥) ∈ 𝑋)
36	16, 15, 27, 28, 29, 31, 32, 34, 35	dvcnvlem 25887	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥))(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))))
37	26, 36	eqbrtrrd 5134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))))
38		reldv 25778	. . . . . . . 8 ⊢ Rel (𝑆 D ^◡𝐹)
39	38	releldmi 5915	. . . . . . 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 3971	. . . . 5 ⊢ (𝜑 → dom (𝑆 D ^◡𝐹) = 𝑌)
42	41	feq2d 6675	. . . 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 6932	. 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 6695	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → Fun (𝑆 D ^◡𝐹))
47		funbrfv 6912	. . . 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 5203	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑆 D ^◡𝐹)‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥)))))
50	44, 49	eqtrd 2765	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 3917 {cpr 4594 class class class wbr 5110 ↦ cmpt 5191 ^◡ccnv 5640 dom cdm 5641 ran crn 5642 Fun wfun 6508 ⟶wf 6510 –1-1-onto→wf1o 6513 ‘cfv 6514 (class class class)co 7390 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 / cdiv 11842 ↾_t crest 17390 TopOpenctopn 17391 ℂ_fldccnfld 21271 TopOnctopon 22804 –cn→ccncf 24776 D cdv 25771

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

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 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-icc 13320 df-fz 13476 df-fzo 13623 df-seq 13974 df-exp 14034 df-hash 14303 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-mulg 19007 df-cntz 19256 df-cmn 19719 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-tms 24217 df-cncf 24778 df-limc 25774 df-dv 25775

This theorem is referenced by: dvcnvre 25931 dvlog 26567

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