dvcnv - Metamath Proof Explorer

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Theorem dvcnv 26039

Description: A weak version of dvcnvre 26081, 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 25968	. . . . 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 25966	. . . . . . . 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 6819	. . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ^◡𝐹:𝑌–1-1-onto→𝑋)
8		f1of 6806	. . . . . . . . 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 25964	. . . . . . . . . 10 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆
12	10, 11	eqsstrrdi 3981	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
13	12, 5	sstrd 3946	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ ℂ)
14	9, 13	fssd 6709	. . . . . . 7 ⊢ (𝜑 → ^◡𝐹:𝑌⟶ℂ)
15		dvcnv.k	. . . . . . . . 9 ⊢ 𝐾 = (𝐽 ↾_t 𝑆)
16		dvcnv.j	. . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
17	16	cnfldtopon 24842	. . . . . . . . . 10 ⊢ 𝐽 ∈ (TopOn‘ℂ)
18		resttopon 23221	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
19	17, 5, 18	sylancr 596	. . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾_t 𝑆) ∈ (TopOn‘𝑆))
20	15, 19	eqeltrid 2866	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆))
21		dvcnv.y	. . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐾)
22		toponss 22987	. . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆)
23	20, 21, 22	syl2anc 593	. . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑆)
24	5, 14, 23	dvbss 25963	. . . . . 6 ⊢ (𝜑 → dom (𝑆 D ^◡𝐹) ⊆ 𝑌)
25		f1ocnvfv2 7261	. . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
26	6, 25	sylan 589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥)) = 𝑥)
27	1	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ {ℝ, ℂ})
28	21	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑌 ∈ 𝐾)
29	6	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋–1-1-onto→𝑌)
30		dvcnv.i	. . . . . . . . . 10 ⊢ (𝜑 → ^◡𝐹 ∈ (𝑌–cn→𝑋))
31	30	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ^◡𝐹 ∈ (𝑌–cn→𝑋))
32	10	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → dom (𝑆 D 𝐹) = 𝑋)
33		dvcnv.z	. . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹))
34	33	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ¬ 0 ∈ ran (𝑆 D 𝐹))
35	9	ffvelcdmda 7065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (^◡𝐹‘𝑥) ∈ 𝑋)
36	16, 15, 27, 28, 29, 31, 32, 34, 35	dvcnvlem 26038	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(^◡𝐹‘𝑥))(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))))
37	26, 36	eqbrtrrd 5124	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑆 D ^◡𝐹)(1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥))))
38		reldv 25932	. . . . . . . 8 ⊢ Rel (𝑆 D ^◡𝐹)
39	38	releldmi 5924	. . . . . . 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 3957	. . . . 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 234	. . 3 ⊢ (𝜑 → (𝑆 D ^◡𝐹):𝑌⟶ℂ)
44	43	feqmptd 6935	. 2 ⊢ (𝜑 → (𝑆 D ^◡𝐹) = (𝑥 ∈ 𝑌 ↦ ((𝑆 D ^◡𝐹)‘𝑥)))
45	3	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑆 D ^◡𝐹):dom (𝑆 D ^◡𝐹)⟶ℂ)
46	45	ffund 6696	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → Fun (𝑆 D ^◡𝐹))
47		funbrfv 6915	. . . 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 5193	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑆 D ^◡𝐹)‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(^◡𝐹‘𝑥)))))
50	44, 49	eqtrd 2797	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 399 = wceq 1560 ∈ wcel 2142 ⊆ wss 3904 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 ^◡ccnv 5646 dom cdm 5647 ran crn 5648 Fun wfun 6515 ⟶wf 6517 –1-1-onto→wf1o 6520 ‘cfv 6521 (class class class)co 7396 ℂcc 11071 ℝcr 11072 0cc0 11073 1c1 11074 / cdiv 11844 ↾_t crest 17449 TopOpenctopn 17450 ℂ_fldccnfld 21424 TopOnctopon 22970 –cn→ccncf 24938 D cdv 25925

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1815 ax-4 1829 ax-5 1930 ax-6 1987 ax-7 2028 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-rep 5227 ax-sep 5246 ax-nul 5256 ax-pow 5322 ax-pr 5390 ax-un 7718 ax-cnex 11129 ax-resscn 11130 ax-1cn 11131 ax-icn 11132 ax-addcl 11133 ax-addrcl 11134 ax-mulcl 11135 ax-mulrcl 11136 ax-mulcom 11137 ax-addass 11138 ax-mulass 11139 ax-distr 11140 ax-i2m1 11141 ax-1ne0 11142 ax-1rid 11143 ax-rnegex 11144 ax-rrecex 11145 ax-cnre 11146 ax-pre-lttri 11147 ax-pre-lttrn 11148 ax-pre-ltadd 11149 ax-pre-mulgt0 11150 ax-pre-sup 11151

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1099 df-3an 1100 df-tru 1563 df-fal 1573 df-ex 1800 df-nf 1804 df-sb 2091 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3456 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4481 df-pw 4557 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4906 df-iun 4951 df-iin 4952 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5542 df-eprel 5547 df-po 5555 df-so 5556 df-fr 5600 df-se 5601 df-we 5602 df-xp 5653 df-rel 5654 df-cnv 5655 df-co 5656 df-dm 5657 df-rn 5658 df-res 5659 df-ima 5660 df-pred 6288 df-ord 6349 df-on 6350 df-lim 6351 df-suc 6352 df-iota 6477 df-fun 6523 df-fn 6524 df-f 6525 df-f1 6526 df-fo 6527 df-f1o 6528 df-fv 6529 df-isom 6530 df-riota 7353 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7660 df-om 7847 df-1st 7970 df-2nd 7971 df-supp 8141 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8678 df-map 8810 df-pm 8811 df-ixp 8880 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-fsupp 9308 df-fi 9357 df-sup 9388 df-inf 9389 df-oi 9458 df-card 9897 df-pnf 11218 df-mnf 11219 df-xr 11220 df-ltxr 11221 df-le 11222 df-sub 11416 df-neg 11417 df-div 11845 df-nn 12211 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12482 df-z 12569 df-dec 12689 df-uz 12840 df-q 12950 df-rp 12994 df-xneg 13114 df-xadd 13115 df-xmul 13116 df-icc 13356 df-fz 13513 df-fzo 13660 df-seq 14015 df-exp 14075 df-hash 14344 df-cj 15126 df-re 15127 df-im 15128 df-sqrt 15262 df-abs 15263 df-struct 17183 df-sets 17200 df-slot 17218 df-ndx 17230 df-base 17246 df-ress 17267 df-plusg 17299 df-mulr 17300 df-starv 17301 df-sca 17302 df-vsca 17303 df-ip 17304 df-tset 17305 df-ple 17306 df-ds 17308 df-unif 17309 df-hom 17310 df-cco 17311 df-rest 17451 df-topn 17452 df-0g 17470 df-gsum 17471 df-topgen 17472 df-pt 17473 df-prds 17476 df-xrs 17532 df-qtop 17537 df-imas 17538 df-xps 17540 df-mre 17614 df-mrc 17615 df-acs 17617 df-mgm 18674 df-sgrp 18753 df-mnd 18769 df-submnd 18818 df-mulg 19110 df-cntz 19357 df-cmn 19822 df-psmet 21416 df-xmet 21417 df-met 21418 df-bl 21419 df-mopn 21420 df-fbas 21421 df-fg 21422 df-cnfld 21425 df-top 22954 df-topon 22971 df-topsp 22993 df-bases 23006 df-cld 23079 df-ntr 23080 df-cls 23081 df-nei 23158 df-lp 23196 df-perf 23197 df-cn 23287 df-cnp 23288 df-haus 23375 df-tx 23622 df-hmeo 23815 df-fil 23906 df-fm 23998 df-flim 23999 df-flf 24000 df-xms 24380 df-ms 24381 df-tms 24382 df-cncf 24940 df-limc 25928 df-dv 25929

This theorem is referenced by: dvcnvre 26081 dvlog 26716

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