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Mirrors > Home > MPE Home > Th. List > dvcnv | Structured version Visualization version GIF version |
Description: A weak version of dvcnvre 24122, 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.) |
Ref | Expression |
---|---|
dvcnv.j | ⊢ 𝐽 = (TopOpen‘ℂfld) |
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)) |
Ref | Expression |
---|---|
dvcnv | ⊢ (𝜑 → (𝑆 D ◡𝐹) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(◡𝐹‘𝑥))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvcnv.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | dvfg 24010 | . . . . 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 24008 | . . . . . . . 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 6369 | . . . . . . . . 9 ⊢ (𝐹:𝑋–1-1-onto→𝑌 → ◡𝐹:𝑌–1-1-onto→𝑋) | |
8 | f1of 6357 | . . . . . . . . 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 24006 | . . . . . . . . . 10 ⊢ dom (𝑆 D 𝐹) ⊆ 𝑆 | |
12 | 10, 11 | syl6eqssr 3853 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
13 | 12, 5 | sstrd 3809 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ⊆ ℂ) |
14 | 9, 13 | fssd 6271 | . . . . . . 7 ⊢ (𝜑 → ◡𝐹:𝑌⟶ℂ) |
15 | dvcnv.k | . . . . . . . . 9 ⊢ 𝐾 = (𝐽 ↾t 𝑆) | |
16 | dvcnv.j | . . . . . . . . . . 11 ⊢ 𝐽 = (TopOpen‘ℂfld) | |
17 | 16 | cnfldtopon 22913 | . . . . . . . . . 10 ⊢ 𝐽 ∈ (TopOn‘ℂ) |
18 | resttopon 21293 | . . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → (𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆)) | |
19 | 17, 5, 18 | sylancr 582 | . . . . . . . . 9 ⊢ (𝜑 → (𝐽 ↾t 𝑆) ∈ (TopOn‘𝑆)) |
20 | 15, 19 | syl5eqel 2883 | . . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑆)) |
21 | dvcnv.y | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝐾) | |
22 | toponss 21059 | . . . . . . . 8 ⊢ ((𝐾 ∈ (TopOn‘𝑆) ∧ 𝑌 ∈ 𝐾) → 𝑌 ⊆ 𝑆) | |
23 | 20, 21, 22 | syl2anc 580 | . . . . . . 7 ⊢ (𝜑 → 𝑌 ⊆ 𝑆) |
24 | 5, 14, 23 | dvbss 24005 | . . . . . 6 ⊢ (𝜑 → dom (𝑆 D ◡𝐹) ⊆ 𝑌) |
25 | f1ocnvfv2 6762 | . . . . . . . . 9 ⊢ ((𝐹:𝑋–1-1-onto→𝑌 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) | |
26 | 6, 25 | sylan 576 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) |
27 | 1 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑆 ∈ {ℝ, ℂ}) |
28 | 21 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑌 ∈ 𝐾) |
29 | 6 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝐹:𝑋–1-1-onto→𝑌) |
30 | dvcnv.i | . . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹 ∈ (𝑌–cn→𝑋)) | |
31 | 30 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ◡𝐹 ∈ (𝑌–cn→𝑋)) |
32 | 10 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → dom (𝑆 D 𝐹) = 𝑋) |
33 | dvcnv.z | . . . . . . . . . 10 ⊢ (𝜑 → ¬ 0 ∈ ran (𝑆 D 𝐹)) | |
34 | 33 | adantr 473 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → ¬ 0 ∈ ran (𝑆 D 𝐹)) |
35 | 9 | ffvelrnda 6586 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (◡𝐹‘𝑥) ∈ 𝑋) |
36 | 16, 15, 27, 28, 29, 31, 32, 34, 35 | dvcnvlem 24079 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝐹‘(◡𝐹‘𝑥))(𝑆 D ◡𝐹)(1 / ((𝑆 D 𝐹)‘(◡𝐹‘𝑥)))) |
37 | 26, 36 | eqbrtrrd 4868 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → 𝑥(𝑆 D ◡𝐹)(1 / ((𝑆 D 𝐹)‘(◡𝐹‘𝑥)))) |
38 | reldv 23974 | . . . . . . . 8 ⊢ Rel (𝑆 D ◡𝐹) | |
39 | 38 | releldmi 5567 | . . . . . . 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 3819 | . . . . 5 ⊢ (𝜑 → dom (𝑆 D ◡𝐹) = 𝑌) |
42 | 41 | feq2d 6243 | . . . 4 ⊢ (𝜑 → ((𝑆 D ◡𝐹):dom (𝑆 D ◡𝐹)⟶ℂ ↔ (𝑆 D ◡𝐹):𝑌⟶ℂ)) |
43 | 3, 42 | mpbid 224 | . . 3 ⊢ (𝜑 → (𝑆 D ◡𝐹):𝑌⟶ℂ) |
44 | 43 | feqmptd 6475 | . 2 ⊢ (𝜑 → (𝑆 D ◡𝐹) = (𝑥 ∈ 𝑌 ↦ ((𝑆 D ◡𝐹)‘𝑥))) |
45 | 3 | adantr 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → (𝑆 D ◡𝐹):dom (𝑆 D ◡𝐹)⟶ℂ) |
46 | 45 | ffund 6261 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑌) → Fun (𝑆 D ◡𝐹)) |
47 | funbrfv 6459 | . . . 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 4938 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝑌 ↦ ((𝑆 D ◡𝐹)‘𝑥)) = (𝑥 ∈ 𝑌 ↦ (1 / ((𝑆 D 𝐹)‘(◡𝐹‘𝑥))))) |
50 | 44, 49 | eqtrd 2834 | 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 385 = wceq 1653 ∈ wcel 2157 ⊆ wss 3770 {cpr 4371 class class class wbr 4844 ↦ cmpt 4923 ◡ccnv 5312 dom cdm 5313 ran crn 5314 Fun wfun 6096 ⟶wf 6098 –1-1-onto→wf1o 6101 ‘cfv 6102 (class class class)co 6879 ℂcc 10223 ℝcr 10224 0cc0 10225 1c1 10226 / cdiv 10977 ↾t crest 16395 TopOpenctopn 16396 ℂfldccnfld 20067 TopOnctopon 21042 –cn→ccncf 23006 D cdv 23967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2378 ax-ext 2778 ax-rep 4965 ax-sep 4976 ax-nul 4984 ax-pow 5036 ax-pr 5098 ax-un 7184 ax-inf2 8789 ax-cnex 10281 ax-resscn 10282 ax-1cn 10283 ax-icn 10284 ax-addcl 10285 ax-addrcl 10286 ax-mulcl 10287 ax-mulrcl 10288 ax-mulcom 10289 ax-addass 10290 ax-mulass 10291 ax-distr 10292 ax-i2m1 10293 ax-1ne0 10294 ax-1rid 10295 ax-rnegex 10296 ax-rrecex 10297 ax-cnre 10298 ax-pre-lttri 10299 ax-pre-lttrn 10300 ax-pre-ltadd 10301 ax-pre-mulgt0 10302 ax-pre-sup 10303 ax-mulf 10305 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2592 df-eu 2610 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-ne 2973 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3388 df-sbc 3635 df-csb 3730 df-dif 3773 df-un 3775 df-in 3777 df-ss 3784 df-pss 3786 df-nul 4117 df-if 4279 df-pw 4352 df-sn 4370 df-pr 4372 df-tp 4374 df-op 4376 df-uni 4630 df-int 4669 df-iun 4713 df-iin 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5221 df-eprel 5226 df-po 5234 df-so 5235 df-fr 5272 df-se 5273 df-we 5274 df-xp 5319 df-rel 5320 df-cnv 5321 df-co 5322 df-dm 5323 df-rn 5324 df-res 5325 df-ima 5326 df-pred 5899 df-ord 5945 df-on 5946 df-lim 5947 df-suc 5948 df-iota 6065 df-fun 6104 df-fn 6105 df-f 6106 df-f1 6107 df-fo 6108 df-f1o 6109 df-fv 6110 df-isom 6111 df-riota 6840 df-ov 6882 df-oprab 6883 df-mpt2 6884 df-of 7132 df-om 7301 df-1st 7402 df-2nd 7403 df-supp 7534 df-wrecs 7646 df-recs 7708 df-rdg 7746 df-1o 7800 df-2o 7801 df-oadd 7804 df-er 7983 df-map 8098 df-pm 8099 df-ixp 8150 df-en 8197 df-dom 8198 df-sdom 8199 df-fin 8200 df-fsupp 8519 df-fi 8560 df-sup 8591 df-inf 8592 df-oi 8658 df-card 9052 df-cda 9279 df-pnf 10366 df-mnf 10367 df-xr 10368 df-ltxr 10369 df-le 10370 df-sub 10559 df-neg 10560 df-div 10978 df-nn 11314 df-2 11375 df-3 11376 df-4 11377 df-5 11378 df-6 11379 df-7 11380 df-8 11381 df-9 11382 df-n0 11580 df-z 11666 df-dec 11783 df-uz 11930 df-q 12033 df-rp 12074 df-xneg 12192 df-xadd 12193 df-xmul 12194 df-icc 12430 df-fz 12580 df-fzo 12720 df-seq 13055 df-exp 13114 df-hash 13370 df-cj 14179 df-re 14180 df-im 14181 df-sqrt 14315 df-abs 14316 df-struct 16185 df-ndx 16186 df-slot 16187 df-base 16189 df-sets 16190 df-ress 16191 df-plusg 16279 df-mulr 16280 df-starv 16281 df-sca 16282 df-vsca 16283 df-ip 16284 df-tset 16285 df-ple 16286 df-ds 16288 df-unif 16289 df-hom 16290 df-cco 16291 df-rest 16397 df-topn 16398 df-0g 16416 df-gsum 16417 df-topgen 16418 df-pt 16419 df-prds 16422 df-xrs 16476 df-qtop 16481 df-imas 16482 df-xps 16484 df-mre 16560 df-mrc 16561 df-acs 16563 df-mgm 17556 df-sgrp 17598 df-mnd 17609 df-submnd 17650 df-mulg 17856 df-cntz 18061 df-cmn 18509 df-psmet 20059 df-xmet 20060 df-met 20061 df-bl 20062 df-mopn 20063 df-fbas 20064 df-fg 20065 df-cnfld 20068 df-top 21026 df-topon 21043 df-topsp 21065 df-bases 21078 df-cld 21151 df-ntr 21152 df-cls 21153 df-nei 21230 df-lp 21268 df-perf 21269 df-cn 21359 df-cnp 21360 df-haus 21447 df-tx 21693 df-hmeo 21886 df-fil 21977 df-fm 22069 df-flim 22070 df-flf 22071 df-xms 22452 df-ms 22453 df-tms 22454 df-cncf 23008 df-limc 23970 df-dv 23971 |
This theorem is referenced by: dvcnvre 24122 dvlog 24737 |
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