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Mirrors > Home > MPE Home > Th. List > imasf1obl | Structured version Visualization version GIF version |
Description: The image of a metric space ball. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
imasf1obl.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasf1obl.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasf1obl.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) |
imasf1obl.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
imasf1obl.e | ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) |
imasf1obl.d | ⊢ 𝐷 = (dist‘𝑈) |
imasf1obl.m | ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) |
imasf1obl.x | ⊢ (𝜑 → 𝑃 ∈ 𝑉) |
imasf1obl.s | ⊢ (𝜑 → 𝑆 ∈ ℝ*) |
Ref | Expression |
---|---|
imasf1obl | ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasf1obl.f | . . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
2 | f1ocnvfv2 6433 | . . . . . . . . . 10 ⊢ ((𝐹:𝑉–1-1-onto→𝐵 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) | |
3 | 1, 2 | sylan 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) |
4 | 3 | oveq2d 6565 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷(𝐹‘(◡𝐹‘𝑥))) = ((𝐹‘𝑃)𝐷𝑥)) |
5 | imasf1obl.u | . . . . . . . . . 10 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
6 | 5 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑈 = (𝐹 “s 𝑅)) |
7 | imasf1obl.v | . . . . . . . . . 10 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
8 | 7 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑉 = (Base‘𝑅)) |
9 | 1 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐹:𝑉–1-1-onto→𝐵) |
10 | imasf1obl.r | . . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
11 | 10 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑅 ∈ 𝑍) |
12 | imasf1obl.e | . . . . . . . . 9 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
13 | imasf1obl.d | . . . . . . . . 9 ⊢ 𝐷 = (dist‘𝑈) | |
14 | imasf1obl.m | . . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) | |
15 | 14 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐸 ∈ (∞Met‘𝑉)) |
16 | imasf1obl.x | . . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ 𝑉) | |
17 | 16 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑃 ∈ 𝑉) |
18 | f1ocnv 6062 | . . . . . . . . . . . 12 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝑉) | |
19 | 1, 18 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → ◡𝐹:𝐵–1-1-onto→𝑉) |
20 | f1of 6050 | . . . . . . . . . . 11 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹:𝐵⟶𝑉) | |
21 | 19, 20 | syl 17 | . . . . . . . . . 10 ⊢ (𝜑 → ◡𝐹:𝐵⟶𝑉) |
22 | 21 | ffvelrnda 6267 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (◡𝐹‘𝑥) ∈ 𝑉) |
23 | 6, 8, 9, 11, 12, 13, 15, 17, 22 | imasdsf1o 21989 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷(𝐹‘(◡𝐹‘𝑥))) = (𝑃𝐸(◡𝐹‘𝑥))) |
24 | 4, 23 | eqtr3d 2646 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((𝐹‘𝑃)𝐷𝑥) = (𝑃𝐸(◡𝐹‘𝑥))) |
25 | 24 | breq1d 4593 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝐹‘𝑃)𝐷𝑥) < 𝑆 ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) |
26 | imasf1obl.s | . . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ ℝ*) | |
27 | 26 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ ℝ*) |
28 | elbl2 22005 | . . . . . . 7 ⊢ (((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑆 ∈ ℝ*) ∧ (𝑃 ∈ 𝑉 ∧ (◡𝐹‘𝑥) ∈ 𝑉)) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆) ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) | |
29 | 15, 27, 17, 22, 28 | syl22anc 1319 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆) ↔ (𝑃𝐸(◡𝐹‘𝑥)) < 𝑆)) |
30 | 25, 29 | bitr4d 270 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (((𝐹‘𝑃)𝐷𝑥) < 𝑆 ↔ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆))) |
31 | 30 | pm5.32da 671 | . . . 4 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆) ↔ (𝑥 ∈ 𝐵 ∧ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆)))) |
32 | 5, 7, 1, 10, 12, 13, 14 | imasf1oxmet 21990 | . . . . 5 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
33 | f1of 6050 | . . . . . . 7 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉⟶𝐵) | |
34 | 1, 33 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐹:𝑉⟶𝐵) |
35 | 34, 16 | ffvelrnd 6268 | . . . . 5 ⊢ (𝜑 → (𝐹‘𝑃) ∈ 𝐵) |
36 | elbl 22003 | . . . . 5 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ (𝐹‘𝑃) ∈ 𝐵 ∧ 𝑆 ∈ ℝ*) → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆))) | |
37 | 32, 35, 26, 36 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ (𝑥 ∈ 𝐵 ∧ ((𝐹‘𝑃)𝐷𝑥) < 𝑆))) |
38 | f1ofn 6051 | . . . . 5 ⊢ (◡𝐹:𝐵–1-1-onto→𝑉 → ◡𝐹 Fn 𝐵) | |
39 | elpreima 6245 | . . . . 5 ⊢ (◡𝐹 Fn 𝐵 → (𝑥 ∈ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) ↔ (𝑥 ∈ 𝐵 ∧ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆)))) | |
40 | 19, 38, 39 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) ↔ (𝑥 ∈ 𝐵 ∧ (◡𝐹‘𝑥) ∈ (𝑃(ball‘𝐸)𝑆)))) |
41 | 31, 37, 40 | 3bitr4d 299 | . . 3 ⊢ (𝜑 → (𝑥 ∈ ((𝐹‘𝑃)(ball‘𝐷)𝑆) ↔ 𝑥 ∈ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)))) |
42 | 41 | eqrdv 2608 | . 2 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
43 | imacnvcnv 5517 | . 2 ⊢ (◡◡𝐹 “ (𝑃(ball‘𝐸)𝑆)) = (𝐹 “ (𝑃(ball‘𝐸)𝑆)) | |
44 | 42, 43 | syl6eq 2660 | 1 ⊢ (𝜑 → ((𝐹‘𝑃)(ball‘𝐷)𝑆) = (𝐹 “ (𝑃(ball‘𝐸)𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 × cxp 5036 ◡ccnv 5037 ↾ cres 5040 “ cima 5041 Fn wfn 5799 ⟶wf 5800 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ℝ*cxr 9952 < clt 9953 Basecbs 15695 distcds 15777 “s cimas 15987 ∞Metcxmt 19552 ballcbl 19554 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-0g 15925 df-gsum 15926 df-xrs 15985 df-imas 15991 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-bl 19562 |
This theorem is referenced by: imasf1oxms 22104 |
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