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Mirrors > Home > MPE Home > Th. List > imasf1omet | Structured version Visualization version GIF version |
Description: The image of a metric is a metric. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
imasf1oxmet.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasf1oxmet.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasf1oxmet.f | ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) |
imasf1oxmet.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
imasf1oxmet.e | ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) |
imasf1oxmet.d | ⊢ 𝐷 = (dist‘𝑈) |
imasf1omet.m | ⊢ (𝜑 → 𝐸 ∈ (Met‘𝑉)) |
Ref | Expression |
---|---|
imasf1omet | ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasf1oxmet.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imasf1oxmet.v | . . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imasf1oxmet.f | . . 3 ⊢ (𝜑 → 𝐹:𝑉–1-1-onto→𝐵) | |
4 | imasf1oxmet.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
5 | imasf1oxmet.e | . . 3 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
6 | imasf1oxmet.d | . . 3 ⊢ 𝐷 = (dist‘𝑈) | |
7 | imasf1omet.m | . . . 4 ⊢ (𝜑 → 𝐸 ∈ (Met‘𝑉)) | |
8 | metxmet 23232 | . . . 4 ⊢ (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉)) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (∞Met‘𝑉)) |
10 | 1, 2, 3, 4, 5, 6, 9 | imasf1oxmet 23273 | . 2 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
11 | f1ofo 6668 | . . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵) | |
12 | 3, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
13 | eqid 2737 | . . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅) | |
14 | 1, 2, 12, 4, 13, 6 | imasdsfn 17019 | . . 3 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵)) |
15 | 1 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑈 = (𝐹 “s 𝑅)) |
16 | 2 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑉 = (Base‘𝑅)) |
17 | 3 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹:𝑉–1-1-onto→𝐵) |
18 | 4 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑅 ∈ 𝑍) |
19 | 9 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐸 ∈ (∞Met‘𝑉)) |
20 | simprl 771 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉) | |
21 | simprr 773 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
22 | 15, 16, 17, 18, 5, 6, 19, 20, 21 | imasdsf1o 23272 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏)) |
23 | metcl 23230 | . . . . . . . . 9 ⊢ ((𝐸 ∈ (Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎𝐸𝑏) ∈ ℝ) | |
24 | 23 | 3expb 1122 | . . . . . . . 8 ⊢ ((𝐸 ∈ (Met‘𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ) |
25 | 7, 24 | sylan 583 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ) |
26 | 22, 25 | eqeltrd 2838 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ) |
27 | 26 | ralrimivva 3112 | . . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ) |
28 | f1ofn 6662 | . . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉) | |
29 | 3, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
30 | oveq2 7221 | . . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎)𝐷𝑦) = ((𝐹‘𝑎)𝐷(𝐹‘𝑏))) | |
31 | 30 | eleq1d 2822 | . . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ)) |
32 | 31 | ralrn 6907 | . . . . . . . 8 ⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ)) |
33 | 29, 32 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ)) |
34 | forn 6636 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
35 | 12, 34 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
36 | 35 | raleqdv 3325 | . . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
37 | 33, 36 | bitr3d 284 | . . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
38 | 37 | ralbidv 3118 | . . . . 5 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
39 | 27, 38 | mpbid 235 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ) |
40 | oveq1 7220 | . . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥𝐷𝑦) = ((𝐹‘𝑎)𝐷𝑦)) | |
41 | 40 | eleq1d 2822 | . . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ∈ ℝ ↔ ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
42 | 41 | ralbidv 3118 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
43 | 42 | ralrn 6907 | . . . . . 6 ⊢ (𝐹 Fn 𝑉 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
44 | 29, 43 | syl 17 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
45 | 35 | raleqdv 3325 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ)) |
46 | 44, 45 | bitr3d 284 | . . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ)) |
47 | 39, 46 | mpbid 235 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ) |
48 | ffnov 7337 | . . 3 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ ↔ (𝐷 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ)) | |
49 | 14, 47, 48 | sylanbrc 586 | . 2 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶ℝ) |
50 | ismet2 23231 | . 2 ⊢ (𝐷 ∈ (Met‘𝐵) ↔ (𝐷 ∈ (∞Met‘𝐵) ∧ 𝐷:(𝐵 × 𝐵)⟶ℝ)) | |
51 | 10, 49, 50 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∀wral 3061 × cxp 5549 ran crn 5552 ↾ cres 5553 Fn wfn 6375 ⟶wf 6376 –onto→wfo 6378 –1-1-onto→wf1o 6379 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 Basecbs 16760 distcds 16811 “s cimas 17009 ∞Metcxmet 20348 Metcmet 20349 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-supp 7904 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-fsupp 8986 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-7 11898 df-8 11899 df-9 11900 df-n0 12091 df-z 12177 df-dec 12294 df-uz 12439 df-rp 12587 df-xneg 12704 df-xadd 12705 df-xmul 12706 df-fz 13096 df-fzo 13239 df-seq 13575 df-hash 13897 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-ip 16820 df-tset 16821 df-ple 16822 df-ds 16824 df-0g 16946 df-gsum 16947 df-xrs 17007 df-imas 17013 df-mre 17089 df-mrc 17090 df-acs 17092 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-mulg 18489 df-cntz 18711 df-cmn 19172 df-xmet 20356 df-met 20357 |
This theorem is referenced by: xpsmet 23280 imasf1oms 23388 |
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