Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 22871 | . . . 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 22912 | . 2 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
11 | f1ofo 6615 | . . . . 5 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹:𝑉–onto→𝐵) | |
12 | 3, 11 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
13 | eqid 2818 | . . . 4 ⊢ (dist‘𝑅) = (dist‘𝑅) | |
14 | 1, 2, 12, 4, 13, 6 | imasdsfn 16775 | . . 3 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵)) |
15 | 1 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑈 = (𝐹 “s 𝑅)) |
16 | 2 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑉 = (Base‘𝑅)) |
17 | 3 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹:𝑉–1-1-onto→𝐵) |
18 | 4 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑅 ∈ 𝑍) |
19 | 9 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐸 ∈ (∞Met‘𝑉)) |
20 | simprl 767 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑎 ∈ 𝑉) | |
21 | simprr 769 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑏 ∈ 𝑉) | |
22 | 15, 16, 17, 18, 5, 6, 19, 20, 21 | imasdsf1o 22911 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏)) |
23 | metcl 22869 | . . . . . . . . 9 ⊢ ((𝐸 ∈ (Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎𝐸𝑏) ∈ ℝ) | |
24 | 23 | 3expb 1112 | . . . . . . . 8 ⊢ ((𝐸 ∈ (Met‘𝑉) ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ) |
25 | 7, 24 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (𝑎𝐸𝑏) ∈ ℝ) |
26 | 22, 25 | eqeltrd 2910 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ) |
27 | 26 | ralrimivva 3188 | . . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ) |
28 | f1ofn 6609 | . . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉) | |
29 | 3, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
30 | oveq2 7153 | . . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎)𝐷𝑦) = ((𝐹‘𝑎)𝐷(𝐹‘𝑏))) | |
31 | 30 | eleq1d 2894 | . . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ)) |
32 | 31 | ralrn 6846 | . . . . . . . 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 6586 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
35 | 12, 34 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
36 | 35 | raleqdv 3413 | . . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
37 | 33, 36 | bitr3d 282 | . . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
38 | 37 | ralbidv 3194 | . . . . 5 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
39 | 27, 38 | mpbid 233 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ) |
40 | oveq1 7152 | . . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥𝐷𝑦) = ((𝐹‘𝑎)𝐷𝑦)) | |
41 | 40 | eleq1d 2894 | . . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ∈ ℝ ↔ ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
42 | 41 | ralbidv 3194 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
43 | 42 | ralrn 6846 | . . . . . 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 3413 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ)) |
46 | 44, 45 | bitr3d 282 | . . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ)) |
47 | 39, 46 | mpbid 233 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ) |
48 | ffnov 7267 | . . 3 ⊢ (𝐷:(𝐵 × 𝐵)⟶ℝ ↔ (𝐷 Fn (𝐵 × 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ)) | |
49 | 14, 47, 48 | sylanbrc 583 | . 2 ⊢ (𝜑 → 𝐷:(𝐵 × 𝐵)⟶ℝ) |
50 | ismet2 22870 | . 2 ⊢ (𝐷 ∈ (Met‘𝐵) ↔ (𝐷 ∈ (∞Met‘𝐵) ∧ 𝐷:(𝐵 × 𝐵)⟶ℝ)) | |
51 | 10, 49, 50 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 × cxp 5546 ran crn 5549 ↾ cres 5550 Fn wfn 6343 ⟶wf 6344 –onto→wfo 6346 –1-1-onto→wf1o 6347 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 Basecbs 16471 distcds 16562 “s cimas 16765 ∞Metcxmet 20458 Metcmet 20459 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12881 df-fzo 13022 df-seq 13358 df-hash 13679 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-0g 16703 df-gsum 16704 df-xrs 16763 df-imas 16769 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-xmet 20466 df-met 20467 |
This theorem is referenced by: xpsmet 22919 imasf1oms 23027 |
Copyright terms: Public domain | W3C validator |