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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 24344 | . . . 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 24385 | . 2 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
| 11 | f1ofo 6855 | . . . . 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 17559 | . . 3 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵)) |
| 15 | 1 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑈 = (𝐹 “s 𝑅)) |
| 16 | 2 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑉 = (Base‘𝑅)) |
| 17 | 3 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝐹:𝑉–1-1-onto→𝐵) |
| 18 | 4 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → 𝑅 ∈ 𝑍) |
| 19 | 9 | adantr 480 | . . . . . . . 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 24384 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) = (𝑎𝐸𝑏)) |
| 23 | metcl 24342 | . . . . . . . . 9 ⊢ ((𝐸 ∈ (Met‘𝑉) ∧ 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) → (𝑎𝐸𝑏) ∈ ℝ) | |
| 24 | 23 | 3expb 1121 | . . . . . . . 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 2841 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ) |
| 27 | 26 | ralrimivva 3202 | . . . . 5 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ) |
| 28 | f1ofn 6849 | . . . . . . . . 9 ⊢ (𝐹:𝑉–1-1-onto→𝐵 → 𝐹 Fn 𝑉) | |
| 29 | 3, 28 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐹 Fn 𝑉) |
| 30 | oveq2 7439 | . . . . . . . . . 10 ⊢ (𝑦 = (𝐹‘𝑏) → ((𝐹‘𝑎)𝐷𝑦) = ((𝐹‘𝑎)𝐷(𝐹‘𝑏))) | |
| 31 | 30 | eleq1d 2826 | . . . . . . . . 9 ⊢ (𝑦 = (𝐹‘𝑏) → (((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ)) |
| 32 | 31 | ralrn 7108 | . . . . . . . 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 6823 | . . . . . . . . 9 ⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) | |
| 35 | 12, 34 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → ran 𝐹 = 𝐵) |
| 36 | 35 | raleqdv 3326 | . . . . . . 7 ⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
| 37 | 33, 36 | bitr3d 281 | . . . . . 6 ⊢ (𝜑 → (∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
| 38 | 37 | ralbidv 3178 | . . . . 5 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ((𝐹‘𝑎)𝐷(𝐹‘𝑏)) ∈ ℝ ↔ ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
| 39 | 27, 38 | mpbid 232 | . . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ) |
| 40 | oveq1 7438 | . . . . . . . . 9 ⊢ (𝑥 = (𝐹‘𝑎) → (𝑥𝐷𝑦) = ((𝐹‘𝑎)𝐷𝑦)) | |
| 41 | 40 | eleq1d 2826 | . . . . . . . 8 ⊢ (𝑥 = (𝐹‘𝑎) → ((𝑥𝐷𝑦) ∈ ℝ ↔ ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
| 42 | 41 | ralbidv 3178 | . . . . . . 7 ⊢ (𝑥 = (𝐹‘𝑎) → (∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ)) |
| 43 | 42 | ralrn 7108 | . . . . . 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 3326 | . . . . 5 ⊢ (𝜑 → (∀𝑥 ∈ ran 𝐹∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ)) |
| 46 | 44, 45 | bitr3d 281 | . . . 4 ⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑦 ∈ 𝐵 ((𝐹‘𝑎)𝐷𝑦) ∈ ℝ ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ)) |
| 47 | 39, 46 | mpbid 232 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥𝐷𝑦) ∈ ℝ) |
| 48 | ffnov 7559 | . . 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 24343 | . 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 × cxp 5683 ran crn 5686 ↾ cres 5687 Fn wfn 6556 ⟶wf 6557 –onto→wfo 6559 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ℝcr 11154 Basecbs 17247 distcds 17306 “s cimas 17549 ∞Metcxmet 21349 Metcmet 21350 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8014 df-2nd 8015 df-supp 8186 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-fsupp 9402 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-0g 17486 df-gsum 17487 df-xrs 17547 df-imas 17553 df-mre 17629 df-mrc 17630 df-acs 17632 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-mulg 19086 df-cntz 19335 df-cmn 19800 df-xmet 21357 df-met 21358 |
| This theorem is referenced by: xpsmet 24392 imasf1oms 24503 |
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