Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > xpsdsfn | Structured version Visualization version GIF version |
Description: Closure of the metric in a binary structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xpsds.t | ⊢ 𝑇 = (𝑅 ×s 𝑆) |
xpsds.x | ⊢ 𝑋 = (Base‘𝑅) |
xpsds.y | ⊢ 𝑌 = (Base‘𝑆) |
xpsds.1 | ⊢ (𝜑 → 𝑅 ∈ 𝑉) |
xpsds.2 | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
xpsds.p | ⊢ 𝑃 = (dist‘𝑇) |
Ref | Expression |
---|---|
xpsdsfn | ⊢ (𝜑 → 𝑃 Fn ((𝑋 × 𝑌) × (𝑋 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpsds.t | . . 3 ⊢ 𝑇 = (𝑅 ×s 𝑆) | |
2 | xpsds.x | . . 3 ⊢ 𝑋 = (Base‘𝑅) | |
3 | xpsds.y | . . 3 ⊢ 𝑌 = (Base‘𝑆) | |
4 | xpsds.1 | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑉) | |
5 | xpsds.2 | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
6 | eqid 2821 | . . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | |
7 | eqid 2821 | . . 3 ⊢ (Scalar‘𝑅) = (Scalar‘𝑅) | |
8 | eqid 2821 | . . 3 ⊢ ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) = ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsval 16826 | . 2 ⊢ (𝜑 → 𝑇 = (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) “s ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
10 | 1, 2, 3, 4, 5, 6, 7, 8 | xpsrnbas 16827 | . 2 ⊢ (𝜑 → ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (Base‘((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}))) |
11 | 6 | xpsff1o2 16825 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})) |
13 | f1ocnv 6613 | . . 3 ⊢ ((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):(𝑋 × 𝑌)–1-1-onto→ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→(𝑋 × 𝑌)) | |
14 | f1ofo 6608 | . . 3 ⊢ (◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–1-1-onto→(𝑋 × 𝑌) → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–onto→(𝑋 × 𝑌)) | |
15 | 12, 13, 14 | 3syl 18 | . 2 ⊢ (𝜑 → ◡(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}):ran (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})–onto→(𝑋 × 𝑌)) |
16 | ovexd 7177 | . 2 ⊢ (𝜑 → ((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) ∈ V) | |
17 | eqid 2821 | . 2 ⊢ (dist‘((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉})) = (dist‘((Scalar‘𝑅)Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉})) | |
18 | xpsds.p | . 2 ⊢ 𝑃 = (dist‘𝑇) | |
19 | 9, 10, 15, 16, 17, 18 | imasdsfn 16770 | 1 ⊢ (𝜑 → 𝑃 Fn ((𝑋 × 𝑌) × (𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Vcvv 3486 ∅c0 4279 {cpr 4555 〈cop 4559 × cxp 5539 ◡ccnv 5540 ran crn 5542 Fn wfn 6336 –onto→wfo 6339 –1-1-onto→wf1o 6340 ‘cfv 6341 (class class class)co 7142 ∈ cmpo 7144 1oc1o 8081 Basecbs 16466 Scalarcsca 16551 distcds 16557 Xscprds 16702 ×s cxps 16762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5176 ax-sep 5189 ax-nul 5196 ax-pow 5252 ax-pr 5316 ax-un 7447 ax-cnex 10579 ax-resscn 10580 ax-1cn 10581 ax-icn 10582 ax-addcl 10583 ax-addrcl 10584 ax-mulcl 10585 ax-mulrcl 10586 ax-mulcom 10587 ax-addass 10588 ax-mulass 10589 ax-distr 10590 ax-i2m1 10591 ax-1ne0 10592 ax-1rid 10593 ax-rnegex 10594 ax-rrecex 10595 ax-cnre 10596 ax-pre-lttri 10597 ax-pre-lttrn 10598 ax-pre-ltadd 10599 ax-pre-mulgt0 10600 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3488 df-sbc 3764 df-csb 3872 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-pss 3942 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-tp 4558 df-op 4560 df-uni 4825 df-int 4863 df-iun 4907 df-br 5053 df-opab 5115 df-mpt 5133 df-tr 5159 df-id 5446 df-eprel 5451 df-po 5460 df-so 5461 df-fr 5500 df-we 5502 df-xp 5547 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-res 5553 df-ima 5554 df-pred 6134 df-ord 6180 df-on 6181 df-lim 6182 df-suc 6183 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-f1 6346 df-fo 6347 df-f1o 6348 df-fv 6349 df-riota 7100 df-ov 7145 df-oprab 7146 df-mpo 7147 df-om 7567 df-1st 7675 df-2nd 7676 df-wrecs 7933 df-recs 7994 df-rdg 8032 df-1o 8088 df-2o 8089 df-oadd 8092 df-er 8275 df-map 8394 df-ixp 8448 df-en 8496 df-dom 8497 df-sdom 8498 df-fin 8499 df-sup 8892 df-inf 8893 df-pnf 10663 df-mnf 10664 df-xr 10665 df-ltxr 10666 df-le 10667 df-sub 10858 df-neg 10859 df-nn 11625 df-2 11687 df-3 11688 df-4 11689 df-5 11690 df-6 11691 df-7 11692 df-8 11693 df-9 11694 df-n0 11885 df-z 11969 df-dec 12086 df-uz 12231 df-fz 12883 df-struct 16468 df-ndx 16469 df-slot 16470 df-base 16472 df-plusg 16561 df-mulr 16562 df-sca 16564 df-vsca 16565 df-ip 16566 df-tset 16567 df-ple 16568 df-ds 16570 df-hom 16572 df-cco 16573 df-prds 16704 df-imas 16764 df-xps 16766 |
This theorem is referenced by: xpsdsfn2 22971 |
Copyright terms: Public domain | W3C validator |