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Mirrors > Home > MPE Home > Th. List > prdsxmet | Structured version Visualization version GIF version |
Description: The product metric is an extended metric. Eliminate disjoint variable conditions from prdsxmetlem 23866. (Contributed by Mario Carneiro, 26-Sep-2015.) |
Ref | Expression |
---|---|
prdsdsf.y | ⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) |
prdsdsf.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsdsf.v | ⊢ 𝑉 = (Base‘𝑅) |
prdsdsf.e | ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) |
prdsdsf.d | ⊢ 𝐷 = (dist‘𝑌) |
prdsdsf.s | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
prdsdsf.i | ⊢ (𝜑 → 𝐼 ∈ 𝑋) |
prdsdsf.r | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍) |
prdsdsf.m | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) |
Ref | Expression |
---|---|
prdsxmet | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsdsf.y | . . 3 ⊢ 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) | |
2 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦𝑅 | |
3 | nfcsb1v 3918 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 | |
4 | csbeq1a 3907 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅) | |
5 | 2, 3, 4 | cbvmpt 5259 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑅) = (𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅) |
6 | 5 | oveq2i 7417 | . . 3 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
7 | 1, 6 | eqtri 2761 | . 2 ⊢ 𝑌 = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
8 | prdsdsf.b | . 2 ⊢ 𝐵 = (Base‘𝑌) | |
9 | eqid 2733 | . 2 ⊢ (Base‘⦋𝑦 / 𝑥⦌𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅) | |
10 | eqid 2733 | . 2 ⊢ ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) | |
11 | prdsdsf.d | . 2 ⊢ 𝐷 = (dist‘𝑌) | |
12 | prdsdsf.s | . 2 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
13 | prdsdsf.i | . 2 ⊢ (𝜑 → 𝐼 ∈ 𝑋) | |
14 | prdsdsf.r | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ 𝑍) | |
15 | 14 | elexd 3495 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ V) |
16 | 15 | ralrimiva 3147 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ V) |
17 | 3 | nfel1 2920 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 ∈ V |
18 | 4 | eleq1d 2819 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑅 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
19 | 17, 18 | rspc 3601 | . . 3 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 𝑅 ∈ V → ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
20 | 16, 19 | mpan9 508 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ⦋𝑦 / 𝑥⦌𝑅 ∈ V) |
21 | prdsdsf.m | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) | |
22 | 21 | ralrimiva 3147 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉)) |
23 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑥dist | |
24 | 23, 3 | nffv 6899 | . . . . . 6 ⊢ Ⅎ𝑥(dist‘⦋𝑦 / 𝑥⦌𝑅) |
25 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑥Base | |
26 | 25, 3 | nffv 6899 | . . . . . . 7 ⊢ Ⅎ𝑥(Base‘⦋𝑦 / 𝑥⦌𝑅) |
27 | 26, 26 | nfxp 5709 | . . . . . 6 ⊢ Ⅎ𝑥((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
28 | 24, 27 | nfres 5982 | . . . . 5 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
29 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑥∞Met | |
30 | 29, 26 | nffv 6899 | . . . . 5 ⊢ Ⅎ𝑥(∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)) |
31 | 28, 30 | nfel 2918 | . . . 4 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) ∈ (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)) |
32 | prdsdsf.e | . . . . . 6 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
33 | 4 | fveq2d 6893 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘⦋𝑦 / 𝑥⦌𝑅)) |
34 | prdsdsf.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑅) | |
35 | 4 | fveq2d 6893 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (Base‘𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
36 | 34, 35 | eqtrid 2785 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑉 = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
37 | 36 | sqxpeqd 5708 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑉 × 𝑉) = ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
38 | 33, 37 | reseq12d 5981 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
39 | 32, 38 | eqtrid 2785 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐸 = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
40 | 36 | fveq2d 6893 | . . . . 5 ⊢ (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅))) |
41 | 39, 40 | eleq12d 2828 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐸 ∈ (∞Met‘𝑉) ↔ ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) ∈ (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
42 | 31, 41 | rspc 3601 | . . 3 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉) → ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) ∈ (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
43 | 22, 42 | mpan9 508 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) ∈ (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅))) |
44 | 7, 8, 9, 10, 11, 12, 13, 20, 43 | prdsxmetlem 23866 | 1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 Vcvv 3475 ⦋csb 3893 ↦ cmpt 5231 × cxp 5674 ↾ cres 5678 ‘cfv 6541 (class class class)co 7406 Basecbs 17141 distcds 17203 Xscprds 17388 ∞Metcxmet 20922 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-map 8819 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-rp 12972 df-xneg 13089 df-xadd 13090 df-xmul 13091 df-icc 13328 df-fz 13482 df-struct 17077 df-slot 17112 df-ndx 17124 df-base 17142 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ds 17216 df-hom 17218 df-cco 17219 df-prds 17390 df-xmet 20930 |
This theorem is referenced by: prdsmet 23868 xpsxmetlem 23877 prdsbl 23992 prdsxmslem1 24029 |
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