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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 22173. (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 2764 | . . . . 5 ⊢ Ⅎ𝑦𝑅 | |
3 | nfcsb1v 3549 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 | |
4 | csbeq1a 3542 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅) | |
5 | 2, 3, 4 | cbvmpt 4749 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑅) = (𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅) |
6 | 5 | oveq2i 6661 | . . 3 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
7 | 1, 6 | eqtri 2644 | . 2 ⊢ 𝑌 = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
8 | prdsdsf.b | . 2 ⊢ 𝐵 = (Base‘𝑌) | |
9 | eqid 2622 | . 2 ⊢ (Base‘⦋𝑦 / 𝑥⦌𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅) | |
10 | eqid 2622 | . 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 | elex 3212 | . . . . 5 ⊢ (𝑅 ∈ 𝑍 → 𝑅 ∈ V) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ V) |
17 | 16 | ralrimiva 2966 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ V) |
18 | 3 | nfel1 2779 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 ∈ V |
19 | 4 | eleq1d 2686 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑅 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
20 | 18, 19 | rspc 3303 | . . 3 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 𝑅 ∈ V → ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
21 | 17, 20 | mpan9 486 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ⦋𝑦 / 𝑥⦌𝑅 ∈ V) |
22 | prdsdsf.m | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) | |
23 | 22 | ralrimiva 2966 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉)) |
24 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑥dist | |
25 | 24, 3 | nffv 6198 | . . . . . 6 ⊢ Ⅎ𝑥(dist‘⦋𝑦 / 𝑥⦌𝑅) |
26 | nfcv 2764 | . . . . . . . 8 ⊢ Ⅎ𝑥Base | |
27 | 26, 3 | nffv 6198 | . . . . . . 7 ⊢ Ⅎ𝑥(Base‘⦋𝑦 / 𝑥⦌𝑅) |
28 | 27, 27 | nfxp 5142 | . . . . . 6 ⊢ Ⅎ𝑥((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
29 | 25, 28 | nfres 5398 | . . . . 5 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
30 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑥∞Met | |
31 | 30, 27 | nffv 6198 | . . . . 5 ⊢ Ⅎ𝑥(∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)) |
32 | 29, 31 | nfel 2777 | . . . 4 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) ∈ (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)) |
33 | prdsdsf.e | . . . . . 6 ⊢ 𝐸 = ((dist‘𝑅) ↾ (𝑉 × 𝑉)) | |
34 | 4 | fveq2d 6195 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘⦋𝑦 / 𝑥⦌𝑅)) |
35 | prdsdsf.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑅) | |
36 | 4 | fveq2d 6195 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (Base‘𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
37 | 35, 36 | syl5eq 2668 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑉 = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
38 | 37 | sqxpeqd 5141 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑉 × 𝑉) = ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
39 | 34, 38 | reseq12d 5397 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
40 | 33, 39 | syl5eq 2668 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐸 = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
41 | 37 | fveq2d 6195 | . . . . 5 ⊢ (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅))) |
42 | 40, 41 | eleq12d 2695 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐸 ∈ (∞Met‘𝑉) ↔ ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) ∈ (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
43 | 32, 42 | rspc 3303 | . . 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)))) |
44 | 23, 43 | mpan9 486 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) ∈ (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅))) |
45 | 7, 8, 9, 10, 11, 12, 13, 21, 44 | prdsxmetlem 22173 | 1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⦋csb 3533 ↦ cmpt 4729 × cxp 5112 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 distcds 15950 Xscprds 16106 ∞Metcxmt 19731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-prds 16108 df-xmet 19739 |
This theorem is referenced by: prdsmet 22175 xpsxmetlem 22184 prdsbl 22296 prdsxmslem1 22333 |
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