Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 21983. (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 2751 | . . . . 5 ⊢ Ⅎ𝑦𝑅 | |
3 | nfcsb1v 3515 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 | |
4 | csbeq1a 3508 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅) | |
5 | 2, 3, 4 | cbvmpt 4677 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑅) = (𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅) |
6 | 5 | oveq2i 6560 | . . 3 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
7 | 1, 6 | eqtri 2632 | . 2 ⊢ 𝑌 = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
8 | prdsdsf.b | . 2 ⊢ 𝐵 = (Base‘𝑌) | |
9 | eqid 2610 | . 2 ⊢ (Base‘⦋𝑦 / 𝑥⦌𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅) | |
10 | eqid 2610 | . 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 3185 | . . . . 5 ⊢ (𝑅 ∈ 𝑍 → 𝑅 ∈ V) | |
16 | 14, 15 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ V) |
17 | 16 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ V) |
18 | 3 | nfel1 2765 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 ∈ V |
19 | 4 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑅 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
20 | 18, 19 | rspc 3276 | . . 3 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 𝑅 ∈ V → ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
21 | 17, 20 | mpan9 485 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ⦋𝑦 / 𝑥⦌𝑅 ∈ V) |
22 | prdsdsf.m | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) | |
23 | 22 | ralrimiva 2949 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉)) |
24 | nfcv 2751 | . . . . . . 7 ⊢ Ⅎ𝑥dist | |
25 | 24, 3 | nffv 6110 | . . . . . 6 ⊢ Ⅎ𝑥(dist‘⦋𝑦 / 𝑥⦌𝑅) |
26 | nfcv 2751 | . . . . . . . 8 ⊢ Ⅎ𝑥Base | |
27 | 26, 3 | nffv 6110 | . . . . . . 7 ⊢ Ⅎ𝑥(Base‘⦋𝑦 / 𝑥⦌𝑅) |
28 | 27, 27 | nfxp 5066 | . . . . . 6 ⊢ Ⅎ𝑥((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
29 | 25, 28 | nfres 5319 | . . . . 5 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
30 | nfcv 2751 | . . . . . 6 ⊢ Ⅎ𝑥∞Met | |
31 | 30, 27 | nffv 6110 | . . . . 5 ⊢ Ⅎ𝑥(∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)) |
32 | 29, 31 | nfel 2763 | . . . 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 6107 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘⦋𝑦 / 𝑥⦌𝑅)) |
35 | prdsdsf.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑅) | |
36 | 4 | fveq2d 6107 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (Base‘𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
37 | 35, 36 | syl5eq 2656 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑉 = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
38 | 37 | sqxpeqd 5065 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑉 × 𝑉) = ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
39 | 34, 38 | reseq12d 5318 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
40 | 33, 39 | syl5eq 2656 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐸 = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
41 | 37 | fveq2d 6107 | . . . . 5 ⊢ (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅))) |
42 | 40, 41 | eleq12d 2682 | . . . 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 3276 | . . 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 485 | . 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 21983 | 1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 ⦋csb 3499 ↦ cmpt 4643 × cxp 5036 ↾ cres 5040 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 distcds 15777 Xscprds 15929 ∞Metcxmt 19552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-icc 12053 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-hom 15793 df-cco 15794 df-prds 15931 df-xmet 19560 |
This theorem is referenced by: prdsmet 21985 xpsxmetlem 21994 prdsbl 22106 prdsxmslem1 22143 |
Copyright terms: Public domain | W3C validator |