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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 22970. (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 2975 | . . . . 5 ⊢ Ⅎ𝑦𝑅 | |
3 | nfcsb1v 3905 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 | |
4 | csbeq1a 3895 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅) | |
5 | 2, 3, 4 | cbvmpt 5158 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑅) = (𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅) |
6 | 5 | oveq2i 7159 | . . 3 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
7 | 1, 6 | eqtri 2842 | . 2 ⊢ 𝑌 = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
8 | prdsdsf.b | . 2 ⊢ 𝐵 = (Base‘𝑌) | |
9 | eqid 2819 | . 2 ⊢ (Base‘⦋𝑦 / 𝑥⦌𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅) | |
10 | eqid 2819 | . 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 3513 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ V) |
16 | 15 | ralrimiva 3180 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ V) |
17 | 3 | nfel1 2992 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 ∈ V |
18 | 4 | eleq1d 2895 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑅 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
19 | 17, 18 | rspc 3609 | . . 3 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 𝑅 ∈ V → ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
20 | 16, 19 | mpan9 509 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ⦋𝑦 / 𝑥⦌𝑅 ∈ V) |
21 | prdsdsf.m | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) | |
22 | 21 | ralrimiva 3180 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉)) |
23 | nfcv 2975 | . . . . . . 7 ⊢ Ⅎ𝑥dist | |
24 | 23, 3 | nffv 6673 | . . . . . 6 ⊢ Ⅎ𝑥(dist‘⦋𝑦 / 𝑥⦌𝑅) |
25 | nfcv 2975 | . . . . . . . 8 ⊢ Ⅎ𝑥Base | |
26 | 25, 3 | nffv 6673 | . . . . . . 7 ⊢ Ⅎ𝑥(Base‘⦋𝑦 / 𝑥⦌𝑅) |
27 | 26, 26 | nfxp 5581 | . . . . . 6 ⊢ Ⅎ𝑥((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
28 | 24, 27 | nfres 5848 | . . . . 5 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
29 | nfcv 2975 | . . . . . 6 ⊢ Ⅎ𝑥∞Met | |
30 | 29, 26 | nffv 6673 | . . . . 5 ⊢ Ⅎ𝑥(∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)) |
31 | 28, 30 | nfel 2990 | . . . 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 6667 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘⦋𝑦 / 𝑥⦌𝑅)) |
34 | prdsdsf.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑅) | |
35 | 4 | fveq2d 6667 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (Base‘𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
36 | 34, 35 | syl5eq 2866 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑉 = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
37 | 36 | sqxpeqd 5580 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑉 × 𝑉) = ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
38 | 33, 37 | reseq12d 5847 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
39 | 32, 38 | syl5eq 2866 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐸 = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
40 | 36 | fveq2d 6667 | . . . . 5 ⊢ (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅))) |
41 | 39, 40 | eleq12d 2905 | . . . 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 3609 | . . 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 509 | . 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 22970 | 1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ∀wral 3136 Vcvv 3493 ⦋csb 3881 ↦ cmpt 5137 × cxp 5546 ↾ cres 5550 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 distcds 16566 Xscprds 16711 ∞Metcxmet 20522 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-sup 8898 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-icc 12737 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-hom 16581 df-cco 16582 df-prds 16713 df-xmet 20530 |
This theorem is referenced by: prdsmet 22972 xpsxmetlem 22981 prdsbl 23093 prdsxmslem1 23130 |
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