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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 24378. (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 2905 | . . . . 5 ⊢ Ⅎ𝑦𝑅 | |
| 3 | nfcsb1v 3923 | . . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 | |
| 4 | csbeq1a 3913 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝑅 = ⦋𝑦 / 𝑥⦌𝑅) | |
| 5 | 2, 3, 4 | cbvmpt 5253 | . . . 4 ⊢ (𝑥 ∈ 𝐼 ↦ 𝑅) = (𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅) |
| 6 | 5 | oveq2i 7442 | . . 3 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ 𝑅)) = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
| 7 | 1, 6 | eqtri 2765 | . 2 ⊢ 𝑌 = (𝑆Xs(𝑦 ∈ 𝐼 ↦ ⦋𝑦 / 𝑥⦌𝑅)) |
| 8 | prdsdsf.b | . 2 ⊢ 𝐵 = (Base‘𝑌) | |
| 9 | eqid 2737 | . 2 ⊢ (Base‘⦋𝑦 / 𝑥⦌𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅) | |
| 10 | eqid 2737 | . 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 3504 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝑅 ∈ V) |
| 16 | 15 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝑅 ∈ V) |
| 17 | 3 | nfel1 2922 | . . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝑅 ∈ V |
| 18 | 4 | eleq1d 2826 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑅 ∈ V ↔ ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
| 19 | 17, 18 | rspc 3610 | . . 3 ⊢ (𝑦 ∈ 𝐼 → (∀𝑥 ∈ 𝐼 𝑅 ∈ V → ⦋𝑦 / 𝑥⦌𝑅 ∈ V)) |
| 20 | 16, 19 | mpan9 506 | . 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → ⦋𝑦 / 𝑥⦌𝑅 ∈ V) |
| 21 | prdsdsf.m | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉)) | |
| 22 | 21 | ralrimiva 3146 | . . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐼 𝐸 ∈ (∞Met‘𝑉)) |
| 23 | nfcv 2905 | . . . . . . 7 ⊢ Ⅎ𝑥dist | |
| 24 | 23, 3 | nffv 6916 | . . . . . 6 ⊢ Ⅎ𝑥(dist‘⦋𝑦 / 𝑥⦌𝑅) |
| 25 | nfcv 2905 | . . . . . . . 8 ⊢ Ⅎ𝑥Base | |
| 26 | 25, 3 | nffv 6916 | . . . . . . 7 ⊢ Ⅎ𝑥(Base‘⦋𝑦 / 𝑥⦌𝑅) |
| 27 | 26, 26 | nfxp 5718 | . . . . . 6 ⊢ Ⅎ𝑥((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
| 28 | 24, 27 | nfres 5999 | . . . . 5 ⊢ Ⅎ𝑥((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
| 29 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑥∞Met | |
| 30 | 29, 26 | nffv 6916 | . . . . 5 ⊢ Ⅎ𝑥(∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅)) |
| 31 | 28, 30 | nfel 2920 | . . . 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 6910 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (dist‘𝑅) = (dist‘⦋𝑦 / 𝑥⦌𝑅)) |
| 34 | prdsdsf.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑅) | |
| 35 | 4 | fveq2d 6910 | . . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (Base‘𝑅) = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
| 36 | 34, 35 | eqtrid 2789 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → 𝑉 = (Base‘⦋𝑦 / 𝑥⦌𝑅)) |
| 37 | 36 | sqxpeqd 5717 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑉 × 𝑉) = ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅))) |
| 38 | 33, 37 | reseq12d 5998 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((dist‘𝑅) ↾ (𝑉 × 𝑉)) = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
| 39 | 32, 38 | eqtrid 2789 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐸 = ((dist‘⦋𝑦 / 𝑥⦌𝑅) ↾ ((Base‘⦋𝑦 / 𝑥⦌𝑅) × (Base‘⦋𝑦 / 𝑥⦌𝑅)))) |
| 40 | 36 | fveq2d 6910 | . . . . 5 ⊢ (𝑥 = 𝑦 → (∞Met‘𝑉) = (∞Met‘(Base‘⦋𝑦 / 𝑥⦌𝑅))) |
| 41 | 39, 40 | eleq12d 2835 | . . . 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 3610 | . . 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 506 | . 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 24378 | 1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⦋csb 3899 ↦ cmpt 5225 × cxp 5683 ↾ cres 5687 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 distcds 17306 Xscprds 17490 ∞Metcxmet 21349 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-ixp 8938 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-7 12334 df-8 12335 df-9 12336 df-n0 12527 df-z 12614 df-dec 12734 df-uz 12879 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-icc 13394 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-hom 17321 df-cco 17322 df-prds 17492 df-xmet 21357 |
| This theorem is referenced by: prdsmet 24380 xpsxmetlem 24389 prdsbl 24504 prdsxmslem1 24541 |
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