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Mirrors > Home > MPE Home > Th. List > prdsxmslem1 | Structured version Visualization version GIF version |
Description: Lemma for prdsms 22838. The distance function of a product structure is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
prdsxms.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsxms.s | ⊢ (𝜑 → 𝑆 ∈ 𝑊) |
prdsxms.i | ⊢ (𝜑 → 𝐼 ∈ Fin) |
prdsxms.d | ⊢ 𝐷 = (dist‘𝑌) |
prdsxms.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsxms.r | ⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp) |
Ref | Expression |
---|---|
prdsxmslem1 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2772 | . . 3 ⊢ (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) | |
2 | eqid 2772 | . . 3 ⊢ (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) | |
3 | eqid 2772 | . . 3 ⊢ (Base‘(𝑅‘𝑘)) = (Base‘(𝑅‘𝑘)) | |
4 | eqid 2772 | . . 3 ⊢ ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) = ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) | |
5 | eqid 2772 | . . 3 ⊢ (dist‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) = (dist‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) | |
6 | prdsxms.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊) | |
7 | prdsxms.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ Fin) | |
8 | prdsxms.r | . . . 4 ⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp) | |
9 | 8 | ffvelrnda 6670 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp) |
10 | 3, 4 | xmsxmet 22763 | . . . 4 ⊢ ((𝑅‘𝑘) ∈ ∞MetSp → ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) ∈ (∞Met‘(Base‘(𝑅‘𝑘)))) |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) ∈ (∞Met‘(Base‘(𝑅‘𝑘)))) |
12 | 1, 2, 3, 4, 5, 6, 7, 9, 11 | prdsxmet 22676 | . 2 ⊢ (𝜑 → (dist‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) ∈ (∞Met‘(Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))) |
13 | prdsxms.d | . . 3 ⊢ 𝐷 = (dist‘𝑌) | |
14 | prdsxms.y | . . . . 5 ⊢ 𝑌 = (𝑆Xs𝑅) | |
15 | 8 | feqmptd 6556 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) |
16 | 15 | oveq2d 6986 | . . . . 5 ⊢ (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
17 | 14, 16 | syl5eq 2820 | . . . 4 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
18 | 17 | fveq2d 6497 | . . 3 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
19 | 13, 18 | syl5eq 2820 | . 2 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
20 | prdsxms.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
21 | 17 | fveq2d 6497 | . . . 4 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
22 | 20, 21 | syl5eq 2820 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
23 | 22 | fveq2d 6497 | . 2 ⊢ (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))) |
24 | 12, 19, 23 | 3eltr4d 2875 | 1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ↦ cmpt 5002 × cxp 5399 ↾ cres 5403 ⟶wf 6178 ‘cfv 6182 (class class class)co 6970 Fincfn 8300 Basecbs 16333 distcds 16424 Xscprds 16569 ∞Metcxmet 20226 ∞MetSpcxms 22624 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-rep 5043 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-1cn 10387 ax-icn 10388 ax-addcl 10389 ax-addrcl 10390 ax-mulcl 10391 ax-mulrcl 10392 ax-mulcom 10393 ax-addass 10394 ax-mulass 10395 ax-distr 10396 ax-i2m1 10397 ax-1ne0 10398 ax-1rid 10399 ax-rnegex 10400 ax-rrecex 10401 ax-cnre 10402 ax-pre-lttri 10403 ax-pre-lttrn 10404 ax-pre-ltadd 10405 ax-pre-mulgt0 10406 ax-pre-sup 10407 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rmo 3090 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-int 4744 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5306 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-1st 7495 df-2nd 7496 df-wrecs 7744 df-recs 7806 df-rdg 7844 df-1o 7899 df-oadd 7903 df-er 8083 df-map 8202 df-ixp 8254 df-en 8301 df-dom 8302 df-sdom 8303 df-fin 8304 df-sup 8695 df-inf 8696 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474 df-sub 10666 df-neg 10667 df-div 11093 df-nn 11434 df-2 11497 df-3 11498 df-4 11499 df-5 11500 df-6 11501 df-7 11502 df-8 11503 df-9 11504 df-n0 11702 df-z 11788 df-dec 11906 df-uz 12053 df-q 12157 df-rp 12199 df-xneg 12318 df-xadd 12319 df-xmul 12320 df-icc 12555 df-fz 12703 df-struct 16335 df-ndx 16336 df-slot 16337 df-base 16339 df-plusg 16428 df-mulr 16429 df-sca 16431 df-vsca 16432 df-ip 16433 df-tset 16434 df-ple 16435 df-ds 16437 df-hom 16439 df-cco 16440 df-topgen 16567 df-prds 16571 df-psmet 20233 df-xmet 20234 df-bl 20236 df-mopn 20237 df-top 21200 df-topon 21217 df-topsp 21239 df-bases 21252 df-xms 22627 |
This theorem is referenced by: prdsxmslem2 22836 prdsxms 22837 |
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