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Mirrors > Home > MPE Home > Th. List > prdsxmslem1 | Structured version Visualization version GIF version |
Description: Lemma for prdsms 23296. 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 2739 | . . 3 ⊢ (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) | |
2 | eqid 2739 | . . 3 ⊢ (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) | |
3 | eqid 2739 | . . 3 ⊢ (Base‘(𝑅‘𝑘)) = (Base‘(𝑅‘𝑘)) | |
4 | eqid 2739 | . . 3 ⊢ ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) = ((dist‘(𝑅‘𝑘)) ↾ ((Base‘(𝑅‘𝑘)) × (Base‘(𝑅‘𝑘)))) | |
5 | eqid 2739 | . . 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 6873 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp) |
10 | 3, 4 | xmsxmet 23221 | . . . 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 23134 | . 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 6749 | . . . . . 6 ⊢ (𝜑 → 𝑅 = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) |
16 | 15 | oveq2d 7198 | . . . . 5 ⊢ (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
17 | 14, 16 | syl5eq 2786 | . . . 4 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) |
18 | 17 | fveq2d 6690 | . . 3 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
19 | 13, 18 | syl5eq 2786 | . 2 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
20 | prdsxms.b | . . . 4 ⊢ 𝐵 = (Base‘𝑌) | |
21 | 17 | fveq2d 6690 | . . . 4 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
22 | 20, 21 | syl5eq 2786 | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))) |
23 | 22 | fveq2d 6690 | . 2 ⊢ (𝜑 → (∞Met‘𝐵) = (∞Met‘(Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))) |
24 | 12, 19, 23 | 3eltr4d 2849 | 1 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ↦ cmpt 5120 × cxp 5533 ↾ cres 5537 ⟶wf 6345 ‘cfv 6349 (class class class)co 7182 Fincfn 8567 Basecbs 16598 distcds 16689 Xscprds 16834 ∞Metcxmet 20214 ∞MetSpcxms 23082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 ax-pre-sup 10705 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-ixp 8520 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-sup 8991 df-inf 8992 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-div 11388 df-nn 11729 df-2 11791 df-3 11792 df-4 11793 df-5 11794 df-6 11795 df-7 11796 df-8 11797 df-9 11798 df-n0 11989 df-z 12075 df-dec 12192 df-uz 12337 df-q 12443 df-rp 12485 df-xneg 12602 df-xadd 12603 df-xmul 12604 df-icc 12840 df-fz 12994 df-struct 16600 df-ndx 16601 df-slot 16602 df-base 16604 df-plusg 16693 df-mulr 16694 df-sca 16696 df-vsca 16697 df-ip 16698 df-tset 16699 df-ple 16700 df-ds 16702 df-hom 16704 df-cco 16705 df-topgen 16832 df-prds 16836 df-psmet 20221 df-xmet 20222 df-bl 20224 df-mopn 20225 df-top 21657 df-topon 21674 df-topsp 21696 df-bases 21709 df-xms 23085 |
This theorem is referenced by: prdsxmslem2 23294 prdsxms 23295 |
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