![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > prdsdsfn | Structured version Visualization version GIF version |
Description: Structure product distance function. (Contributed by Mario Carneiro, 15-Sep-2015.) |
Ref | Expression |
---|---|
prdsbas.p | ⊢ 𝑃 = (𝑆Xs𝑅) |
prdsbas.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsbas.r | ⊢ (𝜑 → 𝑅 ∈ 𝑊) |
prdsbas.b | ⊢ 𝐵 = (Base‘𝑃) |
prdsbas.i | ⊢ (𝜑 → dom 𝑅 = 𝐼) |
prdsds.l | ⊢ 𝐷 = (dist‘𝑃) |
Ref | Expression |
---|---|
prdsdsfn | ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2799 | . . 3 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )) = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )) | |
2 | xrltso 12221 | . . . 4 ⊢ < Or ℝ* | |
3 | 2 | supex 8611 | . . 3 ⊢ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ) ∈ V |
4 | 1, 3 | fnmpt2i 7475 | . 2 ⊢ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )) Fn (𝐵 × 𝐵) |
5 | prdsbas.p | . . . 4 ⊢ 𝑃 = (𝑆Xs𝑅) | |
6 | prdsbas.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
7 | prdsbas.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑊) | |
8 | prdsbas.b | . . . 4 ⊢ 𝐵 = (Base‘𝑃) | |
9 | prdsbas.i | . . . 4 ⊢ (𝜑 → dom 𝑅 = 𝐼) | |
10 | prdsds.l | . . . 4 ⊢ 𝐷 = (dist‘𝑃) | |
11 | 5, 6, 7, 8, 9, 10 | prdsds 16439 | . . 3 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))) |
12 | 11 | fneq1d 6192 | . 2 ⊢ (𝜑 → (𝐷 Fn (𝐵 × 𝐵) ↔ (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < )) Fn (𝐵 × 𝐵))) |
13 | 4, 12 | mpbiri 250 | 1 ⊢ (𝜑 → 𝐷 Fn (𝐵 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 ∪ cun 3767 {csn 4368 ↦ cmpt 4922 × cxp 5310 dom cdm 5312 ran crn 5313 Fn wfn 6096 ‘cfv 6101 (class class class)co 6878 ↦ cmpt2 6880 supcsup 8588 0cc0 10224 ℝ*cxr 10362 < clt 10363 Basecbs 16184 distcds 16276 Xscprds 16421 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-uni 4629 df-int 4668 df-iun 4712 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-om 7300 df-1st 7401 df-2nd 7402 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-ixp 8149 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-sup 8590 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-7 11381 df-8 11382 df-9 11383 df-n0 11581 df-z 11667 df-dec 11784 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-ip 16285 df-tset 16286 df-ple 16287 df-ds 16289 df-hom 16291 df-cco 16292 df-prds 16423 |
This theorem is referenced by: ressprdsds 22504 |
Copyright terms: Public domain | W3C validator |