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Mirrors > Home > MPE Home > Th. List > prdsdsval | Structured version Visualization version GIF version |
Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
prdsbasmpt.y | ⊢ 𝑌 = (𝑆Xs𝑅) |
prdsbasmpt.b | ⊢ 𝐵 = (Base‘𝑌) |
prdsbasmpt.s | ⊢ (𝜑 → 𝑆 ∈ 𝑉) |
prdsbasmpt.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
prdsbasmpt.r | ⊢ (𝜑 → 𝑅 Fn 𝐼) |
prdsplusgval.f | ⊢ (𝜑 → 𝐹 ∈ 𝐵) |
prdsplusgval.g | ⊢ (𝜑 → 𝐺 ∈ 𝐵) |
prdsdsval.d | ⊢ 𝐷 = (dist‘𝑌) |
Ref | Expression |
---|---|
prdsdsval | ⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | prdsbasmpt.y | . . 3 ⊢ 𝑌 = (𝑆Xs𝑅) | |
2 | prdsbasmpt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉) | |
3 | prdsbasmpt.r | . . . 4 ⊢ (𝜑 → 𝑅 Fn 𝐼) | |
4 | prdsbasmpt.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | fnex 6623 | . . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V) | |
6 | 3, 4, 5 | syl2anc 693 | . . 3 ⊢ (𝜑 → 𝑅 ∈ V) |
7 | prdsbasmpt.b | . . 3 ⊢ 𝐵 = (Base‘𝑌) | |
8 | fndm 6129 | . . . 4 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼) | |
9 | 3, 8 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼) |
10 | prdsdsval.d | . . 3 ⊢ 𝐷 = (dist‘𝑌) | |
11 | 1, 2, 6, 7, 9, 10 | prdsds 16338 | . 2 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))) |
12 | fveq1 6330 | . . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥)) | |
13 | fveq1 6330 | . . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥)) | |
14 | 12, 13 | oveqan12d 6810 | . . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)) = ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) |
15 | 14 | adantl 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)) = ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) |
16 | 15 | mpteq2dv 4876 | . . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
17 | 16 | rneqd 5490 | . . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) = ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥)))) |
18 | 17 | uneq1d 3914 | . . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}) = (ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0})) |
19 | 18 | supeq1d 8506 | . 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ*, < )) |
20 | prdsplusgval.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵) | |
21 | prdsplusgval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵) | |
22 | xrltso 12178 | . . . 4 ⊢ < Or ℝ* | |
23 | 22 | supex 8523 | . . 3 ⊢ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ*, < ) ∈ V |
24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ*, < ) ∈ V) |
25 | 11, 19, 20, 21, 24 | ovmpt2d 6933 | 1 ⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1629 ∈ wcel 2143 Vcvv 3348 ∪ cun 3718 {csn 4313 ↦ cmpt 4860 dom cdm 5248 ran crn 5249 Fn wfn 6025 ‘cfv 6030 (class class class)co 6791 supcsup 8500 0cc0 10136 ℝ*cxr 10273 < clt 10274 Basecbs 16070 distcds 16164 Xscprds 16320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1868 ax-4 1883 ax-5 1989 ax-6 2055 ax-7 2091 ax-8 2145 ax-9 2152 ax-10 2172 ax-11 2188 ax-12 2201 ax-13 2406 ax-ext 2749 ax-rep 4901 ax-sep 4911 ax-nul 4919 ax-pow 4970 ax-pr 5033 ax-un 7094 ax-cnex 10192 ax-resscn 10193 ax-1cn 10194 ax-icn 10195 ax-addcl 10196 ax-addrcl 10197 ax-mulcl 10198 ax-mulrcl 10199 ax-mulcom 10200 ax-addass 10201 ax-mulass 10202 ax-distr 10203 ax-i2m1 10204 ax-1ne0 10205 ax-1rid 10206 ax-rnegex 10207 ax-rrecex 10208 ax-cnre 10209 ax-pre-lttri 10210 ax-pre-lttrn 10211 ax-pre-ltadd 10212 ax-pre-mulgt0 10213 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1070 df-3an 1071 df-tru 1632 df-ex 1851 df-nf 1856 df-sb 2048 df-eu 2620 df-mo 2621 df-clab 2756 df-cleq 2762 df-clel 2765 df-nfc 2900 df-ne 2942 df-nel 3045 df-ral 3064 df-rex 3065 df-reu 3066 df-rmo 3067 df-rab 3068 df-v 3350 df-sbc 3585 df-csb 3680 df-dif 3723 df-un 3725 df-in 3727 df-ss 3734 df-pss 3736 df-nul 4061 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4572 df-int 4609 df-iun 4653 df-br 4784 df-opab 4844 df-mpt 4861 df-tr 4884 df-id 5156 df-eprel 5161 df-po 5169 df-so 5170 df-fr 5207 df-we 5209 df-xp 5254 df-rel 5255 df-cnv 5256 df-co 5257 df-dm 5258 df-rn 5259 df-res 5260 df-ima 5261 df-pred 5822 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-riota 6752 df-ov 6794 df-oprab 6795 df-mpt2 6796 df-om 7211 df-1st 7313 df-2nd 7314 df-wrecs 7557 df-recs 7619 df-rdg 7657 df-1o 7711 df-oadd 7715 df-er 7894 df-map 8009 df-ixp 8061 df-en 8108 df-dom 8109 df-sdom 8110 df-fin 8111 df-sup 8502 df-pnf 10276 df-mnf 10277 df-xr 10278 df-ltxr 10279 df-le 10280 df-sub 10468 df-neg 10469 df-nn 11221 df-2 11279 df-3 11280 df-4 11281 df-5 11282 df-6 11283 df-7 11284 df-8 11285 df-9 11286 df-n0 11493 df-z 11578 df-dec 11694 df-uz 11888 df-fz 12533 df-struct 16072 df-ndx 16073 df-slot 16074 df-base 16076 df-plusg 16168 df-mulr 16169 df-sca 16171 df-vsca 16172 df-ip 16173 df-tset 16174 df-ple 16175 df-ds 16178 df-hom 16180 df-cco 16181 df-prds 16322 |
This theorem is referenced by: prdsdsval2 16358 xpsdsval 22412 |
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