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Mirrors > Home > MPE Home > Th. List > imasdsval2 | Structured version Visualization version GIF version |
Description: The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 6-Oct-2020.) |
Ref | Expression |
---|---|
imasbas.u | ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) |
imasbas.v | ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) |
imasbas.f | ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
imasbas.r | ⊢ (𝜑 → 𝑅 ∈ 𝑍) |
imasds.e | ⊢ 𝐸 = (dist‘𝑅) |
imasds.d | ⊢ 𝐷 = (dist‘𝑈) |
imasdsval.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
imasdsval.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
imasdsval.s | ⊢ 𝑆 = {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} |
imasds.u | ⊢ 𝑇 = (𝐸 ↾ (𝑉 × 𝑉)) |
Ref | Expression |
---|---|
imasdsval2 | ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝑇 ∘ 𝑔))), ℝ*, < )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imasbas.u | . . 3 ⊢ (𝜑 → 𝑈 = (𝐹 “s 𝑅)) | |
2 | imasbas.v | . . 3 ⊢ (𝜑 → 𝑉 = (Base‘𝑅)) | |
3 | imasbas.f | . . 3 ⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) | |
4 | imasbas.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑍) | |
5 | imasds.e | . . 3 ⊢ 𝐸 = (dist‘𝑅) | |
6 | imasds.d | . . 3 ⊢ 𝐷 = (dist‘𝑈) | |
7 | imasdsval.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
8 | imasdsval.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
9 | imasdsval.s | . . 3 ⊢ 𝑆 = {ℎ ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1)))))} | |
10 | 1, 2, 3, 4, 5, 6, 7, 8, 9 | imasdsval 17236 | . 2 ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )) |
11 | imasds.u | . . . . . . . . . 10 ⊢ 𝑇 = (𝐸 ↾ (𝑉 × 𝑉)) | |
12 | 11 | coeq1i 5761 | . . . . . . . . 9 ⊢ (𝑇 ∘ 𝑔) = ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔) |
13 | 9 | ssrab3 4014 | . . . . . . . . . . 11 ⊢ 𝑆 ⊆ ((𝑉 × 𝑉) ↑m (1...𝑛)) |
14 | 13 | sseli 3916 | . . . . . . . . . 10 ⊢ (𝑔 ∈ 𝑆 → 𝑔 ∈ ((𝑉 × 𝑉) ↑m (1...𝑛))) |
15 | elmapi 8624 | . . . . . . . . . 10 ⊢ (𝑔 ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) → 𝑔:(1...𝑛)⟶(𝑉 × 𝑉)) | |
16 | frn 6599 | . . . . . . . . . 10 ⊢ (𝑔:(1...𝑛)⟶(𝑉 × 𝑉) → ran 𝑔 ⊆ (𝑉 × 𝑉)) | |
17 | cores 6146 | . . . . . . . . . 10 ⊢ (ran 𝑔 ⊆ (𝑉 × 𝑉) → ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔) = (𝐸 ∘ 𝑔)) | |
18 | 14, 15, 16, 17 | 4syl 19 | . . . . . . . . 9 ⊢ (𝑔 ∈ 𝑆 → ((𝐸 ↾ (𝑉 × 𝑉)) ∘ 𝑔) = (𝐸 ∘ 𝑔)) |
19 | 12, 18 | eqtrid 2790 | . . . . . . . 8 ⊢ (𝑔 ∈ 𝑆 → (𝑇 ∘ 𝑔) = (𝐸 ∘ 𝑔)) |
20 | 19 | oveq2d 7283 | . . . . . . 7 ⊢ (𝑔 ∈ 𝑆 → (ℝ*𝑠 Σg (𝑇 ∘ 𝑔)) = (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) |
21 | 20 | mpteq2ia 5176 | . . . . . 6 ⊢ (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝑇 ∘ 𝑔))) = (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) |
22 | 21 | rneqi 5839 | . . . . 5 ⊢ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝑇 ∘ 𝑔))) = ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) |
23 | 22 | a1i 11 | . . . 4 ⊢ (𝑛 ∈ ℕ → ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝑇 ∘ 𝑔))) = ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔)))) |
24 | 23 | iuneq2i 4945 | . . 3 ⊢ ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝑇 ∘ 𝑔))) = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) |
25 | 24 | infeq1i 9224 | . 2 ⊢ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝑇 ∘ 𝑔))), ℝ*, < ) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ) |
26 | 10, 25 | eqtr4di 2796 | 1 ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝑇 ∘ 𝑔))), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 ⊆ wss 3886 ∪ ciun 4924 ↦ cmpt 5156 × cxp 5582 ran crn 5585 ↾ cres 5586 ∘ ccom 5588 ⟶wf 6422 –onto→wfo 6424 ‘cfv 6426 (class class class)co 7267 1st c1st 7818 2nd c2nd 7819 ↑m cmap 8602 infcinf 9187 1c1 10882 + caddc 10884 ℝ*cxr 11018 < clt 11019 − cmin 11215 ℕcn 11983 ...cfz 13249 Basecbs 16922 distcds 16981 Σg cgsu 17161 ℝ*𝑠cxrs 17221 “s cimas 17225 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-sup 9188 df-inf 9189 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-fz 13250 df-struct 16858 df-slot 16893 df-ndx 16905 df-base 16923 df-plusg 16985 df-mulr 16986 df-sca 16988 df-vsca 16989 df-ip 16990 df-tset 16991 df-ple 16992 df-ds 16994 df-imas 17229 |
This theorem is referenced by: imasdsf1olem 23536 |
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