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Mirrors > Home > MPE Home > Th. List > imasdsval | 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)))))} |
Ref | Expression |
---|---|
imasdsval | ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 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 | 1, 2, 3, 4, 5, 6 | imasds 17560 | . 2 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {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)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ))) |
8 | simplrl 777 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → 𝑥 = 𝑋) | |
9 | 8 | eqeq2d 2746 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ↔ (𝐹‘(1st ‘(ℎ‘1))) = 𝑋)) |
10 | simplrr 778 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑌) | |
11 | 10 | eqeq2d 2746 | . . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ↔ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑌)) |
12 | 9, 11 | 3anbi12d 1436 | . . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → (((𝐹‘(1st ‘(ℎ‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1))))) ↔ ((𝐹‘(1st ‘(ℎ‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(ℎ‘𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(ℎ‘𝑖))) = (𝐹‘(1st ‘(ℎ‘(𝑖 + 1))))))) |
13 | 12 | rabbidv 3441 | . . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → {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)))))} = {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)))))}) |
14 | imasdsval.s | . . . . . . 7 ⊢ 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)))))} | |
15 | 13, 14 | eqtr4di 2793 | . . . . . 6 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → {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)))))} = 𝑆) |
16 | 15 | mpteq1d 5243 | . . . . 5 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → (𝑔 ∈ {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)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) = (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔)))) |
17 | 16 | rneqd 5952 | . . . 4 ⊢ (((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ran (𝑔 ∈ {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)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) = ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔)))) |
18 | 17 | iuneq2dv 5021 | . . 3 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {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)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))) = ∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔)))) |
19 | 18 | infeq1d 9515 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑦 = 𝑌)) → inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ {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)))))} ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )) |
20 | imasdsval.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
21 | imasdsval.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
22 | xrltso 13180 | . . . 4 ⊢ < Or ℝ* | |
23 | 22 | infex 9531 | . . 3 ⊢ inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ) ∈ V |
24 | 23 | a1i 11 | . 2 ⊢ (𝜑 → inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < ) ∈ V) |
25 | 7, 19, 20, 21, 24 | ovmpod 7585 | 1 ⊢ (𝜑 → (𝑋𝐷𝑌) = inf(∪ 𝑛 ∈ ℕ ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Σg (𝐸 ∘ 𝑔))), ℝ*, < )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 Vcvv 3478 ∪ ciun 4996 ↦ cmpt 5231 × cxp 5687 ran crn 5690 ∘ ccom 5693 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 ↑m cmap 8865 infcinf 9479 1c1 11154 + caddc 11156 ℝ*cxr 11292 < clt 11293 − cmin 11490 ℕcn 12264 ...cfz 13544 Basecbs 17245 distcds 17307 Σg cgsu 17487 ℝ*𝑠cxrs 17547 “s cimas 17551 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-imas 17555 |
This theorem is referenced by: imasdsval2 17563 |
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