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Theorem imasdsval 17562
Description: The distance function of an image structure. (Contributed by Mario Carneiro, 20-Aug-2015.) (Revised by AV, 6-Oct-2020.)
Hypotheses
Ref Expression
imasbas.u (𝜑𝑈 = (𝐹s 𝑅))
imasbas.v (𝜑𝑉 = (Base‘𝑅))
imasbas.f (𝜑𝐹:𝑉onto𝐵)
imasbas.r (𝜑𝑅𝑍)
imasds.e 𝐸 = (dist‘𝑅)
imasds.d 𝐷 = (dist‘𝑈)
imasdsval.x (𝜑𝑋𝐵)
imasdsval.y (𝜑𝑌𝐵)
imasdsval.s 𝑆 = { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))}
Assertion
Ref Expression
imasdsval (𝜑 → (𝑋𝐷𝑌) = inf( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < ))
Distinct variable groups:   𝑔,,𝑖,𝑛,𝐹   𝑅,𝑔,,𝑖,𝑛   𝜑,𝑔,,𝑖,𝑛   ,𝑋,𝑛   𝑆,𝑔   𝑔,𝑉,   ,𝑌,𝑛
Allowed substitution hints:   𝐵(𝑔,,𝑖,𝑛)   𝐷(𝑔,,𝑖,𝑛)   𝑆(,𝑖,𝑛)   𝑈(𝑔,,𝑖,𝑛)   𝐸(𝑔,,𝑖,𝑛)   𝑉(𝑖,𝑛)   𝑋(𝑔,𝑖)   𝑌(𝑔,𝑖)   𝑍(𝑔,,𝑖,𝑛)

Proof of Theorem imasdsval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . 3 (𝜑𝑈 = (𝐹s 𝑅))
2 imasbas.v . . 3 (𝜑𝑉 = (Base‘𝑅))
3 imasbas.f . . 3 (𝜑𝐹:𝑉onto𝐵)
4 imasbas.r . . 3 (𝜑𝑅𝑍)
5 imasds.e . . 3 𝐸 = (dist‘𝑅)
6 imasds.d . . 3 𝐷 = (dist‘𝑈)
71, 2, 3, 4, 5, 6imasds 17560 . 2 (𝜑𝐷 = (𝑥𝐵, 𝑦𝐵 ↦ inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < )))
8 simplrl 777 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → 𝑥 = 𝑋)
98eqeq2d 2746 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘(1st ‘(‘1))) = 𝑥 ↔ (𝐹‘(1st ‘(‘1))) = 𝑋))
10 simplrr 778 . . . . . . . . . 10 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → 𝑦 = 𝑌)
1110eqeq2d 2746 . . . . . . . . 9 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ((𝐹‘(2nd ‘(𝑛))) = 𝑦 ↔ (𝐹‘(2nd ‘(𝑛))) = 𝑌))
129, 113anbi12d 1436 . . . . . . . 8 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → (((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1))))) ↔ ((𝐹‘(1st ‘(‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))))
1312rabbidv 3441 . . . . . . 7 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} = { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))})
14 imasdsval.s . . . . . . 7 𝑆 = { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑋 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑌 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))}
1513, 14eqtr4di 2793 . . . . . 6 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} = 𝑆)
1615mpteq1d 5243 . . . . 5 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸𝑔))) = (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))))
1716rneqd 5952 . . . 4 (((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) ∧ 𝑛 ∈ ℕ) → ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸𝑔))) = ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))))
1817iuneq2dv 5021 . . 3 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸𝑔))) = 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))))
1918infeq1d 9515 . 2 ((𝜑 ∧ (𝑥 = 𝑋𝑦 = 𝑌)) → inf( 𝑛 ∈ ℕ ran (𝑔 ∈ { ∈ ((𝑉 × 𝑉) ↑m (1...𝑛)) ∣ ((𝐹‘(1st ‘(‘1))) = 𝑥 ∧ (𝐹‘(2nd ‘(𝑛))) = 𝑦 ∧ ∀𝑖 ∈ (1...(𝑛 − 1))(𝐹‘(2nd ‘(𝑖))) = (𝐹‘(1st ‘(‘(𝑖 + 1)))))} ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < ) = inf( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < ))
20 imasdsval.x . 2 (𝜑𝑋𝐵)
21 imasdsval.y . 2 (𝜑𝑌𝐵)
22 xrltso 13180 . . . 4 < Or ℝ*
2322infex 9531 . . 3 inf( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < ) ∈ V
2423a1i 11 . 2 (𝜑 → inf( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))), ℝ*, < ) ∈ V)
257, 19, 20, 21, 24ovmpod 7585 1 (𝜑 → (𝑋𝐷𝑌) = inf( 𝑛 ∈ ℕ ran (𝑔𝑆 ↦ (ℝ*𝑠 Σg (𝐸𝑔))), 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  ontowfo 6561  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  m cmap 8865  infcinf 9479  1c1 11154   + caddc 11156  *cxr 11292   < clt 11293  cmin 11490  cn 12264  ...cfz 13544  Basecbs 17245  distcds 17307   Σg cgsu 17487  *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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