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Theorem imasdsval 17398
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 (πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))
imasbas.v (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))
imasbas.f (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)
imasbas.r (πœ‘ β†’ 𝑅 ∈ 𝑍)
imasds.e 𝐸 = (distβ€˜π‘…)
imasds.d 𝐷 = (distβ€˜π‘ˆ)
imasdsval.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
imasdsval.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
imasdsval.s 𝑆 = {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = 𝑋 ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = π‘Œ ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))}
Assertion
Ref Expression
imasdsval (πœ‘ β†’ (π‘‹π·π‘Œ) = inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))
Distinct variable groups:   𝑔,β„Ž,𝑖,𝑛,𝐹   𝑅,𝑔,β„Ž,𝑖,𝑛   πœ‘,𝑔,β„Ž,𝑖,𝑛   β„Ž,𝑋,𝑛   𝑆,𝑔   𝑔,𝑉,β„Ž   β„Ž,π‘Œ,𝑛
Allowed substitution hints:   𝐡(𝑔,β„Ž,𝑖,𝑛)   𝐷(𝑔,β„Ž,𝑖,𝑛)   𝑆(β„Ž,𝑖,𝑛)   π‘ˆ(𝑔,β„Ž,𝑖,𝑛)   𝐸(𝑔,β„Ž,𝑖,𝑛)   𝑉(𝑖,𝑛)   𝑋(𝑔,𝑖)   π‘Œ(𝑔,𝑖)   𝑍(𝑔,β„Ž,𝑖,𝑛)

Proof of Theorem imasdsval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imasbas.u . . 3 (πœ‘ β†’ π‘ˆ = (𝐹 β€œs 𝑅))
2 imasbas.v . . 3 (πœ‘ β†’ 𝑉 = (Baseβ€˜π‘…))
3 imasbas.f . . 3 (πœ‘ β†’ 𝐹:𝑉–onto→𝐡)
4 imasbas.r . . 3 (πœ‘ β†’ 𝑅 ∈ 𝑍)
5 imasds.e . . 3 𝐸 = (distβ€˜π‘…)
6 imasds.d . . 3 𝐷 = (distβ€˜π‘ˆ)
71, 2, 3, 4, 5, 6imasds 17396 . 2 (πœ‘ β†’ 𝐷 = (π‘₯ ∈ 𝐡, 𝑦 ∈ 𝐡 ↦ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < )))
8 simplrl 776 . . . . . . . . . 10 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ π‘₯ = 𝑋)
98eqeq2d 2748 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ↔ (πΉβ€˜(1st β€˜(β„Žβ€˜1))) = 𝑋))
10 simplrr 777 . . . . . . . . . 10 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ 𝑦 = π‘Œ)
1110eqeq2d 2748 . . . . . . . . 9 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ ((πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ↔ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = π‘Œ))
129, 113anbi12d 1438 . . . . . . . 8 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ (((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1))))) ↔ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = 𝑋 ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = π‘Œ ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))))
1312rabbidv 3416 . . . . . . 7 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} = {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = 𝑋 ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = π‘Œ ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))})
14 imasdsval.s . . . . . . 7 𝑆 = {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = 𝑋 ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = π‘Œ ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))}
1513, 14eqtr4di 2795 . . . . . 6 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} = 𝑆)
1615mpteq1d 5201 . . . . 5 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))) = (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))))
1716rneqd 5894 . . . 4 (((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) ∧ 𝑛 ∈ β„•) β†’ ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))) = ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))))
1817iuneq2dv 4979 . . 3 ((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))) = βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))))
1918infeq1d 9414 . 2 ((πœ‘ ∧ (π‘₯ = 𝑋 ∧ 𝑦 = π‘Œ)) β†’ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ {β„Ž ∈ ((𝑉 Γ— 𝑉) ↑m (1...𝑛)) ∣ ((πΉβ€˜(1st β€˜(β„Žβ€˜1))) = π‘₯ ∧ (πΉβ€˜(2nd β€˜(β„Žβ€˜π‘›))) = 𝑦 ∧ βˆ€π‘– ∈ (1...(𝑛 βˆ’ 1))(πΉβ€˜(2nd β€˜(β„Žβ€˜π‘–))) = (πΉβ€˜(1st β€˜(β„Žβ€˜(𝑖 + 1)))))} ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ) = inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))
20 imasdsval.x . 2 (πœ‘ β†’ 𝑋 ∈ 𝐡)
21 imasdsval.y . 2 (πœ‘ β†’ π‘Œ ∈ 𝐡)
22 xrltso 13061 . . . 4 < Or ℝ*
2322infex 9430 . . 3 inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ) ∈ V
2423a1i 11 . 2 (πœ‘ β†’ inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ) ∈ V)
257, 19, 20, 21, 24ovmpod 7508 1 (πœ‘ β†’ (π‘‹π·π‘Œ) = inf(βˆͺ 𝑛 ∈ β„• ran (𝑔 ∈ 𝑆 ↦ (ℝ*𝑠 Ξ£g (𝐸 ∘ 𝑔))), ℝ*, < ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408  Vcvv 3446  βˆͺ ciun 4955   ↦ cmpt 5189   Γ— cxp 5632  ran crn 5635   ∘ ccom 5638  β€“ontoβ†’wfo 6495  β€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921   ↑m cmap 8766  infcinf 9378  1c1 11053   + caddc 11055  β„*cxr 11189   < clt 11190   βˆ’ cmin 11386  β„•cn 12154  ...cfz 13425  Basecbs 17084  distcds 17143   Ξ£g cgsu 17323  β„*𝑠cxrs 17383   β€œs cimas 17387
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-sup 9379  df-inf 9380  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-3 12218  df-4 12219  df-5 12220  df-6 12221  df-7 12222  df-8 12223  df-9 12224  df-n0 12415  df-z 12501  df-dec 12620  df-uz 12765  df-fz 13426  df-struct 17020  df-slot 17055  df-ndx 17067  df-base 17085  df-plusg 17147  df-mulr 17148  df-sca 17150  df-vsca 17151  df-ip 17152  df-tset 17153  df-ple 17154  df-ds 17156  df-imas 17391
This theorem is referenced by:  imasdsval2  17399
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