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Theorem reldmprds 17399
Description: The structure product is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Thierry Arnoux, 15-Jun-2019.) (Revised by Zhi Wang, 18-Aug-2024.)
Assertion
Ref Expression
reldmprds Rel dom Xs

Proof of Theorem reldmprds
Dummy variables π‘Ž 𝑐 𝑑 𝑒 𝑓 𝑔 β„Ž 𝑠 π‘Ÿ π‘₯ 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prds 17398 . 2 Xs = (𝑠 ∈ V, π‘Ÿ ∈ V ↦ ⦋Xπ‘₯ ∈ dom π‘Ÿ(Baseβ€˜(π‘Ÿβ€˜π‘₯)) / π‘£β¦Œβ¦‹(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ Xπ‘₯ ∈ dom π‘Ÿ((π‘“β€˜π‘₯)(Hom β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) / β„Žβ¦Œ(({⟨(Baseβ€˜ndx), π‘£βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(+gβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(.rβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(.rβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩} βˆͺ {⟨(Scalarβ€˜ndx), π‘ βŸ©, ⟨( ·𝑠 β€˜ndx), (𝑓 ∈ (Baseβ€˜π‘ ), 𝑔 ∈ 𝑣 ↦ (π‘₯ ∈ dom π‘Ÿ ↦ (𝑓( ·𝑠 β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))))⟩, ⟨(Β·π‘–β€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Ξ£g (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(Β·π‘–β€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯)))))⟩}) βˆͺ ({⟨(TopSetβ€˜ndx), (∏tβ€˜(TopOpen ∘ π‘Ÿ))⟩, ⟨(leβ€˜ndx), {βŸ¨π‘“, π‘”βŸ© ∣ ({𝑓, 𝑔} βŠ† 𝑣 ∧ βˆ€π‘₯ ∈ dom π‘Ÿ(π‘“β€˜π‘₯)(leβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))}⟩, ⟨(distβ€˜ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘“β€˜π‘₯)(distβ€˜(π‘Ÿβ€˜π‘₯))(π‘”β€˜π‘₯))) βˆͺ {0}), ℝ*, < ))⟩} βˆͺ {⟨(Hom β€˜ndx), β„ŽβŸ©, ⟨(compβ€˜ndx), (π‘Ž ∈ (𝑣 Γ— 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ ((2nd β€˜π‘Ž)β„Žπ‘), 𝑒 ∈ (β„Žβ€˜π‘Ž) ↦ (π‘₯ ∈ dom π‘Ÿ ↦ ((π‘‘β€˜π‘₯)(⟨((1st β€˜π‘Ž)β€˜π‘₯), ((2nd β€˜π‘Ž)β€˜π‘₯)⟩(compβ€˜(π‘Ÿβ€˜π‘₯))(π‘β€˜π‘₯))(π‘’β€˜π‘₯)))))⟩})))
21reldmmpo 7546 1 Rel dom Xs
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 395  βˆ€wral 3060  Vcvv 3473  β¦‹csb 3893   βˆͺ cun 3946   βŠ† wss 3948  {csn 4628  {cpr 4630  {ctp 4632  βŸ¨cop 4634   class class class wbr 5148  {copab 5210   ↦ cmpt 5231   Γ— cxp 5674  dom cdm 5676  ran crn 5677   ∘ ccom 5680  Rel wrel 5681  β€˜cfv 6543  (class class class)co 7412   ∈ cmpo 7414  1st c1st 7977  2nd c2nd 7978  Xcixp 8895  supcsup 9439  0cc0 11114  β„*cxr 11252   < clt 11253  ndxcnx 17131  Basecbs 17149  +gcplusg 17202  .rcmulr 17203  Scalarcsca 17205   ·𝑠 cvsca 17206  Β·π‘–cip 17207  TopSetcts 17208  lecple 17209  distcds 17211  Hom chom 17213  compcco 17214  TopOpenctopn 17372  βˆtcpt 17389   Ξ£g cgsu 17391  Xscprds 17396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-rel 5683  df-dm 5686  df-oprab 7416  df-mpo 7417  df-prds 17398
This theorem is referenced by:  dsmmval  21509  dsmmval2  21511  dsmmbas2  21512  dsmmfi  21513
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