Theorem reldmprds 16781

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 𝑎 𝑐 𝑑 𝑒 𝑓 𝑔 ℎ 𝑠 𝑟 𝑥 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-prds 16780	. 2 ⊢ Xs = (𝑠 ∈ V, 𝑟 ∈ V ↦ ⦋X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) / 𝑣⦌⦋(𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ X𝑥 ∈ dom 𝑟((𝑓‘𝑥)(Hom ‘(𝑟‘𝑥))(𝑔‘𝑥))) / ℎ⦌(({⟨(Base‘ndx), 𝑣⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(+g‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(.r‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑠⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑠), 𝑔 ∈ 𝑣 ↦ (𝑥 ∈ dom 𝑟 ↦ (𝑓( ·𝑠 ‘(𝑟‘𝑥))(𝑔‘𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ (𝑠 Σg (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(·𝑖‘(𝑟‘𝑥))(𝑔‘𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑟))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑣 ∧ ∀𝑥 ∈ dom 𝑟(𝑓‘𝑥)(le‘(𝑟‘𝑥))(𝑔‘𝑥))}⟩, ⟨(dist‘ndx), (𝑓 ∈ 𝑣, 𝑔 ∈ 𝑣 ↦ sup((ran (𝑥 ∈ dom 𝑟 ↦ ((𝑓‘𝑥)(dist‘(𝑟‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), ℎ⟩, ⟨(comp‘ndx), (𝑎 ∈ (𝑣 × 𝑣), 𝑐 ∈ 𝑣 ↦ (𝑑 ∈ ((2nd ‘𝑎)ℎ𝑐), 𝑒 ∈ (ℎ‘𝑎) ↦ (𝑥 ∈ dom 𝑟 ↦ ((𝑑‘𝑥)(⟨((1st ‘𝑎)‘𝑥), ((2nd ‘𝑎)‘𝑥)⟩(comp‘(𝑟‘𝑥))(𝑐‘𝑥))(𝑒‘𝑥)))))⟩})))
2	1	reldmmpo 7281	1 ⊢ Rel dom Xs

Syntax hints:   ∧ wa 400  ∀wral 3071  Vcvv 3410  ⦋csb 3806   ∪ cun 3857   ⊆ wss 3859  {csn 4523  {cpr 4525  {ctp 4527  ⟨cop 4529   class class class wbr 5033  {copab 5095   ↦ cmpt 5113   × cxp 5523  dom cdm 5525  ran crn 5526   ∘ ccom 5529  Rel wrel 5530  ‘cfv 6336  (class class class)co 7151   ∈ cmpo 7153  1^st c1st 7692  2^nd c2nd 7693  Xcixp 8480  supcsup 8938  0cc0 10576  ℝ^*cxr 10713   < clt 10714  ndxcnx 16539  Basecbs 16542  +_gcplusg 16624  ._rcmulr 16625  Scalarcsca 16627   ·_𝑠 cvsca 16628  ·_𝑖cip 16629  TopSetcts 16630  lecple 16631  distcds 16633  Hom chom 16635  compcco 16636  TopOpenctopn 16754  ∏_tcpt 16771   Σ_g cgsu 16773  X_scprds 16778

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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pr 5299

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-v 3412  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-nul 4227  df-if 4422  df-sn 4524  df-pr 4526  df-op 4530  df-br 5034  df-opab 5096  df-xp 5531  df-rel 5532  df-dm 5535  df-oprab 7155  df-mpo 7156  df-prds 16780

This theorem is referenced by:  dsmmval  20500  dsmmval2  20502  dsmmbas2  20503  dsmmfi  20504