MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpsval Structured version   Visualization version   GIF version

Theorem xpsval 17615
Description: Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.) (Revised by Jim Kingdon, 25-Sep-2023.)
Hypotheses
Ref Expression
xpsval.t 𝑇 = (𝑅 ×s 𝑆)
xpsval.x 𝑋 = (Base‘𝑅)
xpsval.y 𝑌 = (Base‘𝑆)
xpsval.1 (𝜑𝑅𝑉)
xpsval.2 (𝜑𝑆𝑊)
xpsval.f 𝐹 = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
xpsval.k 𝐺 = (Scalar‘𝑅)
xpsval.u 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
Assertion
Ref Expression
xpsval (𝜑𝑇 = (𝐹s 𝑈))
Distinct variable groups:   𝑥,𝑦   𝑥,𝑊   𝑥,𝑋,𝑦   𝑥,𝑅   𝑥,𝑌,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑦)   𝑆(𝑥,𝑦)   𝑇(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑦)

Proof of Theorem xpsval
Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsval.t . 2 𝑇 = (𝑅 ×s 𝑆)
2 xpsval.1 . . . 4 (𝜑𝑅𝑉)
32elexd 3504 . . 3 (𝜑𝑅 ∈ V)
4 xpsval.2 . . . 4 (𝜑𝑆𝑊)
54elexd 3504 . . 3 (𝜑𝑆 ∈ V)
6 fveq2 6906 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
7 xpsval.x . . . . . . . . 9 𝑋 = (Base‘𝑅)
86, 7eqtr4di 2795 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
9 fveq2 6906 . . . . . . . . 9 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
10 xpsval.y . . . . . . . . 9 𝑌 = (Base‘𝑆)
119, 10eqtr4di 2795 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌)
12 mpoeq12 7506 . . . . . . . 8 (((Base‘𝑟) = 𝑋 ∧ (Base‘𝑠) = 𝑌) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
138, 11, 12syl2an 596 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
14 xpsval.f . . . . . . 7 𝐹 = (𝑥𝑋, 𝑦𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
1513, 14eqtr4di 2795 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = 𝐹)
1615cnveqd 5886 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = 𝐹)
17 fveq2 6906 . . . . . . . . 9 (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅))
1817adantr 480 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅))
19 xpsval.k . . . . . . . 8 𝐺 = (Scalar‘𝑅)
2018, 19eqtr4di 2795 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
21 simpl 482 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
2221opeq2d 4880 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ⟨∅, 𝑟⟩ = ⟨∅, 𝑅⟩)
23 simpr 484 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑠 = 𝑆)
2423opeq2d 4880 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ⟨1o, 𝑠⟩ = ⟨1o, 𝑆⟩)
2522, 24preq12d 4741 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → {⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩} = {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
2620, 25oveq12d 7449 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩}) = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
27 xpsval.u . . . . . 6 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
2826, 27eqtr4di 2795 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩}) = 𝑈)
2916, 28oveq12d 7449 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})) = (𝐹s 𝑈))
30 df-xps 17555 . . . 4 ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
31 ovex 7464 . . . 4 (𝐹s 𝑈) ∈ V
3229, 30, 31ovmpoa 7588 . . 3 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×s 𝑆) = (𝐹s 𝑈))
333, 5, 32syl2anc 584 . 2 (𝜑 → (𝑅 ×s 𝑆) = (𝐹s 𝑈))
341, 33eqtrid 2789 1 (𝜑𝑇 = (𝐹s 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {cpr 4628  cop 4632  ccnv 5684  cfv 6561  (class class class)co 7431  cmpo 7433  1oc1o 8499  Basecbs 17247  Scalarcsca 17300  Xscprds 17490  s cimas 17549   ×s cxps 17551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xps 17555
This theorem is referenced by:  xpsbas  17617  xpsadd  17619  xpsmul  17620  xpssca  17621  xpsvsca  17622  xpsless  17623  xpsle  17624  xpsmnd  18790  xpsgrp  19077  xpsrngd  20176  xpsringd  20329  xpstps  23818  xpstopnlem2  23819  xpsdsfn  24387  xpsxmet  24390  xpsdsval  24391  xpsmet  24392  xpsxms  24547  xpsms  24548
  Copyright terms: Public domain W3C validator