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

Theorem xpsval 17617
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 3502 . . 3 (𝜑𝑅 ∈ V)
4 xpsval.2 . . . 4 (𝜑𝑆𝑊)
54elexd 3502 . . 3 (𝜑𝑆 ∈ V)
6 fveq2 6907 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
7 xpsval.x . . . . . . . . 9 𝑋 = (Base‘𝑅)
86, 7eqtr4di 2793 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
9 fveq2 6907 . . . . . . . . 9 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
10 xpsval.y . . . . . . . . 9 𝑌 = (Base‘𝑆)
119, 10eqtr4di 2793 . . . . . . . 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 2793 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = 𝐹)
1615cnveqd 5889 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = 𝐹)
17 fveq2 6907 . . . . . . . . 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 2793 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
21 simpl 482 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑟 = 𝑅)
2221opeq2d 4885 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ⟨∅, 𝑟⟩ = ⟨∅, 𝑅⟩)
23 simpr 484 . . . . . . . . 9 ((𝑟 = 𝑅𝑠 = 𝑆) → 𝑠 = 𝑆)
2423opeq2d 4885 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ⟨1o, 𝑠⟩ = ⟨1o, 𝑆⟩)
2522, 24preq12d 4746 . . . . . . 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 2793 . . . . 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 17557 . . . 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 2787 1 (𝜑𝑇 = (𝐹s 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {cpr 4633  cop 4637  ccnv 5688  cfv 6563  (class class class)co 7431  cmpo 7433  1oc1o 8498  Basecbs 17245  Scalarcsca 17301  Xscprds 17492  s cimas 17551   ×s cxps 17553
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-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xps 17557
This theorem is referenced by:  xpsbas  17619  xpsadd  17621  xpsmul  17622  xpssca  17623  xpsvsca  17624  xpsless  17625  xpsle  17626  xpsmnd  18803  xpsgrp  19090  xpsrngd  20197  xpsringd  20346  xpstps  23834  xpstopnlem2  23835  xpsdsfn  24403  xpsxmet  24406  xpsdsval  24407  xpsmet  24408  xpsxms  24563  xpsms  24564
  Copyright terms: Public domain W3C validator