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

Theorem xpsval 17453
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 Γ—s 𝑆)
xpsval.x 𝑋 = (Baseβ€˜π‘…)
xpsval.y π‘Œ = (Baseβ€˜π‘†)
xpsval.1 (πœ‘ β†’ 𝑅 ∈ 𝑉)
xpsval.2 (πœ‘ β†’ 𝑆 ∈ π‘Š)
xpsval.f 𝐹 = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©})
xpsval.k 𝐺 = (Scalarβ€˜π‘…)
xpsval.u π‘ˆ = (𝐺Xs{βŸ¨βˆ…, π‘…βŸ©, ⟨1o, π‘†βŸ©})
Assertion
Ref Expression
xpsval (πœ‘ β†’ 𝑇 = (◑𝐹 β€œs π‘ˆ))
Distinct variable groups:   π‘₯,𝑦   π‘₯,π‘Š   π‘₯,𝑋,𝑦   π‘₯,𝑅   π‘₯,π‘Œ,𝑦
Allowed substitution hints:   πœ‘(π‘₯,𝑦)   𝑅(𝑦)   𝑆(π‘₯,𝑦)   𝑇(π‘₯,𝑦)   π‘ˆ(π‘₯,𝑦)   𝐹(π‘₯,𝑦)   𝐺(π‘₯,𝑦)   𝑉(π‘₯,𝑦)   π‘Š(𝑦)

Proof of Theorem xpsval
Dummy variables 𝑠 π‘Ÿ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xpsval.t . 2 𝑇 = (𝑅 Γ—s 𝑆)
2 xpsval.1 . . . 4 (πœ‘ β†’ 𝑅 ∈ 𝑉)
32elexd 3466 . . 3 (πœ‘ β†’ 𝑅 ∈ V)
4 xpsval.2 . . . 4 (πœ‘ β†’ 𝑆 ∈ π‘Š)
54elexd 3466 . . 3 (πœ‘ β†’ 𝑆 ∈ V)
6 fveq2 6843 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
7 xpsval.x . . . . . . . . 9 𝑋 = (Baseβ€˜π‘…)
86, 7eqtr4di 2795 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝑋)
9 fveq2 6843 . . . . . . . . 9 (𝑠 = 𝑆 β†’ (Baseβ€˜π‘ ) = (Baseβ€˜π‘†))
10 xpsval.y . . . . . . . . 9 π‘Œ = (Baseβ€˜π‘†)
119, 10eqtr4di 2795 . . . . . . . 8 (𝑠 = 𝑆 β†’ (Baseβ€˜π‘ ) = π‘Œ)
12 mpoeq12 7431 . . . . . . . 8 (((Baseβ€˜π‘Ÿ) = 𝑋 ∧ (Baseβ€˜π‘ ) = π‘Œ) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}))
138, 11, 12syl2an 597 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}))
14 xpsval.f . . . . . . 7 𝐹 = (π‘₯ ∈ 𝑋, 𝑦 ∈ π‘Œ ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©})
1513, 14eqtr4di 2795 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) = 𝐹)
1615cnveqd 5832 . . . . 5 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) = ◑𝐹)
17 fveq2 6843 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (Scalarβ€˜π‘Ÿ) = (Scalarβ€˜π‘…))
1817adantr 482 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (Scalarβ€˜π‘Ÿ) = (Scalarβ€˜π‘…))
19 xpsval.k . . . . . . . 8 𝐺 = (Scalarβ€˜π‘…)
2018, 19eqtr4di 2795 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (Scalarβ€˜π‘Ÿ) = 𝐺)
21 simpl 484 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ π‘Ÿ = 𝑅)
2221opeq2d 4838 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ βŸ¨βˆ…, π‘ŸβŸ© = βŸ¨βˆ…, π‘…βŸ©)
23 simpr 486 . . . . . . . . 9 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ 𝑠 = 𝑆)
2423opeq2d 4838 . . . . . . . 8 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ ⟨1o, π‘ βŸ© = ⟨1o, π‘†βŸ©)
2522, 24preq12d 4703 . . . . . . 7 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ {βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©} = {βŸ¨βˆ…, π‘…βŸ©, ⟨1o, π‘†βŸ©})
2620, 25oveq12d 7376 . . . . . 6 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©}) = (𝐺Xs{βŸ¨βˆ…, π‘…βŸ©, ⟨1o, π‘†βŸ©}))
27 xpsval.u . . . . . 6 π‘ˆ = (𝐺Xs{βŸ¨βˆ…, π‘…βŸ©, ⟨1o, π‘†βŸ©})
2826, 27eqtr4di 2795 . . . . 5 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©}) = π‘ˆ)
2916, 28oveq12d 7376 . . . 4 ((π‘Ÿ = 𝑅 ∧ 𝑠 = 𝑆) β†’ (β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) β€œs ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©})) = (◑𝐹 β€œs π‘ˆ))
30 df-xps 17393 . . . 4 Γ—s = (π‘Ÿ ∈ V, 𝑠 ∈ V ↦ (β—‘(π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Baseβ€˜π‘ ) ↦ {βŸ¨βˆ…, π‘₯⟩, ⟨1o, π‘¦βŸ©}) β€œs ((Scalarβ€˜π‘Ÿ)Xs{βŸ¨βˆ…, π‘ŸβŸ©, ⟨1o, π‘ βŸ©})))
31 ovex 7391 . . . 4 (◑𝐹 β€œs π‘ˆ) ∈ V
3229, 30, 31ovmpoa 7511 . . 3 ((𝑅 ∈ V ∧ 𝑆 ∈ V) β†’ (𝑅 Γ—s 𝑆) = (◑𝐹 β€œs π‘ˆ))
333, 5, 32syl2anc 585 . 2 (πœ‘ β†’ (𝑅 Γ—s 𝑆) = (◑𝐹 β€œs π‘ˆ))
341, 33eqtrid 2789 1 (πœ‘ β†’ 𝑇 = (◑𝐹 β€œs π‘ˆ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3446  βˆ…c0 4283  {cpr 4589  βŸ¨cop 4593  β—‘ccnv 5633  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1oc1o 8406  Basecbs 17084  Scalarcsca 17137  Xscprds 17328   β€œs cimas 17387   Γ—s cxps 17389
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-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  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-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-xps 17393
This theorem is referenced by:  xpsbas  17455  xpsadd  17457  xpsmul  17458  xpssca  17459  xpsvsca  17460  xpsless  17461  xpsle  17462  xpsmnd  18597  xpsgrp  18867  xpstps  23164  xpstopnlem2  23165  xpsdsfn  23733  xpsxmet  23736  xpsdsval  23737  xpsmet  23738  xpsxms  23893  xpsms  23894
  Copyright terms: Public domain W3C validator