Theorem xpsval 17614

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	⊢ (𝜑 → 𝑅 ∈ 𝑉)
xpsval.2	⊢ (𝜑 → 𝑆 ∈ 𝑊)
xpsval.f	⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
xpsval.k	⊢ 𝐺 = (Scalar‘𝑅)
xpsval.u	⊢ 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})

Assertion

Ref	Expression
xpsval	⊢ (𝜑 → 𝑇 = (◡𝐹 “s 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.

Step	Hyp	Ref	Expression
1		xpsval.t	. 2 ⊢ 𝑇 = (𝑅 ×s 𝑆)
2		xpsval.1	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑉)
3	2	elexd 3480	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
4		xpsval.2	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
5	4	elexd 3480	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
6		fveq2 6871	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
7		xpsval.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑅)
8	6, 7	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
9		fveq2 6871	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
10		xpsval.y	. . . . . . . . 9 ⊢ 𝑌 = (Base‘𝑆)
11	9, 10	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌)
12		mpoeq12 7473	. . . . . . . 8 ⊢ (((Base‘𝑟) = 𝑋 ∧ (Base‘𝑠) = 𝑌) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
13	8, 11, 12	syl2an 607	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}))
14		xpsval.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩})
15	13, 14	eqtr4di 2818	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = 𝐹)
16	15	cnveqd 5852	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = ◡𝐹)
17		fveq2 6871	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅))
18	17	adantr 485	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅))
19		xpsval.k	. . . . . . . 8 ⊢ 𝐺 = (Scalar‘𝑅)
20	18, 19	eqtr4di 2818	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
21		simpl 487	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
22	21	opeq2d 4841	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⟨∅, 𝑟⟩ = ⟨∅, 𝑅⟩)
23		simpr 489	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
24	23	opeq2d 4841	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⟨1o, 𝑠⟩ = ⟨1o, 𝑆⟩)
25	22, 24	preq12d 4703	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩} = {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
26	20, 25	oveq12d 7418	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩}) = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩}))
27		xpsval.u	. . . . . 6 ⊢ 𝑈 = (𝐺Xs{⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
28	26, 27	eqtr4di 2818	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩}) = 𝑈)
29	16, 28	oveq12d 7418	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})) = (◡𝐹 “s 𝑈))
30		df-xps 17554	. . . 4 ⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
31		ovex 7433	. . . 4 ⊢ (◡𝐹 “s 𝑈) ∈ V
32	29, 30, 31	ovmpoa 7555	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
33	3, 5, 32	syl2anc 595	. 2 ⊢ (𝜑 → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
34	1, 33	eqtrid 2812	1 ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ∅c0 4288  {cpr 4587  ⟨cop 4591  ^◡ccnv 5651  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  1_oc1o 8434  Basecbs 17259  Scalarcsca 17303  X_scprds 17488   “_s cimas 17548   ×_s cxps 17550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-xps 17554

This theorem is referenced by:  xpsbas  17616  xpsadd  17618  xpsmul  17619  xpssca  17620  xpsvsca  17621  xpsless  17622  xpsle  17623  xpsmnd  18825  xpsgrp  19116  xpsrngd  20248  xpsringd  20405  xpstps  23928  xpstopnlem2  23929  xpsdsfn  24495  xpsxmet  24498  xpsdsval  24499  xpsmet  24500  xpsxms  24652  xpsms  24653