Theorem xpsval 17649

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 3481	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
4		xpsval.2	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
5	4	elexd 3481	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
6		fveq2 6888	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
7		xpsval.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑅)
8	6, 7	eqtr4di 2819	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
9		fveq2 6888	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
10		xpsval.y	. . . . . . . . 9 ⊢ 𝑌 = (Base‘𝑆)
11	9, 10	eqtr4di 2819	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌)
12		mpoeq12 7496	. . . . . . . 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 608	. . . . . . 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 2819	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = 𝐹)
16	15	cnveqd 5866	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) = ◡𝐹)
17		fveq2 6888	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅))
18	17	adantr 486	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅))
19		xpsval.k	. . . . . . . 8 ⊢ 𝐺 = (Scalar‘𝑅)
20	18, 19	eqtr4di 2819	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
21		simpl 488	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅)
22	21	opeq2d 4850	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⟨∅, 𝑟⟩ = ⟨∅, 𝑅⟩)
23		simpr 490	. . . . . . . . 9 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆)
24	23	opeq2d 4850	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ⟨1o, 𝑠⟩ = ⟨1o, 𝑆⟩)
25	22, 24	preq12d 4712	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩} = {⟨∅, 𝑅⟩, ⟨1o, 𝑆⟩})
26	20, 25	oveq12d 7441	. . . . . 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 2819	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩}) = 𝑈)
29	16, 28	oveq12d 7441	. . . 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 17589	. . . 4 ⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {⟨∅, 𝑥⟩, ⟨1o, 𝑦⟩}) “s ((Scalar‘𝑟)Xs{⟨∅, 𝑟⟩, ⟨1o, 𝑠⟩})))
31		ovex 7456	. . . 4 ⊢ (◡𝐹 “s 𝑈) ∈ V
32	29, 30, 31	ovmpoa 7578	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
33	3, 5, 32	syl2anc 596	. 2 ⊢ (𝜑 → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
34	1, 33	eqtrid 2813	1 ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  Vcvv 3458  ∅c0 4289  {cpr 4596  ⟨cop 4600  ^◡ccnv 5665  ‘cfv 6543  (class class class)co 7423   ∈ cmpo 7425  1_oc1o 8455  Basecbs 17294  Scalarcsca 17338  X_scprds 17523   “_s cimas 17583   ×_s cxps 17585

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6499  df-fun 6545  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-xps 17589

This theorem is used by:  xpsbas  17651  xpsadd  17653  xpsmul  17654  xpssca  17655  xpsvsca  17656  xpsless  17657  xpsle  17658  xpsmnd  18866  xpsgrp  19156  xpsrngd  20288  xpsringd  20447  xpstps  24004  xpstopnlem2  24005  xpsdsfn  24571  xpsxmet  24574  xpsdsval  24575  xpsmet  24576  xpsxms  24728  xpsms  24729