Theorem xpsval 16055

Description: Value of the binary structure product function. (Contributed by Mario Carneiro, 14-Aug-2015.)

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} +𝑐 {𝑦}))
xpsval.k	⊢ 𝐺 = (Scalar‘𝑅)
xpsval.u	⊢ 𝑈 = (𝐺Xs◡({𝑅} +𝑐 {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		elex 3185	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
5		xpsval.2	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
6		elex 3185	. . . 4 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
8		fveq2 6103	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
9		xpsval.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑅)
10	8, 9	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
11		fveq2 6103	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
12		xpsval.y	. . . . . . . . 9 ⊢ 𝑌 = (Base‘𝑆)
13	11, 12	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌)
14		mpt2eq12 6613	. . . . . . . 8 ⊢ (((Base‘𝑟) = 𝑋 ∧ (Base‘𝑠) = 𝑌) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
15	10, 13, 14	syl2an 493	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦})))
16		xpsval.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ ◡({𝑥} +𝑐 {𝑦}))
17	15, 16	syl6eqr 2662	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) = 𝐹)
18	17	cnveqd 5220	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) = ◡𝐹)
19		fveq2 6103	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅))
20	19	adantr 480	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅))
21		xpsval.k	. . . . . . . 8 ⊢ 𝐺 = (Scalar‘𝑅)
22	20, 21	syl6eqr 2662	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
23		sneq 4135	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅})
24		sneq 4135	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → {𝑠} = {𝑆})
25	23, 24	oveqan12d 6568	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ({𝑟} +𝑐 {𝑠}) = ({𝑅} +𝑐 {𝑆}))
26	25	cnveqd 5220	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡({𝑟} +𝑐 {𝑠}) = ◡({𝑅} +𝑐 {𝑆}))
27	22, 26	oveq12d 6567	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs◡({𝑟} +𝑐 {𝑠})) = (𝐺Xs◡({𝑅} +𝑐 {𝑆})))
28		xpsval.u	. . . . . 6 ⊢ 𝑈 = (𝐺Xs◡({𝑅} +𝑐 {𝑆}))
29	27, 28	syl6eqr 2662	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs◡({𝑟} +𝑐 {𝑠})) = 𝑈)
30	18, 29	oveq12d 6567	. . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑟)Xs◡({𝑟} +𝑐 {𝑠}))) = (◡𝐹 “s 𝑈))
31		df-xps 15993	. . . 4 ⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑟)Xs◡({𝑟} +𝑐 {𝑠}))))
32		ovex 6577	. . . 4 ⊢ (◡𝐹 “s 𝑈) ∈ V
33	30, 31, 32	ovmpt2a 6689	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
34	4, 7, 33	syl2anc 691	. 2 ⊢ (𝜑 → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
35	1, 34	syl5eq 2656	1 ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ^◡ccnv 5037  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551   +_𝑐 ccda 8872  Basecbs 15695  Scalarcsca 15771  X_scprds 15929   “_s cimas 15987   ×_s cxps 15989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-xps 15993

This theorem is referenced by:  xpsbas  16057  xpsadd  16059  xpsmul  16060  xpssca  16061  xpsvsca  16062  xpsless  16063  xpsle  16064  xpsmnd  17153  xpsgrp  17357  xpstps  21423  xpstopnlem2  21424  xpsdsfn  21992  xpsxmet  21995  xpsdsval  21996  xpsmet  21997  xpsxms  22149  xpsms  22150