Theorem xpsval 16232

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 3212	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
5		xpsval.2	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
6		elex 3212	. . . 4 ⊢ (𝑆 ∈ 𝑊 → 𝑆 ∈ V)
7	5, 6	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
8		fveq2 6191	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
9		xpsval.x	. . . . . . . . 9 ⊢ 𝑋 = (Base‘𝑅)
10	8, 9	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
11		fveq2 6191	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
12		xpsval.y	. . . . . . . . 9 ⊢ 𝑌 = (Base‘𝑆)
13	11, 12	syl6eqr 2674	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌)
14		mpt2eq12 6715	. . . . . . . 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 494	. . . . . . 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 2674	. . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) = 𝐹)
18	17	cnveqd 5298	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) = ◡𝐹)
19		fveq2 6191	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅))
20	19	adantr 481	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅))
21		xpsval.k	. . . . . . . 8 ⊢ 𝐺 = (Scalar‘𝑅)
22	20, 21	syl6eqr 2674	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
23		sneq 4187	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅})
24		sneq 4187	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → {𝑠} = {𝑆})
25	23, 24	oveqan12d 6669	. . . . . . . 8 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ({𝑟} +𝑐 {𝑠}) = ({𝑅} +𝑐 {𝑆}))
26	25	cnveqd 5298	. . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡({𝑟} +𝑐 {𝑠}) = ◡({𝑅} +𝑐 {𝑆}))
27	22, 26	oveq12d 6668	. . . . . 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 2674	. . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs◡({𝑟} +𝑐 {𝑠})) = 𝑈)
30	18, 29	oveq12d 6668	. . . 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 16170	. . . 4 ⊢ ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ◡({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑟)Xs◡({𝑟} +𝑐 {𝑠}))))
32		ovex 6678	. . . 4 ⊢ (◡𝐹 “s 𝑈) ∈ V
33	30, 31, 32	ovmpt2a 6791	. . 3 ⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
34	4, 7, 33	syl2anc 693	. 2 ⊢ (𝜑 → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈))
35	1, 34	syl5eq 2668	1 ⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈))

Syntax hints:   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ^◡ccnv 5113  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652   +_𝑐 ccda 8989  Basecbs 15857  Scalarcsca 15944  X_scprds 16106   “_s cimas 16164   ×_s cxps 16166

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-xps 16170

This theorem is referenced by:  xpsbas  16234  xpsadd  16236  xpsmul  16237  xpssca  16238  xpsvsca  16239  xpsless  16240  xpsle  16241  xpsmnd  17330  xpsgrp  17534  xpstps  21613  xpstopnlem2  21614  xpsdsfn  22182  xpsxmet  22185  xpsdsval  22186  xpsmet  22187  xpsxms  22339  xpsms  22340