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Theorem xpsval 16153
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 (𝜑𝑅𝑉)
xpsval.2 (𝜑𝑆𝑊)
xpsval.f 𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
xpsval.k 𝐺 = (Scalar‘𝑅)
xpsval.u 𝑈 = (𝐺Xs({𝑅} +𝑐 {𝑆}))
Assertion
Ref Expression
xpsval (𝜑𝑇 = (𝐹s 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.
StepHypRef Expression
1 xpsval.t . 2 𝑇 = (𝑅 ×s 𝑆)
2 xpsval.1 . . . 4 (𝜑𝑅𝑉)
3 elex 3198 . . . 4 (𝑅𝑉𝑅 ∈ V)
42, 3syl 17 . . 3 (𝜑𝑅 ∈ V)
5 xpsval.2 . . . 4 (𝜑𝑆𝑊)
6 elex 3198 . . . 4 (𝑆𝑊𝑆 ∈ V)
75, 6syl 17 . . 3 (𝜑𝑆 ∈ V)
8 fveq2 6148 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
9 xpsval.x . . . . . . . . 9 𝑋 = (Base‘𝑅)
108, 9syl6eqr 2673 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
11 fveq2 6148 . . . . . . . . 9 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
12 xpsval.y . . . . . . . . 9 𝑌 = (Base‘𝑆)
1311, 12syl6eqr 2673 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌)
14 mpt2eq12 6668 . . . . . . . 8 (((Base‘𝑟) = 𝑋 ∧ (Base‘𝑠) = 𝑌) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))
1510, 13, 14syl2an 494 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))
16 xpsval.f . . . . . . 7 𝐹 = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦}))
1715, 16syl6eqr 2673 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) = 𝐹)
1817cnveqd 5258 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) = 𝐹)
19 fveq2 6148 . . . . . . . . 9 (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅))
2019adantr 481 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅))
21 xpsval.k . . . . . . . 8 𝐺 = (Scalar‘𝑅)
2220, 21syl6eqr 2673 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
23 sneq 4158 . . . . . . . . 9 (𝑟 = 𝑅 → {𝑟} = {𝑅})
24 sneq 4158 . . . . . . . . 9 (𝑠 = 𝑆 → {𝑠} = {𝑆})
2523, 24oveqan12d 6623 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ({𝑟} +𝑐 {𝑠}) = ({𝑅} +𝑐 {𝑆}))
2625cnveqd 5258 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → ({𝑟} +𝑐 {𝑠}) = ({𝑅} +𝑐 {𝑆}))
2722, 26oveq12d 6622 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → ((Scalar‘𝑟)Xs({𝑟} +𝑐 {𝑠})) = (𝐺Xs({𝑅} +𝑐 {𝑆})))
28 xpsval.u . . . . . 6 𝑈 = (𝐺Xs({𝑅} +𝑐 {𝑆}))
2927, 28syl6eqr 2673 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ((Scalar‘𝑟)Xs({𝑟} +𝑐 {𝑠})) = 𝑈)
3018, 29oveq12d 6622 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ((𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑟)Xs({𝑟} +𝑐 {𝑠}))) = (𝐹s 𝑈))
31 df-xps 16091 . . . 4 ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑟)Xs({𝑟} +𝑐 {𝑠}))))
32 ovex 6632 . . . 4 (𝐹s 𝑈) ∈ V
3330, 31, 32ovmpt2a 6744 . . 3 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×s 𝑆) = (𝐹s 𝑈))
344, 7, 33syl2anc 692 . 2 (𝜑 → (𝑅 ×s 𝑆) = (𝐹s 𝑈))
351, 34syl5eq 2667 1 (𝜑𝑇 = (𝐹s 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148  ccnv 5073  cfv 5847  (class class class)co 6604  cmpt2 6606   +𝑐 ccda 8933  Basecbs 15781  Scalarcsca 15865  Xscprds 16027  s cimas 16085   ×s cxps 16087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-xps 16091
This theorem is referenced by:  xpsbas  16155  xpsadd  16157  xpsmul  16158  xpssca  16159  xpsvsca  16160  xpsless  16161  xpsle  16162  xpsmnd  17251  xpsgrp  17455  xpstps  21523  xpstopnlem2  21524  xpsdsfn  22092  xpsxmet  22095  xpsdsval  22096  xpsmet  22097  xpsxms  22249  xpsms  22250
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