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Theorem xpsval 16437
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 3406 . . . 4 (𝑅𝑉𝑅 ∈ V)
42, 3syl 17 . . 3 (𝜑𝑅 ∈ V)
5 xpsval.2 . . . 4 (𝜑𝑆𝑊)
6 elex 3406 . . . 4 (𝑆𝑊𝑆 ∈ V)
75, 6syl 17 . . 3 (𝜑𝑆 ∈ V)
8 fveq2 6408 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
9 xpsval.x . . . . . . . . 9 𝑋 = (Base‘𝑅)
108, 9syl6eqr 2858 . . . . . . . 8 (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋)
11 fveq2 6408 . . . . . . . . 9 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
12 xpsval.y . . . . . . . . 9 𝑌 = (Base‘𝑆)
1311, 12syl6eqr 2858 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌)
14 mpt2eq12 6945 . . . . . . . 8 (((Base‘𝑟) = 𝑋 ∧ (Base‘𝑠) = 𝑌) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) = (𝑥𝑋, 𝑦𝑌({𝑥} +𝑐 {𝑦})))
1510, 13, 14syl2an 585 . . . . . . 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 2858 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) = 𝐹)
1817cnveqd 5499 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) = 𝐹)
19 fveq2 6408 . . . . . . . . 9 (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅))
2019adantr 468 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅))
21 xpsval.k . . . . . . . 8 𝐺 = (Scalar‘𝑅)
2220, 21syl6eqr 2858 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺)
23 sneq 4380 . . . . . . . . 9 (𝑟 = 𝑅 → {𝑟} = {𝑅})
24 sneq 4380 . . . . . . . . 9 (𝑠 = 𝑆 → {𝑠} = {𝑆})
2523, 24oveqan12d 6893 . . . . . . . 8 ((𝑟 = 𝑅𝑠 = 𝑆) → ({𝑟} +𝑐 {𝑠}) = ({𝑅} +𝑐 {𝑆}))
2625cnveqd 5499 . . . . . . 7 ((𝑟 = 𝑅𝑠 = 𝑆) → ({𝑟} +𝑐 {𝑠}) = ({𝑅} +𝑐 {𝑆}))
2722, 26oveq12d 6892 . . . . . 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 2858 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ((Scalar‘𝑟)Xs({𝑟} +𝑐 {𝑠})) = 𝑈)
3018, 29oveq12d 6892 . . . 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 16375 . . . 4 ×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ ((𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ ({𝑥} +𝑐 {𝑦})) “s ((Scalar‘𝑟)Xs({𝑟} +𝑐 {𝑠}))))
32 ovex 6906 . . . 4 (𝐹s 𝑈) ∈ V
3330, 31, 32ovmpt2a 7021 . . 3 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×s 𝑆) = (𝐹s 𝑈))
344, 7, 33syl2anc 575 . 2 (𝜑 → (𝑅 ×s 𝑆) = (𝐹s 𝑈))
351, 34syl5eq 2852 1 (𝜑𝑇 = (𝐹s 𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1637  wcel 2156  Vcvv 3391  {csn 4370  ccnv 5310  cfv 6101  (class class class)co 6874  cmpt2 6876   +𝑐 ccda 9274  Basecbs 16068  Scalarcsca 16156  Xscprds 16311  s cimas 16369   ×s cxps 16371
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6877  df-oprab 6878  df-mpt2 6879  df-xps 16375
This theorem is referenced by:  xpsbas  16439  xpsadd  16441  xpsmul  16442  xpssca  16443  xpsvsca  16444  xpsless  16445  xpsle  16446  xpsmnd  17535  xpsgrp  17739  xpstps  21827  xpstopnlem2  21828  xpsdsfn  22395  xpsxmet  22398  xpsdsval  22399  xpsmet  22400  xpsxms  22552  xpsms  22553
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