![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pwsval | Structured version Visualization version GIF version |
Description: Value of a structure power. (Contributed by Mario Carneiro, 11-Jan-2015.) |
Ref | Expression |
---|---|
pwsval.y | ⊢ 𝑌 = (𝑅 ↑s 𝐼) |
pwsval.f | ⊢ 𝐹 = (Scalar‘𝑅) |
Ref | Expression |
---|---|
pwsval | ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsval.y | . 2 ⊢ 𝑌 = (𝑅 ↑s 𝐼) | |
2 | elex 3498 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | elex 3498 | . . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
4 | simpl 482 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑅) | |
5 | 4 | fveq2d 6910 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅)) |
6 | pwsval.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑅) | |
7 | 5, 6 | eqtr4di 2792 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹) |
8 | id 22 | . . . . . 6 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼) | |
9 | sneq 4640 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅}) | |
10 | xpeq12 5713 | . . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) | |
11 | 8, 9, 10 | syl2anr 597 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) |
12 | 7, 11 | oveq12d 7448 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅}))) |
13 | df-pws 17495 | . . . 4 ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) | |
14 | ovex 7463 | . . . 4 ⊢ (𝐹Xs(𝐼 × {𝑅})) ∈ V | |
15 | 12, 13, 14 | ovmpoa 7587 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
16 | 2, 3, 15 | syl2an 596 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
17 | 1, 16 | eqtrid 2786 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 Vcvv 3477 {csn 4630 × cxp 5686 ‘cfv 6562 (class class class)co 7430 Scalarcsca 17300 Xscprds 17491 ↑s cpws 17492 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-pws 17495 |
This theorem is referenced by: pwsbas 17533 pwsplusgval 17536 pwsmulrval 17537 pwsle 17538 pwsvscafval 17540 pwssca 17542 pwsmnd 18797 pws0g 18798 pwspjmhm 18855 pwsgrp 19082 pwsinvg 19083 pwscmn 19895 pwsabl 19896 pwsgsum 20014 pwsring 20337 pws1 20338 pwscrng 20339 pwsmgp 20340 pwslmod 20985 frlmpws 21787 frlmlss 21788 frlmpwsfi 21789 frlmbas 21792 frlmip 21815 pwstps 23653 resspwsds 24397 pwsxms 24560 pwsms 24561 rrxprds 25436 cnpwstotbnd 37783 repwsmet 37820 rrnequiv 37821 |
Copyright terms: Public domain | W3C validator |