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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 3212 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | elex 3212 | . . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
4 | simpl 473 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑅) | |
5 | 4 | fveq2d 6195 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅)) |
6 | pwsval.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑅) | |
7 | 5, 6 | syl6eqr 2674 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹) |
8 | id 22 | . . . . . 6 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼) | |
9 | sneq 4187 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅}) | |
10 | xpeq12 5134 | . . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) | |
11 | 8, 9, 10 | syl2anr 495 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) |
12 | 7, 11 | oveq12d 6668 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅}))) |
13 | df-pws 16110 | . . . 4 ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) | |
14 | ovex 6678 | . . . 4 ⊢ (𝐹Xs(𝐼 × {𝑅})) ∈ V | |
15 | 12, 13, 14 | ovmpt2a 6791 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
16 | 2, 3, 15 | syl2an 494 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
17 | 1, 16 | syl5eq 2668 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 {csn 4177 × cxp 5112 ‘cfv 5888 (class class class)co 6650 Scalarcsca 15944 Xscprds 16106 ↑s cpws 16107 |
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-pws 16110 |
This theorem is referenced by: pwsbas 16147 pwsplusgval 16150 pwsmulrval 16151 pwsle 16152 pwsvscafval 16154 pwssca 16156 pwsmnd 17325 pws0g 17326 pwspjmhm 17368 pwsgrp 17527 pwsinvg 17528 pwscmn 18266 pwsabl 18267 pwsgsum 18378 pwsring 18615 pws1 18616 pwscrng 18617 pwsmgp 18618 pwslmod 18970 frlmpws 20094 frlmlss 20095 frlmpwsfi 20096 frlmbas 20099 frlmip 20117 pwstps 21433 resspwsds 22177 pwsxms 22337 pwsms 22338 rrxprds 23177 cnpwstotbnd 33596 repwsmet 33633 rrnequiv 33634 |
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