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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 3499 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | elex 3499 | . . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
4 | simpl 485 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑅) | |
5 | 4 | fveq2d 6655 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅)) |
6 | pwsval.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑅) | |
7 | 5, 6 | syl6eqr 2873 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹) |
8 | id 22 | . . . . . 6 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼) | |
9 | sneq 4558 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅}) | |
10 | xpeq12 5561 | . . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) | |
11 | 8, 9, 10 | syl2anr 598 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) |
12 | 7, 11 | oveq12d 7155 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅}))) |
13 | df-pws 16701 | . . . 4 ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) | |
14 | ovex 7170 | . . . 4 ⊢ (𝐹Xs(𝐼 × {𝑅})) ∈ V | |
15 | 12, 13, 14 | ovmpoa 7286 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
16 | 2, 3, 15 | syl2an 597 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
17 | 1, 16 | syl5eq 2867 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3481 {csn 4548 × cxp 5534 ‘cfv 6336 (class class class)co 7137 Scalarcsca 16546 Xscprds 16697 ↑s cpws 16698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-sep 5184 ax-nul 5191 ax-pr 5311 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ral 3138 df-rex 3139 df-rab 3142 df-v 3483 df-sbc 3759 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-br 5048 df-opab 5110 df-id 5441 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-iota 6295 df-fun 6338 df-fv 6344 df-ov 7140 df-oprab 7141 df-mpo 7142 df-pws 16701 |
This theorem is referenced by: pwsbas 16738 pwsplusgval 16741 pwsmulrval 16742 pwsle 16743 pwsvscafval 16745 pwssca 16747 pwsmnd 17924 pws0g 17925 pwspjmhm 17972 pwsgrp 18189 pwsinvg 18190 pwscmn 18961 pwsabl 18962 pwsgsum 19080 pwsring 19343 pws1 19344 pwscrng 19345 pwsmgp 19346 pwslmod 19720 frlmpws 20872 frlmlss 20873 frlmpwsfi 20874 frlmbas 20877 frlmip 20900 pwstps 22216 resspwsds 22960 pwsxms 23120 pwsms 23121 rrxprds 23970 cnpwstotbnd 35102 repwsmet 35139 rrnequiv 35140 |
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