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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 3429 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | elex 3429 | . . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
4 | simpl 476 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑅) | |
5 | 4 | fveq2d 6441 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅)) |
6 | pwsval.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑅) | |
7 | 5, 6 | syl6eqr 2879 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹) |
8 | id 22 | . . . . . 6 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼) | |
9 | sneq 4409 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅}) | |
10 | xpeq12 5371 | . . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) | |
11 | 8, 9, 10 | syl2anr 590 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) |
12 | 7, 11 | oveq12d 6928 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅}))) |
13 | df-pws 16470 | . . . 4 ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) | |
14 | ovex 6942 | . . . 4 ⊢ (𝐹Xs(𝐼 × {𝑅})) ∈ V | |
15 | 12, 13, 14 | ovmpt2a 7056 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
16 | 2, 3, 15 | syl2an 589 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
17 | 1, 16 | syl5eq 2873 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 Vcvv 3414 {csn 4399 × cxp 5344 ‘cfv 6127 (class class class)co 6910 Scalarcsca 16315 Xscprds 16466 ↑s cpws 16467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-iota 6090 df-fun 6129 df-fv 6135 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-pws 16470 |
This theorem is referenced by: pwsbas 16507 pwsplusgval 16510 pwsmulrval 16511 pwsle 16512 pwsvscafval 16514 pwssca 16516 pwsmnd 17685 pws0g 17686 pwspjmhm 17728 pwsgrp 17888 pwsinvg 17889 pwscmn 18626 pwsabl 18627 pwsgsum 18738 pwsring 18976 pws1 18977 pwscrng 18978 pwsmgp 18979 pwslmod 19336 frlmpws 20464 frlmlss 20465 frlmpwsfi 20466 frlmbas 20469 frlmip 20491 pwstps 21811 resspwsds 22554 pwsxms 22714 pwsms 22715 rrxprds 23564 cnpwstotbnd 34133 repwsmet 34170 rrnequiv 34171 |
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