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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 3492 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
3 | elex 3492 | . . 3 ⊢ (𝐼 ∈ 𝑊 → 𝐼 ∈ V) | |
4 | simpl 482 | . . . . . . 7 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → 𝑟 = 𝑅) | |
5 | 4 | fveq2d 6895 | . . . . . 6 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = (Scalar‘𝑅)) |
6 | pwsval.f | . . . . . 6 ⊢ 𝐹 = (Scalar‘𝑅) | |
7 | 5, 6 | eqtr4di 2789 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹) |
8 | id 22 | . . . . . 6 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼) | |
9 | sneq 4638 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅}) | |
10 | xpeq12 5701 | . . . . . 6 ⊢ ((𝑖 = 𝐼 ∧ {𝑟} = {𝑅}) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) | |
11 | 8, 9, 10 | syl2anr 596 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (𝑖 × {𝑟}) = (𝐼 × {𝑅})) |
12 | 7, 11 | oveq12d 7430 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅}))) |
13 | df-pws 17402 | . . . 4 ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) | |
14 | ovex 7445 | . . . 4 ⊢ (𝐹Xs(𝐼 × {𝑅})) ∈ V | |
15 | 12, 13, 14 | ovmpoa 7566 | . . 3 ⊢ ((𝑅 ∈ V ∧ 𝐼 ∈ V) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
16 | 2, 3, 15 | syl2an 595 | . 2 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑅 ↑s 𝐼) = (𝐹Xs(𝐼 × {𝑅}))) |
17 | 1, 16 | eqtrid 2783 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2105 Vcvv 3473 {csn 4628 × cxp 5674 ‘cfv 6543 (class class class)co 7412 Scalarcsca 17207 Xscprds 17398 ↑s cpws 17399 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-pws 17402 |
This theorem is referenced by: pwsbas 17440 pwsplusgval 17443 pwsmulrval 17444 pwsle 17445 pwsvscafval 17447 pwssca 17449 pwsmnd 18697 pws0g 18698 pwspjmhm 18750 pwsgrp 18975 pwsinvg 18976 pwscmn 19776 pwsabl 19777 pwsgsum 19895 pwsring 20216 pws1 20217 pwscrng 20218 pwsmgp 20219 pwslmod 20729 frlmpws 21528 frlmlss 21529 frlmpwsfi 21530 frlmbas 21533 frlmip 21556 pwstps 23367 resspwsds 24111 pwsxms 24274 pwsms 24275 rrxprds 25150 cnpwstotbnd 36981 repwsmet 37018 rrnequiv 37019 |
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