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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 3501 | . . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V) | |
| 3 | elex 3501 | . . 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 2795 | . . . . 5 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → (Scalar‘𝑟) = 𝐹) |
| 8 | id 22 | . . . . . 6 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼) | |
| 9 | sneq 4636 | . . . . . 6 ⊢ (𝑟 = 𝑅 → {𝑟} = {𝑅}) | |
| 10 | xpeq12 5710 | . . . . . 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 7449 | . . . 4 ⊢ ((𝑟 = 𝑅 ∧ 𝑖 = 𝐼) → ((Scalar‘𝑟)Xs(𝑖 × {𝑟})) = (𝐹Xs(𝐼 × {𝑅}))) |
| 13 | df-pws 17494 | . . . 4 ⊢ ↑s = (𝑟 ∈ V, 𝑖 ∈ V ↦ ((Scalar‘𝑟)Xs(𝑖 × {𝑟}))) | |
| 14 | ovex 7464 | . . . 4 ⊢ (𝐹Xs(𝐼 × {𝑅})) ∈ V | |
| 15 | 12, 13, 14 | ovmpoa 7588 | . . 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 2789 | 1 ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑌 = (𝐹Xs(𝐼 × {𝑅}))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 {csn 4626 × cxp 5683 ‘cfv 6561 (class class class)co 7431 Scalarcsca 17300 Xscprds 17490 ↑s cpws 17491 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pws 17494 |
| This theorem is referenced by: pwsbas 17532 pwsplusgval 17535 pwsmulrval 17536 pwsle 17537 pwsvscafval 17539 pwssca 17541 pwsmnd 18785 pws0g 18786 pwspjmhm 18843 pwsgrp 19070 pwsinvg 19071 pwscmn 19881 pwsabl 19882 pwsgsum 20000 pwsring 20321 pws1 20322 pwscrng 20323 pwsmgp 20324 pwslmod 20968 frlmpws 21770 frlmlss 21771 frlmpwsfi 21772 frlmbas 21775 frlmip 21798 pwstps 23638 resspwsds 24382 pwsxms 24545 pwsms 24546 rrxprds 25423 cnpwstotbnd 37804 repwsmet 37841 rrnequiv 37842 |
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