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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 | β’ π = (π βs πΌ) |
pwsval.f | β’ πΉ = (Scalarβπ ) |
Ref | Expression |
---|---|
pwsval | β’ ((π β π β§ πΌ β π) β π = (πΉXs(πΌ Γ {π }))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwsval.y | . 2 β’ π = (π βs πΌ) | |
2 | elex 3487 | . . 3 β’ (π β π β π β V) | |
3 | elex 3487 | . . 3 β’ (πΌ β π β πΌ β V) | |
4 | simpl 482 | . . . . . . 7 β’ ((π = π β§ π = πΌ) β π = π ) | |
5 | 4 | fveq2d 6888 | . . . . . 6 β’ ((π = π β§ π = πΌ) β (Scalarβπ) = (Scalarβπ )) |
6 | pwsval.f | . . . . . 6 β’ πΉ = (Scalarβπ ) | |
7 | 5, 6 | eqtr4di 2784 | . . . . 5 β’ ((π = π β§ π = πΌ) β (Scalarβπ) = πΉ) |
8 | id 22 | . . . . . 6 β’ (π = πΌ β π = πΌ) | |
9 | sneq 4633 | . . . . . 6 β’ (π = π β {π} = {π }) | |
10 | xpeq12 5694 | . . . . . 6 β’ ((π = πΌ β§ {π} = {π }) β (π Γ {π}) = (πΌ Γ {π })) | |
11 | 8, 9, 10 | syl2anr 596 | . . . . 5 β’ ((π = π β§ π = πΌ) β (π Γ {π}) = (πΌ Γ {π })) |
12 | 7, 11 | oveq12d 7422 | . . . 4 β’ ((π = π β§ π = πΌ) β ((Scalarβπ)Xs(π Γ {π})) = (πΉXs(πΌ Γ {π }))) |
13 | df-pws 17402 | . . . 4 β’ βs = (π β V, π β V β¦ ((Scalarβπ)Xs(π Γ {π}))) | |
14 | ovex 7437 | . . . 4 β’ (πΉXs(πΌ Γ {π })) β V | |
15 | 12, 13, 14 | ovmpoa 7558 | . . 3 β’ ((π β V β§ πΌ β V) β (π βs πΌ) = (πΉXs(πΌ Γ {π }))) |
16 | 2, 3, 15 | syl2an 595 | . 2 β’ ((π β π β§ πΌ β π) β (π βs πΌ) = (πΉXs(πΌ Γ {π }))) |
17 | 1, 16 | eqtrid 2778 | 1 β’ ((π β π β§ πΌ β π) β π = (πΉXs(πΌ Γ {π }))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 {csn 4623 Γ cxp 5667 βcfv 6536 (class class class)co 7404 Scalarcsca 17207 Xscprds 17398 βs cpws 17399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6488 df-fun 6538 df-fv 6544 df-ov 7407 df-oprab 7408 df-mpo 7409 df-pws 17402 |
This theorem is referenced by: pwsbas 17440 pwsplusgval 17443 pwsmulrval 17444 pwsle 17445 pwsvscafval 17447 pwssca 17449 pwsmnd 18700 pws0g 18701 pwspjmhm 18753 pwsgrp 18978 pwsinvg 18979 pwscmn 19781 pwsabl 19782 pwsgsum 19900 pwsring 20221 pws1 20222 pwscrng 20223 pwsmgp 20224 pwslmod 20815 frlmpws 21641 frlmlss 21642 frlmpwsfi 21643 frlmbas 21646 frlmip 21669 pwstps 23485 resspwsds 24229 pwsxms 24392 pwsms 24393 rrxprds 25268 cnpwstotbnd 37176 repwsmet 37213 rrnequiv 37214 |
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