![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3490 | . . 3 β’ (π β π β π β V) | |
3 | elex 3490 | . . 3 β’ (πΌ β π β πΌ β V) | |
4 | simpl 481 | . . . . . . 7 β’ ((π = π β§ π = πΌ) β π = π ) | |
5 | 4 | fveq2d 6904 | . . . . . 6 β’ ((π = π β§ π = πΌ) β (Scalarβπ) = (Scalarβπ )) |
6 | pwsval.f | . . . . . 6 β’ πΉ = (Scalarβπ ) | |
7 | 5, 6 | eqtr4di 2785 | . . . . 5 β’ ((π = π β§ π = πΌ) β (Scalarβπ) = πΉ) |
8 | id 22 | . . . . . 6 β’ (π = πΌ β π = πΌ) | |
9 | sneq 4640 | . . . . . 6 β’ (π = π β {π} = {π }) | |
10 | xpeq12 5705 | . . . . . 6 β’ ((π = πΌ β§ {π} = {π }) β (π Γ {π}) = (πΌ Γ {π })) | |
11 | 8, 9, 10 | syl2anr 595 | . . . . 5 β’ ((π = π β§ π = πΌ) β (π Γ {π}) = (πΌ Γ {π })) |
12 | 7, 11 | oveq12d 7442 | . . . 4 β’ ((π = π β§ π = πΌ) β ((Scalarβπ)Xs(π Γ {π})) = (πΉXs(πΌ Γ {π }))) |
13 | df-pws 17436 | . . . 4 β’ βs = (π β V, π β V β¦ ((Scalarβπ)Xs(π Γ {π}))) | |
14 | ovex 7457 | . . . 4 β’ (πΉXs(πΌ Γ {π })) β V | |
15 | 12, 13, 14 | ovmpoa 7580 | . . 3 β’ ((π β V β§ πΌ β V) β (π βs πΌ) = (πΉXs(πΌ Γ {π }))) |
16 | 2, 3, 15 | syl2an 594 | . 2 β’ ((π β π β§ πΌ β π) β (π βs πΌ) = (πΉXs(πΌ Γ {π }))) |
17 | 1, 16 | eqtrid 2779 | 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 394 = wceq 1533 β wcel 2098 Vcvv 3471 {csn 4630 Γ cxp 5678 βcfv 6551 (class class class)co 7424 Scalarcsca 17241 Xscprds 17432 βs cpws 17433 |
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 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-iota 6503 df-fun 6553 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-pws 17436 |
This theorem is referenced by: pwsbas 17474 pwsplusgval 17477 pwsmulrval 17478 pwsle 17479 pwsvscafval 17481 pwssca 17483 pwsmnd 18734 pws0g 18735 pwspjmhm 18787 pwsgrp 19013 pwsinvg 19014 pwscmn 19823 pwsabl 19824 pwsgsum 19942 pwsring 20265 pws1 20266 pwscrng 20267 pwsmgp 20268 pwslmod 20859 frlmpws 21689 frlmlss 21690 frlmpwsfi 21691 frlmbas 21694 frlmip 21717 pwstps 23552 resspwsds 24296 pwsxms 24459 pwsms 24460 rrxprds 25335 cnpwstotbnd 37275 repwsmet 37312 rrnequiv 37313 |
Copyright terms: Public domain | W3C validator |