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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 3464 | . . 3 β’ (π β π β π β V) | |
3 | elex 3464 | . . 3 β’ (πΌ β π β πΌ β V) | |
4 | simpl 484 | . . . . . . 7 β’ ((π = π β§ π = πΌ) β π = π ) | |
5 | 4 | fveq2d 6847 | . . . . . 6 β’ ((π = π β§ π = πΌ) β (Scalarβπ) = (Scalarβπ )) |
6 | pwsval.f | . . . . . 6 β’ πΉ = (Scalarβπ ) | |
7 | 5, 6 | eqtr4di 2795 | . . . . 5 β’ ((π = π β§ π = πΌ) β (Scalarβπ) = πΉ) |
8 | id 22 | . . . . . 6 β’ (π = πΌ β π = πΌ) | |
9 | sneq 4597 | . . . . . 6 β’ (π = π β {π} = {π }) | |
10 | xpeq12 5659 | . . . . . 6 β’ ((π = πΌ β§ {π} = {π }) β (π Γ {π}) = (πΌ Γ {π })) | |
11 | 8, 9, 10 | syl2anr 598 | . . . . 5 β’ ((π = π β§ π = πΌ) β (π Γ {π}) = (πΌ Γ {π })) |
12 | 7, 11 | oveq12d 7376 | . . . 4 β’ ((π = π β§ π = πΌ) β ((Scalarβπ)Xs(π Γ {π})) = (πΉXs(πΌ Γ {π }))) |
13 | df-pws 17332 | . . . 4 β’ βs = (π β V, π β V β¦ ((Scalarβπ)Xs(π Γ {π}))) | |
14 | ovex 7391 | . . . 4 β’ (πΉXs(πΌ Γ {π })) β V | |
15 | 12, 13, 14 | ovmpoa 7511 | . . 3 β’ ((π β V β§ πΌ β V) β (π βs πΌ) = (πΉXs(πΌ Γ {π }))) |
16 | 2, 3, 15 | syl2an 597 | . 2 β’ ((π β π β§ πΌ β π) β (π βs πΌ) = (πΉXs(πΌ Γ {π }))) |
17 | 1, 16 | eqtrid 2789 | 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 397 = wceq 1542 β wcel 2107 Vcvv 3446 {csn 4587 Γ cxp 5632 βcfv 6497 (class class class)co 7358 Scalarcsca 17137 Xscprds 17328 βs cpws 17329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-sbc 3741 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-iota 6449 df-fun 6499 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-pws 17332 |
This theorem is referenced by: pwsbas 17370 pwsplusgval 17373 pwsmulrval 17374 pwsle 17375 pwsvscafval 17377 pwssca 17379 pwsmnd 18592 pws0g 18593 pwspjmhm 18641 pwsgrp 18860 pwsinvg 18861 pwscmn 19642 pwsabl 19643 pwsgsum 19760 pwsring 20040 pws1 20041 pwscrng 20042 pwsmgp 20043 pwslmod 20434 frlmpws 21159 frlmlss 21160 frlmpwsfi 21161 frlmbas 21164 frlmip 21187 pwstps 22984 resspwsds 23728 pwsxms 23891 pwsms 23892 rrxprds 24756 cnpwstotbnd 36259 repwsmet 36296 rrnequiv 36297 |
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