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Mirrors > Home > MPE Home > Th. List > lspfval | Structured version Visualization version GIF version |
Description: The span function for a left vector space (or a left module). (df-span 30293 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
Ref | Expression |
---|---|
lspval.v | β’ π = (Baseβπ) |
lspval.s | β’ π = (LSubSpβπ) |
lspval.n | β’ π = (LSpanβπ) |
Ref | Expression |
---|---|
lspfval | β’ (π β π β π = (π β π« π β¦ β© {π‘ β π β£ π β π‘})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lspval.n | . 2 β’ π = (LSpanβπ) | |
2 | elex 3462 | . . 3 β’ (π β π β π β V) | |
3 | fveq2 6843 | . . . . . . 7 β’ (π€ = π β (Baseβπ€) = (Baseβπ)) | |
4 | lspval.v | . . . . . . 7 β’ π = (Baseβπ) | |
5 | 3, 4 | eqtr4di 2791 | . . . . . 6 β’ (π€ = π β (Baseβπ€) = π) |
6 | 5 | pweqd 4578 | . . . . 5 β’ (π€ = π β π« (Baseβπ€) = π« π) |
7 | fveq2 6843 | . . . . . . . 8 β’ (π€ = π β (LSubSpβπ€) = (LSubSpβπ)) | |
8 | lspval.s | . . . . . . . 8 β’ π = (LSubSpβπ) | |
9 | 7, 8 | eqtr4di 2791 | . . . . . . 7 β’ (π€ = π β (LSubSpβπ€) = π) |
10 | 9 | rabeqdv 3421 | . . . . . 6 β’ (π€ = π β {π‘ β (LSubSpβπ€) β£ π β π‘} = {π‘ β π β£ π β π‘}) |
11 | 10 | inteqd 4913 | . . . . 5 β’ (π€ = π β β© {π‘ β (LSubSpβπ€) β£ π β π‘} = β© {π‘ β π β£ π β π‘}) |
12 | 6, 11 | mpteq12dv 5197 | . . . 4 β’ (π€ = π β (π β π« (Baseβπ€) β¦ β© {π‘ β (LSubSpβπ€) β£ π β π‘}) = (π β π« π β¦ β© {π‘ β π β£ π β π‘})) |
13 | df-lsp 20448 | . . . 4 β’ LSpan = (π€ β V β¦ (π β π« (Baseβπ€) β¦ β© {π‘ β (LSubSpβπ€) β£ π β π‘})) | |
14 | 4 | fvexi 6857 | . . . . . 6 β’ π β V |
15 | 14 | pwex 5336 | . . . . 5 β’ π« π β V |
16 | 15 | mptex 7174 | . . . 4 β’ (π β π« π β¦ β© {π‘ β π β£ π β π‘}) β V |
17 | 12, 13, 16 | fvmpt 6949 | . . 3 β’ (π β V β (LSpanβπ) = (π β π« π β¦ β© {π‘ β π β£ π β π‘})) |
18 | 2, 17 | syl 17 | . 2 β’ (π β π β (LSpanβπ) = (π β π« π β¦ β© {π‘ β π β£ π β π‘})) |
19 | 1, 18 | eqtrid 2785 | 1 β’ (π β π β π = (π β π« π β¦ β© {π‘ β π β£ π β π‘})) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 {crab 3406 Vcvv 3444 β wss 3911 π« cpw 4561 β© cint 4908 β¦ cmpt 5189 βcfv 6497 Basecbs 17088 LSubSpclss 20407 LSpanclspn 20447 |
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 2704 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-int 4909 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-lsp 20448 |
This theorem is referenced by: lspf 20450 lspval 20451 00lsp 20457 mrclsp 20465 lsppropd 20494 |
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