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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 31066 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 3487 | . . 3 β’ (π β π β π β V) | |
3 | fveq2 6884 | . . . . . . 7 β’ (π€ = π β (Baseβπ€) = (Baseβπ)) | |
4 | lspval.v | . . . . . . 7 β’ π = (Baseβπ) | |
5 | 3, 4 | eqtr4di 2784 | . . . . . 6 β’ (π€ = π β (Baseβπ€) = π) |
6 | 5 | pweqd 4614 | . . . . 5 β’ (π€ = π β π« (Baseβπ€) = π« π) |
7 | fveq2 6884 | . . . . . . . 8 β’ (π€ = π β (LSubSpβπ€) = (LSubSpβπ)) | |
8 | lspval.s | . . . . . . . 8 β’ π = (LSubSpβπ) | |
9 | 7, 8 | eqtr4di 2784 | . . . . . . 7 β’ (π€ = π β (LSubSpβπ€) = π) |
10 | 9 | rabeqdv 3441 | . . . . . 6 β’ (π€ = π β {π‘ β (LSubSpβπ€) β£ π β π‘} = {π‘ β π β£ π β π‘}) |
11 | 10 | inteqd 4948 | . . . . 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 5232 | . . . 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 20816 | . . . 4 β’ LSpan = (π€ β V β¦ (π β π« (Baseβπ€) β¦ β© {π‘ β (LSubSpβπ€) β£ π β π‘})) | |
14 | 4 | fvexi 6898 | . . . . . 6 β’ π β V |
15 | 14 | pwex 5371 | . . . . 5 β’ π« π β V |
16 | 15 | mptex 7219 | . . . 4 β’ (π β π« π β¦ β© {π‘ β π β£ π β π‘}) β V |
17 | 12, 13, 16 | fvmpt 6991 | . . 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 2778 | 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 1533 β wcel 2098 {crab 3426 Vcvv 3468 β wss 3943 π« cpw 4597 β© cint 4943 β¦ cmpt 5224 βcfv 6536 Basecbs 17150 LSubSpclss 20775 LSpanclspn 20815 |
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-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 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-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-lsp 20816 |
This theorem is referenced by: lspf 20818 lspval 20819 00lsp 20825 mrclsp 20833 lsppropd 20863 |
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