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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 31139 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 3492 | . . 3 β’ (π β π β π β V) | |
3 | fveq2 6902 | . . . . . . 7 β’ (π€ = π β (Baseβπ€) = (Baseβπ)) | |
4 | lspval.v | . . . . . . 7 β’ π = (Baseβπ) | |
5 | 3, 4 | eqtr4di 2786 | . . . . . 6 β’ (π€ = π β (Baseβπ€) = π) |
6 | 5 | pweqd 4623 | . . . . 5 β’ (π€ = π β π« (Baseβπ€) = π« π) |
7 | fveq2 6902 | . . . . . . . 8 β’ (π€ = π β (LSubSpβπ€) = (LSubSpβπ)) | |
8 | lspval.s | . . . . . . . 8 β’ π = (LSubSpβπ) | |
9 | 7, 8 | eqtr4di 2786 | . . . . . . 7 β’ (π€ = π β (LSubSpβπ€) = π) |
10 | 9 | rabeqdv 3446 | . . . . . 6 β’ (π€ = π β {π‘ β (LSubSpβπ€) β£ π β π‘} = {π‘ β π β£ π β π‘}) |
11 | 10 | inteqd 4958 | . . . . 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 5243 | . . . 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 20863 | . . . 4 β’ LSpan = (π€ β V β¦ (π β π« (Baseβπ€) β¦ β© {π‘ β (LSubSpβπ€) β£ π β π‘})) | |
14 | 4 | fvexi 6916 | . . . . . 6 β’ π β V |
15 | 14 | pwex 5384 | . . . . 5 β’ π« π β V |
16 | 15 | mptex 7241 | . . . 4 β’ (π β π« π β¦ β© {π‘ β π β£ π β π‘}) β V |
17 | 12, 13, 16 | fvmpt 7010 | . . 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 2780 | 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 3430 Vcvv 3473 β wss 3949 π« cpw 4606 β© cint 4953 β¦ cmpt 5235 βcfv 6553 Basecbs 17187 LSubSpclss 20822 LSpanclspn 20862 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 |
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 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-lsp 20863 |
This theorem is referenced by: lspf 20865 lspval 20866 00lsp 20872 mrclsp 20880 lsppropd 20910 |
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