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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindspropd | Structured version Visualization version GIF version |
Description: Property deduction for linearly independent sets. (Contributed by Thierry Arnoux, 16-Jul-2023.) |
Ref | Expression |
---|---|
lindfpropd.1 | β’ (π β (BaseβπΎ) = (BaseβπΏ)) |
lindfpropd.2 | β’ (π β (Baseβ(ScalarβπΎ)) = (Baseβ(ScalarβπΏ))) |
lindfpropd.3 | β’ (π β (0gβ(ScalarβπΎ)) = (0gβ(ScalarβπΏ))) |
lindfpropd.4 | β’ ((π β§ (π₯ β (BaseβπΎ) β§ π¦ β (BaseβπΎ))) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) |
lindfpropd.5 | β’ ((π β§ (π₯ β (Baseβ(ScalarβπΎ)) β§ π¦ β (BaseβπΎ))) β (π₯( Β·π βπΎ)π¦) β (BaseβπΎ)) |
lindfpropd.6 | β’ ((π β§ (π₯ β (Baseβ(ScalarβπΎ)) β§ π¦ β (BaseβπΎ))) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) |
lindfpropd.k | β’ (π β πΎ β π) |
lindfpropd.l | β’ (π β πΏ β π) |
Ref | Expression |
---|---|
lindspropd | β’ (π β (LIndSβπΎ) = (LIndSβπΏ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindfpropd.1 | . . . . 5 β’ (π β (BaseβπΎ) = (BaseβπΏ)) | |
2 | 1 | sseq2d 3977 | . . . 4 β’ (π β (π§ β (BaseβπΎ) β π§ β (BaseβπΏ))) |
3 | lindfpropd.2 | . . . . 5 β’ (π β (Baseβ(ScalarβπΎ)) = (Baseβ(ScalarβπΏ))) | |
4 | lindfpropd.3 | . . . . 5 β’ (π β (0gβ(ScalarβπΎ)) = (0gβ(ScalarβπΏ))) | |
5 | lindfpropd.4 | . . . . 5 β’ ((π β§ (π₯ β (BaseβπΎ) β§ π¦ β (BaseβπΎ))) β (π₯(+gβπΎ)π¦) = (π₯(+gβπΏ)π¦)) | |
6 | lindfpropd.5 | . . . . 5 β’ ((π β§ (π₯ β (Baseβ(ScalarβπΎ)) β§ π¦ β (BaseβπΎ))) β (π₯( Β·π βπΎ)π¦) β (BaseβπΎ)) | |
7 | lindfpropd.6 | . . . . 5 β’ ((π β§ (π₯ β (Baseβ(ScalarβπΎ)) β§ π¦ β (BaseβπΎ))) β (π₯( Β·π βπΎ)π¦) = (π₯( Β·π βπΏ)π¦)) | |
8 | lindfpropd.k | . . . . 5 β’ (π β πΎ β π) | |
9 | lindfpropd.l | . . . . 5 β’ (π β πΏ β π) | |
10 | vex 3448 | . . . . . . 7 β’ π§ β V | |
11 | 10 | a1i 11 | . . . . . 6 β’ (π β π§ β V) |
12 | 11 | resiexd 7167 | . . . . 5 β’ (π β ( I βΎ π§) β V) |
13 | 1, 3, 4, 5, 6, 7, 8, 9, 12 | lindfpropd 32217 | . . . 4 β’ (π β (( I βΎ π§) LIndF πΎ β ( I βΎ π§) LIndF πΏ)) |
14 | 2, 13 | anbi12d 632 | . . 3 β’ (π β ((π§ β (BaseβπΎ) β§ ( I βΎ π§) LIndF πΎ) β (π§ β (BaseβπΏ) β§ ( I βΎ π§) LIndF πΏ))) |
15 | eqid 2733 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
16 | 15 | islinds 21231 | . . . 4 β’ (πΎ β π β (π§ β (LIndSβπΎ) β (π§ β (BaseβπΎ) β§ ( I βΎ π§) LIndF πΎ))) |
17 | 8, 16 | syl 17 | . . 3 β’ (π β (π§ β (LIndSβπΎ) β (π§ β (BaseβπΎ) β§ ( I βΎ π§) LIndF πΎ))) |
18 | eqid 2733 | . . . . 5 β’ (BaseβπΏ) = (BaseβπΏ) | |
19 | 18 | islinds 21231 | . . . 4 β’ (πΏ β π β (π§ β (LIndSβπΏ) β (π§ β (BaseβπΏ) β§ ( I βΎ π§) LIndF πΏ))) |
20 | 9, 19 | syl 17 | . . 3 β’ (π β (π§ β (LIndSβπΏ) β (π§ β (BaseβπΏ) β§ ( I βΎ π§) LIndF πΏ))) |
21 | 14, 17, 20 | 3bitr4d 311 | . 2 β’ (π β (π§ β (LIndSβπΎ) β π§ β (LIndSβπΏ))) |
22 | 21 | eqrdv 2731 | 1 β’ (π β (LIndSβπΎ) = (LIndSβπΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 Vcvv 3444 β wss 3911 class class class wbr 5106 I cid 5531 βΎ cres 5636 βcfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 Scalarcsca 17141 Β·π cvsca 17142 0gc0g 17326 LIndF clindf 21226 LIndSclinds 21227 |
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-ov 7361 df-lss 20408 df-lsp 20448 df-lindf 21228 df-linds 21229 |
This theorem is referenced by: fedgmullem2 32382 |
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