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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 4009 | . . . 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 3472 | . . . . . . 7 β’ π§ β V | |
11 | 10 | a1i 11 | . . . . . 6 β’ (π β π§ β V) |
12 | 11 | resiexd 7213 | . . . . 5 β’ (π β ( I βΎ π§) β V) |
13 | 1, 3, 4, 5, 6, 7, 8, 9, 12 | lindfpropd 33004 | . . . 4 β’ (π β (( I βΎ π§) LIndF πΎ β ( I βΎ π§) LIndF πΏ)) |
14 | 2, 13 | anbi12d 630 | . . 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 2726 | . . . . 5 β’ (BaseβπΎ) = (BaseβπΎ) | |
16 | 15 | islinds 21704 | . . . 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 2726 | . . . . 5 β’ (BaseβπΏ) = (BaseβπΏ) | |
19 | 18 | islinds 21704 | . . . 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 2724 | 1 β’ (π β (LIndSβπΎ) = (LIndSβπΏ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 = wceq 1533 β wcel 2098 Vcvv 3468 β wss 3943 class class class wbr 5141 I cid 5566 βΎ cres 5671 βcfv 6537 (class class class)co 7405 Basecbs 17153 +gcplusg 17206 Scalarcsca 17209 Β·π cvsca 17210 0gc0g 17394 LIndF clindf 21699 LIndSclinds 21700 |
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 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-lss 20779 df-lsp 20819 df-lindf 21701 df-linds 21702 |
This theorem is referenced by: fedgmullem2 33233 |
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