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Mirrors > Home > MPE Home > Th. List > lcomf | Structured version Visualization version GIF version |
Description: A linear-combination sum is a function. (Contributed by Stefan O'Rear, 28-Feb-2015.) |
Ref | Expression |
---|---|
lcomf.f | β’ πΉ = (Scalarβπ) |
lcomf.k | β’ πΎ = (BaseβπΉ) |
lcomf.s | β’ Β· = ( Β·π βπ) |
lcomf.b | β’ π΅ = (Baseβπ) |
lcomf.w | β’ (π β π β LMod) |
lcomf.g | β’ (π β πΊ:πΌβΆπΎ) |
lcomf.h | β’ (π β π»:πΌβΆπ΅) |
lcomf.i | β’ (π β πΌ β π) |
Ref | Expression |
---|---|
lcomf | β’ (π β (πΊ βf Β· π»):πΌβΆπ΅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lcomf.w | . . 3 β’ (π β π β LMod) | |
2 | lcomf.b | . . . . 5 β’ π΅ = (Baseβπ) | |
3 | lcomf.f | . . . . 5 β’ πΉ = (Scalarβπ) | |
4 | lcomf.s | . . . . 5 β’ Β· = ( Β·π βπ) | |
5 | lcomf.k | . . . . 5 β’ πΎ = (BaseβπΉ) | |
6 | 2, 3, 4, 5 | lmodvscl 20761 | . . . 4 β’ ((π β LMod β§ π₯ β πΎ β§ π¦ β π΅) β (π₯ Β· π¦) β π΅) |
7 | 6 | 3expb 1118 | . . 3 β’ ((π β LMod β§ (π₯ β πΎ β§ π¦ β π΅)) β (π₯ Β· π¦) β π΅) |
8 | 1, 7 | sylan 579 | . 2 β’ ((π β§ (π₯ β πΎ β§ π¦ β π΅)) β (π₯ Β· π¦) β π΅) |
9 | lcomf.g | . 2 β’ (π β πΊ:πΌβΆπΎ) | |
10 | lcomf.h | . 2 β’ (π β π»:πΌβΆπ΅) | |
11 | lcomf.i | . 2 β’ (π β πΌ β π) | |
12 | inidm 4219 | . 2 β’ (πΌ β© πΌ) = πΌ | |
13 | 8, 9, 10, 11, 11, 12 | off 7703 | 1 β’ (π β (πΊ βf Β· π»):πΌβΆπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 βΆwf 6544 βcfv 6548 (class class class)co 7420 βf cof 7683 Basecbs 17180 Scalarcsca 17236 Β·π cvsca 17237 LModclmod 20743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-lmod 20745 |
This theorem is referenced by: lcomfsupp 20785 frlmup2 21733 islindf4 21772 fedgmullem2 33328 |
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