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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 20720 | . . . 4 β’ ((π β LMod β§ π₯ β πΎ β§ π¦ β π΅) β (π₯ Β· π¦) β π΅) |
7 | 6 | 3expb 1117 | . . 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 4211 | . 2 β’ (πΌ β© πΌ) = πΌ | |
13 | 8, 9, 10, 11, 11, 12 | off 7682 | 1 β’ (π β (πΊ βf Β· π»):πΌβΆπ΅) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1533 β wcel 2098 βΆwf 6530 βcfv 6534 (class class class)co 7402 βf cof 7662 Basecbs 17149 Scalarcsca 17205 Β·π cvsca 17206 LModclmod 20702 |
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 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-lmod 20704 |
This theorem is referenced by: lcomfsupp 20744 frlmup2 21683 islindf4 21722 fedgmullem2 33223 |
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