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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindslinindimp2lem3 | Structured version Visualization version GIF version |
Description: Lemma 3 for lindslinindsimp2 46634. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.) |
Ref | Expression |
---|---|
lindslinind.r | β’ π = (Scalarβπ) |
lindslinind.b | β’ π΅ = (Baseβπ ) |
lindslinind.0 | β’ 0 = (0gβπ ) |
lindslinind.z | β’ π = (0gβπ) |
lindslinind.y | β’ π = ((invgβπ )β(πβπ₯)) |
lindslinind.g | β’ πΊ = (π βΎ (π β {π₯})) |
Ref | Expression |
---|---|
lindslinindimp2lem3 | β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π) β§ (π β (π΅ βm π) β§ π finSupp 0 )) β πΊ finSupp 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lindslinind.g | . 2 β’ πΊ = (π βΎ (π β {π₯})) | |
2 | simp3r 1203 | . . 3 β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π) β§ (π β (π΅ βm π) β§ π finSupp 0 )) β π finSupp 0 ) | |
3 | lindslinind.0 | . . . . 5 β’ 0 = (0gβπ ) | |
4 | 3 | fvexi 6860 | . . . 4 β’ 0 β V |
5 | 4 | a1i 11 | . . 3 β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π) β§ (π β (π΅ βm π) β§ π finSupp 0 )) β 0 β V) |
6 | 2, 5 | fsuppres 9338 | . 2 β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π) β§ (π β (π΅ βm π) β§ π finSupp 0 )) β (π βΎ (π β {π₯})) finSupp 0 ) |
7 | 1, 6 | eqbrtrid 5144 | 1 β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π) β§ (π β (π΅ βm π) β§ π finSupp 0 )) β πΊ finSupp 0 ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3447 β cdif 3911 β wss 3914 {csn 4590 class class class wbr 5109 βΎ cres 5639 βcfv 6500 (class class class)co 7361 βm cmap 8771 finSupp cfsupp 9311 Basecbs 17091 Scalarcsca 17144 0gc0g 17329 invgcminusg 18757 LModclmod 20365 |
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-sep 5260 ax-nul 5267 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 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 3449 df-sbc 3744 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-supp 8097 df-1o 8416 df-en 8890 df-fin 8893 df-fsupp 9312 |
This theorem is referenced by: lindslinindimp2lem4 46632 |
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