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Mirrors > Home > MPE Home > Th. List > Mathboxes > lindslinindimp2lem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for lindslinindsimp2 46634. (Contributed by AV, 25-Apr-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 |
---|---|
lindslinindimp2lem2 | β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π β§ π β (π΅ βm π))) β πΊ β (π΅ βm (π β {π₯}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elmapi 8793 | . . . . . 6 β’ (π β (π΅ βm π) β π:πβΆπ΅) | |
2 | 1 | 3ad2ant3 1136 | . . . . 5 β’ ((π β (Baseβπ) β§ π₯ β π β§ π β (π΅ βm π)) β π:πβΆπ΅) |
3 | 2 | adantl 483 | . . . 4 β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π β§ π β (π΅ βm π))) β π:πβΆπ΅) |
4 | difss 4095 | . . . 4 β’ (π β {π₯}) β π | |
5 | fssres 6712 | . . . 4 β’ ((π:πβΆπ΅ β§ (π β {π₯}) β π) β (π βΎ (π β {π₯})):(π β {π₯})βΆπ΅) | |
6 | 3, 4, 5 | sylancl 587 | . . 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 βΎ (π β {π₯})):(π β {π₯})βΆπ΅) |
7 | lindslinind.g | . . . 4 β’ πΊ = (π βΎ (π β {π₯})) | |
8 | 7 | feq1i 6663 | . . 3 β’ (πΊ:(π β {π₯})βΆπ΅ β (π βΎ (π β {π₯})):(π β {π₯})βΆπ΅) |
9 | 6, 8 | sylibr 233 | . 2 β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π β§ π β (π΅ βm π))) β πΊ:(π β {π₯})βΆπ΅) |
10 | lindslinind.b | . . . 4 β’ π΅ = (Baseβπ ) | |
11 | 10 | fvexi 6860 | . . 3 β’ π΅ β V |
12 | difexg 5288 | . . . 4 β’ (π β π β (π β {π₯}) β V) | |
13 | 12 | ad2antrr 725 | . . 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 β {π₯}) β V) |
14 | elmapg 8784 | . . 3 β’ ((π΅ β V β§ (π β {π₯}) β V) β (πΊ β (π΅ βm (π β {π₯})) β πΊ:(π β {π₯})βΆπ΅)) | |
15 | 11, 13, 14 | sylancr 588 | . 2 β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π β§ π β (π΅ βm π))) β (πΊ β (π΅ βm (π β {π₯})) β πΊ:(π β {π₯})βΆπ΅)) |
16 | 9, 15 | mpbird 257 | 1 β’ (((π β π β§ π β LMod) β§ (π β (Baseβπ) β§ π₯ β π β§ π β (π΅ βm π))) β πΊ β (π΅ βm (π β {π₯}))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β§ w3a 1088 = wceq 1542 β wcel 2107 Vcvv 3447 β cdif 3911 β wss 3914 {csn 4590 βΎ cres 5639 βΆwf 6496 βcfv 6500 (class class class)co 7361 βm cmap 8771 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-pow 5324 ax-pr 5388 ax-un 7676 |
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-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 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-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-1st 7925 df-2nd 7926 df-map 8773 |
This theorem is referenced by: lindslinindimp2lem4 46632 lindslinindsimp2lem5 46633 |
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