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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflnegl | Structured version Visualization version GIF version |
Description: A functional plus its negative is the zero functional. (This is specialized for the purpose of proving ldualgrp 37658, and we do not define a general operation here.) (Contributed by NM, 22-Oct-2014.) |
Ref | Expression |
---|---|
lflnegcl.v | β’ π = (Baseβπ) |
lflnegcl.r | β’ π = (Scalarβπ) |
lflnegcl.i | β’ πΌ = (invgβπ ) |
lflnegcl.n | β’ π = (π₯ β π β¦ (πΌβ(πΊβπ₯))) |
lflnegcl.f | β’ πΉ = (LFnlβπ) |
lflnegcl.w | β’ (π β π β LMod) |
lflnegcl.g | β’ (π β πΊ β πΉ) |
lflnegl.p | β’ + = (+gβπ ) |
lflnegl.o | β’ 0 = (0gβπ ) |
Ref | Expression |
---|---|
lflnegl | β’ (π β (π βf + πΊ) = (π Γ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflnegcl.v | . . . 4 β’ π = (Baseβπ) | |
2 | 1 | fvexi 6860 | . . 3 β’ π β V |
3 | 2 | a1i 11 | . 2 β’ (π β π β V) |
4 | lflnegcl.w | . . 3 β’ (π β π β LMod) | |
5 | lflnegcl.g | . . 3 β’ (π β πΊ β πΉ) | |
6 | lflnegcl.r | . . . 4 β’ π = (Scalarβπ) | |
7 | eqid 2733 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
8 | lflnegcl.f | . . . 4 β’ πΉ = (LFnlβπ) | |
9 | 6, 7, 1, 8 | lflf 37575 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆ(Baseβπ )) |
10 | 4, 5, 9 | syl2anc 585 | . 2 β’ (π β πΊ:πβΆ(Baseβπ )) |
11 | lflnegl.o | . . . 4 β’ 0 = (0gβπ ) | |
12 | 11 | fvexi 6860 | . . 3 β’ 0 β V |
13 | 12 | a1i 11 | . 2 β’ (π β 0 β V) |
14 | lflnegcl.i | . . . 4 β’ πΌ = (invgβπ ) | |
15 | 6 | lmodring 20373 | . . . . 5 β’ (π β LMod β π β Ring) |
16 | ringgrp 19977 | . . . . 5 β’ (π β Ring β π β Grp) | |
17 | 4, 15, 16 | 3syl 18 | . . . 4 β’ (π β π β Grp) |
18 | 7, 14, 17 | grpinvf1o 18825 | . . 3 β’ (π β πΌ:(Baseβπ )β1-1-ontoβ(Baseβπ )) |
19 | f1of 6788 | . . 3 β’ (πΌ:(Baseβπ )β1-1-ontoβ(Baseβπ ) β πΌ:(Baseβπ )βΆ(Baseβπ )) | |
20 | 18, 19 | syl 17 | . 2 β’ (π β πΌ:(Baseβπ )βΆ(Baseβπ )) |
21 | lflnegcl.n | . . 3 β’ π = (π₯ β π β¦ (πΌβ(πΊβπ₯))) | |
22 | 21 | a1i 11 | . 2 β’ (π β π = (π₯ β π β¦ (πΌβ(πΊβπ₯)))) |
23 | lflnegl.p | . . . 4 β’ + = (+gβπ ) | |
24 | 7, 23, 11, 14 | grplinv 18808 | . . 3 β’ ((π β Grp β§ π¦ β (Baseβπ )) β ((πΌβπ¦) + π¦) = 0 ) |
25 | 17, 24 | sylan 581 | . 2 β’ ((π β§ π¦ β (Baseβπ )) β ((πΌβπ¦) + π¦) = 0 ) |
26 | 3, 10, 13, 20, 22, 25 | caofinvl 7651 | 1 β’ (π β (π βf + πΊ) = (π Γ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3447 {csn 4590 β¦ cmpt 5192 Γ cxp 5635 βΆwf 6496 β1-1-ontoβwf1o 6499 βcfv 6500 (class class class)co 7361 βf cof 7619 Basecbs 17091 +gcplusg 17141 Scalarcsca 17144 0gc0g 17329 Grpcgrp 18756 invgcminusg 18757 Ringcrg 19972 LModclmod 20365 LFnlclfn 37569 |
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-rep 5246 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-rmo 3352 df-reu 3353 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-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-map 8773 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-ring 19974 df-lmod 20367 df-lfl 37570 |
This theorem is referenced by: ldualgrplem 37657 |
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