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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 38510, 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 6896 | . . 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 2724 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
8 | lflnegcl.f | . . . 4 β’ πΉ = (LFnlβπ) | |
9 | 6, 7, 1, 8 | lflf 38427 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆ(Baseβπ )) |
10 | 4, 5, 9 | syl2anc 583 | . 2 β’ (π β πΊ:πβΆ(Baseβπ )) |
11 | lflnegl.o | . . . 4 β’ 0 = (0gβπ ) | |
12 | 11 | fvexi 6896 | . . 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 20706 | . . . . 5 β’ (π β LMod β π β Ring) |
16 | ringgrp 20135 | . . . . 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 18930 | . . 3 β’ (π β πΌ:(Baseβπ )β1-1-ontoβ(Baseβπ )) |
19 | f1of 6824 | . . 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 18911 | . . 3 β’ ((π β Grp β§ π¦ β (Baseβπ )) β ((πΌβπ¦) + π¦) = 0 ) |
25 | 17, 24 | sylan 579 | . 2 β’ ((π β§ π¦ β (Baseβπ )) β ((πΌβπ¦) + π¦) = 0 ) |
26 | 3, 10, 13, 20, 22, 25 | caofinvl 7694 | 1 β’ (π β (π βf + πΊ) = (π Γ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3466 {csn 4621 β¦ cmpt 5222 Γ cxp 5665 βΆwf 6530 β1-1-ontoβwf1o 6533 βcfv 6534 (class class class)co 7402 βf cof 7662 Basecbs 17145 +gcplusg 17198 Scalarcsca 17201 0gc0g 17386 Grpcgrp 18855 invgcminusg 18856 Ringcrg 20130 LModclmod 20698 LFnlclfn 38421 |
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-pow 5354 ax-pr 5418 ax-un 7719 |
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-rmo 3368 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-pw 4597 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-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-map 8819 df-0g 17388 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 df-minusg 18859 df-ring 20132 df-lmod 20700 df-lfl 38422 |
This theorem is referenced by: ldualgrplem 38509 |
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