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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lfladd0l | Structured version Visualization version GIF version |
Description: Functional addition with the zero functional. (Contributed by NM, 21-Oct-2014.) |
Ref | Expression |
---|---|
lfladd0l.v | β’ π = (Baseβπ) |
lfladd0l.r | β’ π = (Scalarβπ) |
lfladd0l.p | β’ + = (+gβπ ) |
lfladd0l.o | β’ 0 = (0gβπ ) |
lfladd0l.f | β’ πΉ = (LFnlβπ) |
lfladd0l.w | β’ (π β π β LMod) |
lfladd0l.g | β’ (π β πΊ β πΉ) |
Ref | Expression |
---|---|
lfladd0l | β’ (π β ((π Γ { 0 }) βf + πΊ) = πΊ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lfladd0l.v | . . . 4 β’ π = (Baseβπ) | |
2 | 1 | fvexi 6916 | . . 3 β’ π β V |
3 | 2 | a1i 11 | . 2 β’ (π β π β V) |
4 | lfladd0l.w | . . 3 β’ (π β π β LMod) | |
5 | lfladd0l.g | . . 3 β’ (π β πΊ β πΉ) | |
6 | lfladd0l.r | . . . 4 β’ π = (Scalarβπ) | |
7 | eqid 2728 | . . . 4 β’ (Baseβπ ) = (Baseβπ ) | |
8 | lfladd0l.f | . . . 4 β’ πΉ = (LFnlβπ) | |
9 | 6, 7, 1, 8 | lflf 38575 | . . 3 β’ ((π β LMod β§ πΊ β πΉ) β πΊ:πβΆ(Baseβπ )) |
10 | 4, 5, 9 | syl2anc 582 | . 2 β’ (π β πΊ:πβΆ(Baseβπ )) |
11 | lfladd0l.o | . . . 4 β’ 0 = (0gβπ ) | |
12 | 11 | fvexi 6916 | . . 3 β’ 0 β V |
13 | 12 | a1i 11 | . 2 β’ (π β 0 β V) |
14 | 6 | lmodring 20765 | . . . 4 β’ (π β LMod β π β Ring) |
15 | ringgrp 20192 | . . . 4 β’ (π β Ring β π β Grp) | |
16 | 4, 14, 15 | 3syl 18 | . . 3 β’ (π β π β Grp) |
17 | lfladd0l.p | . . . 4 β’ + = (+gβπ ) | |
18 | 7, 17, 11 | grplid 18938 | . . 3 β’ ((π β Grp β§ π β (Baseβπ )) β ( 0 + π) = π) |
19 | 16, 18 | sylan 578 | . 2 β’ ((π β§ π β (Baseβπ )) β ( 0 + π) = π) |
20 | 3, 10, 13, 19 | caofid0l 7724 | 1 β’ (π β ((π Γ { 0 }) βf + πΊ) = πΊ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1533 β wcel 2098 Vcvv 3473 {csn 4632 Γ cxp 5680 βΆwf 6549 βcfv 6553 (class class class)co 7426 βf cof 7690 Basecbs 17189 +gcplusg 17242 Scalarcsca 17245 0gc0g 17430 Grpcgrp 18904 Ringcrg 20187 LModclmod 20757 LFnlclfn 38569 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7692 df-map 8855 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-ring 20189 df-lmod 20759 df-lfl 38570 |
This theorem is referenced by: ldualgrplem 38657 ldual0v 38662 |
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