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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > lflsc0N | Structured version Visualization version GIF version |
Description: The scalar product with the zero functional is the zero functional. (Contributed by NM, 7-Oct-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
lflsc0.v | β’ π = (Baseβπ) |
lflsc0.d | β’ π· = (Scalarβπ) |
lflsc0.k | β’ πΎ = (Baseβπ·) |
lflsc0.t | β’ Β· = (.rβπ·) |
lflsc0.o | β’ 0 = (0gβπ·) |
lflsc0.w | β’ (π β π β LMod) |
lflsc0.x | β’ (π β π β πΎ) |
Ref | Expression |
---|---|
lflsc0N | β’ (π β ((π Γ { 0 }) βf Β· (π Γ {π})) = (π Γ { 0 })) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lflsc0.v | . . . . 5 β’ π = (Baseβπ) | |
2 | 1 | fvexi 6860 | . . . 4 β’ π β V |
3 | 2 | a1i 11 | . . 3 β’ (π β π β V) |
4 | lflsc0.w | . . . . 5 β’ (π β π β LMod) | |
5 | lflsc0.d | . . . . . 6 β’ π· = (Scalarβπ) | |
6 | 5 | lmodring 20373 | . . . . 5 β’ (π β LMod β π· β Ring) |
7 | 4, 6 | syl 17 | . . . 4 β’ (π β π· β Ring) |
8 | lflsc0.k | . . . . 5 β’ πΎ = (Baseβπ·) | |
9 | lflsc0.o | . . . . 5 β’ 0 = (0gβπ·) | |
10 | 8, 9 | ring0cl 19998 | . . . 4 β’ (π· β Ring β 0 β πΎ) |
11 | 7, 10 | syl 17 | . . 3 β’ (π β 0 β πΎ) |
12 | lflsc0.x | . . 3 β’ (π β π β πΎ) | |
13 | 3, 11, 12 | ofc12 7649 | . 2 β’ (π β ((π Γ { 0 }) βf Β· (π Γ {π})) = (π Γ {( 0 Β· π)})) |
14 | lflsc0.t | . . . . . 6 β’ Β· = (.rβπ·) | |
15 | 8, 14, 9 | ringlz 20019 | . . . . 5 β’ ((π· β Ring β§ π β πΎ) β ( 0 Β· π) = 0 ) |
16 | 7, 12, 15 | syl2anc 585 | . . . 4 β’ (π β ( 0 Β· π) = 0 ) |
17 | 16 | sneqd 4602 | . . 3 β’ (π β {( 0 Β· π)} = { 0 }) |
18 | 17 | xpeq2d 5667 | . 2 β’ (π β (π Γ {( 0 Β· π)}) = (π Γ { 0 })) |
19 | 13, 18 | eqtrd 2773 | 1 β’ (π β ((π Γ { 0 }) βf Β· (π Γ {π})) = (π Γ { 0 })) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 Vcvv 3447 {csn 4590 Γ cxp 5635 βcfv 6500 (class class class)co 7361 βf cof 7619 Basecbs 17091 .rcmulr 17142 Scalarcsca 17144 0gc0g 17329 Ringcrg 19972 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-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
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-nel 3047 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-pss 3933 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-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-pred 6257 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-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-plusg 17154 df-0g 17331 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-grp 18759 df-minusg 18760 df-mgp 19905 df-ring 19974 df-lmod 20367 |
This theorem is referenced by: (None) |
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