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Mirrors > Home > MPE Home > Th. List > Mathboxes > baerlem5a | Structured version Visualization version GIF version |
Description: An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. First equation of part (5) in [Baer] p. 46. (Contributed by NM, 10-Apr-2015.) |
Ref | Expression |
---|---|
baerlem3.v | ⊢ 𝑉 = (Base‘𝑊) |
baerlem3.m | ⊢ − = (-g‘𝑊) |
baerlem3.o | ⊢ 0 = (0g‘𝑊) |
baerlem3.s | ⊢ ⊕ = (LSSum‘𝑊) |
baerlem3.n | ⊢ 𝑁 = (LSpan‘𝑊) |
baerlem3.w | ⊢ (𝜑 → 𝑊 ∈ LVec) |
baerlem3.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
baerlem3.c | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
baerlem3.d | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
baerlem3.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
baerlem3.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
baerlem5a.p | ⊢ + = (+g‘𝑊) |
Ref | Expression |
---|---|
baerlem5a | ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)}) ⊕ (𝑁‘{𝑌})))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | baerlem3.v | . 2 ⊢ 𝑉 = (Base‘𝑊) | |
2 | baerlem3.m | . 2 ⊢ − = (-g‘𝑊) | |
3 | baerlem3.o | . 2 ⊢ 0 = (0g‘𝑊) | |
4 | baerlem3.s | . 2 ⊢ ⊕ = (LSSum‘𝑊) | |
5 | baerlem3.n | . 2 ⊢ 𝑁 = (LSpan‘𝑊) | |
6 | baerlem3.w | . 2 ⊢ (𝜑 → 𝑊 ∈ LVec) | |
7 | baerlem3.x | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
8 | baerlem3.c | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
9 | baerlem3.d | . 2 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) | |
10 | baerlem3.y | . 2 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
11 | baerlem3.z | . 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
12 | baerlem5a.p | . 2 ⊢ + = (+g‘𝑊) | |
13 | eqid 2738 | . 2 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
14 | eqid 2738 | . 2 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
15 | eqid 2738 | . 2 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
16 | eqid 2738 | . 2 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
17 | eqid 2738 | . 2 ⊢ (-g‘(Scalar‘𝑊)) = (-g‘(Scalar‘𝑊)) | |
18 | eqid 2738 | . 2 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
19 | eqid 2738 | . 2 ⊢ (invg‘(Scalar‘𝑊)) = (invg‘(Scalar‘𝑊)) | |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 | baerlem5alem2 39722 | 1 ⊢ (𝜑 → (𝑁‘{(𝑋 − (𝑌 + 𝑍))}) = (((𝑁‘{(𝑋 − 𝑌)}) ⊕ (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 − 𝑍)}) ⊕ (𝑁‘{𝑌})))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3885 ∩ cin 3887 {csn 4563 {cpr 4565 ‘cfv 6435 (class class class)co 7277 Basecbs 16910 +gcplusg 16960 Scalarcsca 16963 ·𝑠 cvsca 16964 0gc0g 17148 invgcminusg 18576 -gcsg 18577 LSSumclsm 19237 LSpanclspn 20231 LVecclvec 20362 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5211 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8040 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 |
This theorem is referenced by: baerlem5amN 39727 baerlem5abmN 39729 mapdh6lem1N 39744 hdmap1l6lem1 39818 |
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