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Theorem baerlem5bmN 40583
Description: An equality that holds when 𝑋, π‘Œ, 𝑍 are independent (non-colinear) vectors. Subtraction version of second equation of part (5) in [Baer] p. 46. TODO: This is the subtraction version, may not be needed. TODO: delete if baerlem5abmN 40584 is used. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
baerlem3.v 𝑉 = (Baseβ€˜π‘Š)
baerlem3.m βˆ’ = (-gβ€˜π‘Š)
baerlem3.o 0 = (0gβ€˜π‘Š)
baerlem3.s βŠ• = (LSSumβ€˜π‘Š)
baerlem3.n 𝑁 = (LSpanβ€˜π‘Š)
baerlem3.w (πœ‘ β†’ π‘Š ∈ LVec)
baerlem3.x (πœ‘ β†’ 𝑋 ∈ 𝑉)
baerlem3.c (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
baerlem3.d (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
baerlem3.y (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
baerlem3.z (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
baerlem5a.p + = (+gβ€˜π‘Š)
Assertion
Ref Expression
baerlem5bmN (πœ‘ β†’ (π‘β€˜{(π‘Œ βˆ’ 𝑍)}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}) βŠ• (π‘β€˜{𝑋}))))
Proof of Theorem baerlem5bmN
StepHypRef Expression
1 baerlem3.y . . . . . 6 (πœ‘ β†’ π‘Œ ∈ (𝑉 βˆ– { 0 }))
21eldifad 3960 . . . . 5 (πœ‘ β†’ π‘Œ ∈ 𝑉)
3 baerlem3.z . . . . . 6 (πœ‘ β†’ 𝑍 ∈ (𝑉 βˆ– { 0 }))
43eldifad 3960 . . . . 5 (πœ‘ β†’ 𝑍 ∈ 𝑉)
5 baerlem3.v . . . . . 6 𝑉 = (Baseβ€˜π‘Š)
6 baerlem5a.p . . . . . 6 + = (+gβ€˜π‘Š)
7 eqid 2732 . . . . . 6 (invgβ€˜π‘Š) = (invgβ€˜π‘Š)
8 baerlem3.m . . . . . 6 βˆ’ = (-gβ€˜π‘Š)
95, 6, 7, 8grpsubval 18869 . . . . 5 ((π‘Œ ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) β†’ (π‘Œ βˆ’ 𝑍) = (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘)))
102, 4, 9syl2anc 584 . . . 4 (πœ‘ β†’ (π‘Œ βˆ’ 𝑍) = (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘)))
1110sneqd 4640 . . 3 (πœ‘ β†’ {(π‘Œ βˆ’ 𝑍)} = {(π‘Œ + ((invgβ€˜π‘Š)β€˜π‘))})
1211fveq2d 6895 . 2 (πœ‘ β†’ (π‘β€˜{(π‘Œ βˆ’ 𝑍)}) = (π‘β€˜{(π‘Œ + ((invgβ€˜π‘Š)β€˜π‘))}))
13 baerlem3.o . . 3 0 = (0gβ€˜π‘Š)
14 baerlem3.s . . 3 βŠ• = (LSSumβ€˜π‘Š)
15 baerlem3.n . . 3 𝑁 = (LSpanβ€˜π‘Š)
16 baerlem3.w . . 3 (πœ‘ β†’ π‘Š ∈ LVec)
17 baerlem3.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝑉)
18 lveclmod 20716 . . . . . 6 (π‘Š ∈ LVec β†’ π‘Š ∈ LMod)
1916, 18syl 17 . . . . 5 (πœ‘ β†’ π‘Š ∈ LMod)
205, 7lmodvnegcl 20512 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑍 ∈ 𝑉) β†’ ((invgβ€˜π‘Š)β€˜π‘) ∈ 𝑉)
2119, 4, 20syl2anc 584 . . . 4 (πœ‘ β†’ ((invgβ€˜π‘Š)β€˜π‘) ∈ 𝑉)
22 eqid 2732 . . . . . 6 (LSubSpβ€˜π‘Š) = (LSubSpβ€˜π‘Š)
235, 22, 15, 19, 2, 4lspprcl 20588 . . . . . 6 (πœ‘ β†’ (π‘β€˜{π‘Œ, 𝑍}) ∈ (LSubSpβ€˜π‘Š))
24 baerlem3.c . . . . . 6 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, 𝑍}))
2513, 22, 19, 23, 17, 24lssneln0 20562 . . . . 5 (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
265, 15, 16, 17, 2, 4, 24lspindpi 20744 . . . . . 6 (πœ‘ β†’ ((π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}) ∧ (π‘β€˜{𝑋}) β‰  (π‘β€˜{𝑍})))
2726simpld 495 . . . . 5 (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))
285, 13, 15, 16, 25, 2, 27lspsnne1 20729 . . . 4 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ}))
29 baerlem3.d . . . . . . . 8 (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{𝑍}))
3029necomd 2996 . . . . . . 7 (πœ‘ β†’ (π‘β€˜{𝑍}) β‰  (π‘β€˜{π‘Œ}))
315, 13, 15, 16, 3, 2, 30lspsnne1 20729 . . . . . 6 (πœ‘ β†’ Β¬ 𝑍 ∈ (π‘β€˜{π‘Œ}))
325, 15, 16, 17, 4, 2, 31, 24lspexchn2 20743 . . . . 5 (πœ‘ β†’ Β¬ 𝑍 ∈ (π‘β€˜{π‘Œ, 𝑋}))
33 lmodgrp 20477 . . . . . . . . 9 (π‘Š ∈ LMod β†’ π‘Š ∈ Grp)
3419, 33syl 17 . . . . . . . 8 (πœ‘ β†’ π‘Š ∈ Grp)
3534adantr 481 . . . . . . 7 ((πœ‘ ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ π‘Š ∈ Grp)
364adantr 481 . . . . . . 7 ((πœ‘ ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ 𝑍 ∈ 𝑉)
375, 7grpinvinv 18889 . . . . . . 7 ((π‘Š ∈ Grp ∧ 𝑍 ∈ 𝑉) β†’ ((invgβ€˜π‘Š)β€˜((invgβ€˜π‘Š)β€˜π‘)) = 𝑍)
3835, 36, 37syl2anc 584 . . . . . 6 ((πœ‘ ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ ((invgβ€˜π‘Š)β€˜((invgβ€˜π‘Š)β€˜π‘)) = 𝑍)
3919adantr 481 . . . . . . 7 ((πœ‘ ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ π‘Š ∈ LMod)
405, 22, 15, 19, 2, 17lspprcl 20588 . . . . . . . 8 (πœ‘ β†’ (π‘β€˜{π‘Œ, 𝑋}) ∈ (LSubSpβ€˜π‘Š))
4140adantr 481 . . . . . . 7 ((πœ‘ ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ (π‘β€˜{π‘Œ, 𝑋}) ∈ (LSubSpβ€˜π‘Š))
42 simpr 485 . . . . . . 7 ((πœ‘ ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋}))
4322, 7lssvnegcl 20566 . . . . . . 7 ((π‘Š ∈ LMod ∧ (π‘β€˜{π‘Œ, 𝑋}) ∈ (LSubSpβ€˜π‘Š) ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ ((invgβ€˜π‘Š)β€˜((invgβ€˜π‘Š)β€˜π‘)) ∈ (π‘β€˜{π‘Œ, 𝑋}))
4439, 41, 42, 43syl3anc 1371 . . . . . 6 ((πœ‘ ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ ((invgβ€˜π‘Š)β€˜((invgβ€˜π‘Š)β€˜π‘)) ∈ (π‘β€˜{π‘Œ, 𝑋}))
4538, 44eqeltrrd 2834 . . . . 5 ((πœ‘ ∧ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋})) β†’ 𝑍 ∈ (π‘β€˜{π‘Œ, 𝑋}))
4632, 45mtand 814 . . . 4 (πœ‘ β†’ Β¬ ((invgβ€˜π‘Š)β€˜π‘) ∈ (π‘β€˜{π‘Œ, 𝑋}))
475, 15, 16, 21, 17, 2, 28, 46lspexchn2 20743 . . 3 (πœ‘ β†’ Β¬ 𝑋 ∈ (π‘β€˜{π‘Œ, ((invgβ€˜π‘Š)β€˜π‘)}))
485, 7, 15lspsnneg 20616 . . . . 5 ((π‘Š ∈ LMod ∧ 𝑍 ∈ 𝑉) β†’ (π‘β€˜{((invgβ€˜π‘Š)β€˜π‘)}) = (π‘β€˜{𝑍}))
4919, 4, 48syl2anc 584 . . . 4 (πœ‘ β†’ (π‘β€˜{((invgβ€˜π‘Š)β€˜π‘)}) = (π‘β€˜{𝑍}))
5029, 49neeqtrrd 3015 . . 3 (πœ‘ β†’ (π‘β€˜{π‘Œ}) β‰  (π‘β€˜{((invgβ€˜π‘Š)β€˜π‘)}))
515, 13, 7grpinvnzcl 18894 . . . 4 ((π‘Š ∈ Grp ∧ 𝑍 ∈ (𝑉 βˆ– { 0 })) β†’ ((invgβ€˜π‘Š)β€˜π‘) ∈ (𝑉 βˆ– { 0 }))
5234, 3, 51syl2anc 584 . . 3 (πœ‘ β†’ ((invgβ€˜π‘Š)β€˜π‘) ∈ (𝑉 βˆ– { 0 }))
535, 8, 13, 14, 15, 16, 17, 47, 50, 1, 52, 6baerlem5b 40581 . 2 (πœ‘ β†’ (π‘β€˜{(π‘Œ + ((invgβ€˜π‘Š)β€˜π‘))}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{((invgβ€˜π‘Š)β€˜π‘)})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘)))}) βŠ• (π‘β€˜{𝑋}))))
5449oveq2d 7424 . . 3 (πœ‘ β†’ ((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{((invgβ€˜π‘Š)β€˜π‘)})) = ((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})))
5510eqcomd 2738 . . . . . . 7 (πœ‘ β†’ (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘)) = (π‘Œ βˆ’ 𝑍))
5655oveq2d 7424 . . . . . 6 (πœ‘ β†’ (𝑋 βˆ’ (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘))) = (𝑋 βˆ’ (π‘Œ βˆ’ 𝑍)))
5756sneqd 4640 . . . . 5 (πœ‘ β†’ {(𝑋 βˆ’ (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘)))} = {(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))})
5857fveq2d 6895 . . . 4 (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘)))}) = (π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}))
5958oveq1d 7423 . . 3 (πœ‘ β†’ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘)))}) βŠ• (π‘β€˜{𝑋})) = ((π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}) βŠ• (π‘β€˜{𝑋})))
6054, 59ineq12d 4213 . 2 (πœ‘ β†’ (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{((invgβ€˜π‘Š)β€˜π‘)})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ + ((invgβ€˜π‘Š)β€˜π‘)))}) βŠ• (π‘β€˜{𝑋}))) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}) βŠ• (π‘β€˜{𝑋}))))
6112, 53, 603eqtrd 2776 1 (πœ‘ β†’ (π‘β€˜{(π‘Œ βˆ’ 𝑍)}) = (((π‘β€˜{π‘Œ}) βŠ• (π‘β€˜{𝑍})) ∩ ((π‘β€˜{(𝑋 βˆ’ (π‘Œ βˆ’ 𝑍))}) βŠ• (π‘β€˜{𝑋}))))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βˆ– cdif 3945   ∩ cin 3947  {csn 4628  {cpr 4630  β€˜cfv 6543  (class class class)co 7408  Basecbs 17143  +gcplusg 17196  0gc0g 17384  Grpcgrp 18818  invgcminusg 18819  -gcsg 18820  LSSumclsm 19501  LModclmod 20470  LSubSpclss 20541  LSpanclspn 20581  LVecclvec 20712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-tpos 8210  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-3 12275  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-mulr 17210  df-0g 17386  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-submnd 18671  df-grp 18821  df-minusg 18822  df-sbg 18823  df-subg 19002  df-cntz 19180  df-lsm 19503  df-cmn 19649  df-abl 19650  df-mgp 19987  df-ur 20004  df-ring 20057  df-oppr 20149  df-dvdsr 20170  df-unit 20171  df-invr 20201  df-drng 20358  df-lmod 20472  df-lss 20542  df-lsp 20582  df-lvec 20713
This theorem is referenced by: (None)
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