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Theorem baerlem5abmN 36025
 Description: An equality that holds when 𝑋, 𝑌, 𝑍 are independent (non-colinear) vectors. Subtraction versions of first and second equations of part (5) in [Baer] p. 46, conjoined to share commonality in their proofs. TODO: Delete if not be needed. (Contributed by NM, 24-May-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
baerlem3.v 𝑉 = (Base‘𝑊)
baerlem3.m = (-g𝑊)
baerlem3.o 0 = (0g𝑊)
baerlem3.s = (LSSum‘𝑊)
baerlem3.n 𝑁 = (LSpan‘𝑊)
baerlem3.w (𝜑𝑊 ∈ LVec)
baerlem3.x (𝜑𝑋𝑉)
baerlem3.c (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
baerlem3.d (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
baerlem3.y (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
baerlem3.z (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))
baerlem5a.p + = (+g𝑊)
Assertion
Ref Expression
baerlem5abmN (𝜑 → ((𝑁‘{(𝑋 (𝑌 𝑍))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌}))) ∧ (𝑁‘{(𝑌 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋})))))

Proof of Theorem baerlem5abmN
StepHypRef Expression
1 baerlem3.y . . . . . . . 8 (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
21eldifad 3552 . . . . . . 7 (𝜑𝑌𝑉)
3 baerlem3.z . . . . . . . 8 (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))
43eldifad 3552 . . . . . . 7 (𝜑𝑍𝑉)
5 baerlem3.v . . . . . . . 8 𝑉 = (Base‘𝑊)
6 baerlem5a.p . . . . . . . 8 + = (+g𝑊)
7 eqid 2610 . . . . . . . 8 (invg𝑊) = (invg𝑊)
8 baerlem3.m . . . . . . . 8 = (-g𝑊)
95, 6, 7, 8grpsubval 17288 . . . . . . 7 ((𝑌𝑉𝑍𝑉) → (𝑌 𝑍) = (𝑌 + ((invg𝑊)‘𝑍)))
102, 4, 9syl2anc 691 . . . . . 6 (𝜑 → (𝑌 𝑍) = (𝑌 + ((invg𝑊)‘𝑍)))
1110oveq2d 6565 . . . . 5 (𝜑 → (𝑋 (𝑌 𝑍)) = (𝑋 (𝑌 + ((invg𝑊)‘𝑍))))
1211sneqd 4137 . . . 4 (𝜑 → {(𝑋 (𝑌 𝑍))} = {(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))})
1312fveq2d 6107 . . 3 (𝜑 → (𝑁‘{(𝑋 (𝑌 𝑍))}) = (𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}))
14 baerlem3.o . . . 4 0 = (0g𝑊)
15 baerlem3.s . . . 4 = (LSSum‘𝑊)
16 baerlem3.n . . . 4 𝑁 = (LSpan‘𝑊)
17 baerlem3.w . . . 4 (𝜑𝑊 ∈ LVec)
18 baerlem3.x . . . 4 (𝜑𝑋𝑉)
19 lveclmod 18927 . . . . . . 7 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
2017, 19syl 17 . . . . . 6 (𝜑𝑊 ∈ LMod)
215, 7lmodvnegcl 18727 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → ((invg𝑊)‘𝑍) ∈ 𝑉)
2220, 4, 21syl2anc 691 . . . . 5 (𝜑 → ((invg𝑊)‘𝑍) ∈ 𝑉)
23 eqid 2610 . . . . . . 7 (LSubSp‘𝑊) = (LSubSp‘𝑊)
245, 23, 16, 20, 2, 4lspprcl 18799 . . . . . . 7 (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑊))
25 baerlem3.c . . . . . . 7 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
265, 14, 23, 20, 24, 18, 25lssneln0 18773 . . . . . 6 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
275, 16, 17, 18, 2, 4, 25lspindpi 18953 . . . . . . 7 (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
2827simpld 474 . . . . . 6 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
295, 14, 16, 17, 26, 2, 28lspsnne1 18938 . . . . 5 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
30 baerlem3.d . . . . . . . . 9 (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
3130necomd 2837 . . . . . . . 8 (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}))
325, 14, 16, 17, 3, 2, 31lspsnne1 18938 . . . . . . 7 (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑌}))
335, 16, 17, 18, 4, 2, 32, 25lspexchn2 18952 . . . . . 6 (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑌, 𝑋}))
34 lmodgrp 18693 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
3517, 19, 343syl 18 . . . . . . . . 9 (𝜑𝑊 ∈ Grp)
3635adantr 480 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → 𝑊 ∈ Grp)
374adantr 480 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → 𝑍𝑉)
385, 7grpinvinv 17305 . . . . . . . 8 ((𝑊 ∈ Grp ∧ 𝑍𝑉) → ((invg𝑊)‘((invg𝑊)‘𝑍)) = 𝑍)
3936, 37, 38syl2anc 691 . . . . . . 7 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → ((invg𝑊)‘((invg𝑊)‘𝑍)) = 𝑍)
4020adantr 480 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → 𝑊 ∈ LMod)
415, 23, 16, 20, 2, 18lspprcl 18799 . . . . . . . . 9 (𝜑 → (𝑁‘{𝑌, 𝑋}) ∈ (LSubSp‘𝑊))
4241adantr 480 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → (𝑁‘{𝑌, 𝑋}) ∈ (LSubSp‘𝑊))
43 simpr 476 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋}))
4423, 7lssvnegcl 18777 . . . . . . . 8 ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌, 𝑋}) ∈ (LSubSp‘𝑊) ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → ((invg𝑊)‘((invg𝑊)‘𝑍)) ∈ (𝑁‘{𝑌, 𝑋}))
4540, 42, 43, 44syl3anc 1318 . . . . . . 7 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → ((invg𝑊)‘((invg𝑊)‘𝑍)) ∈ (𝑁‘{𝑌, 𝑋}))
4639, 45eqeltrrd 2689 . . . . . 6 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → 𝑍 ∈ (𝑁‘{𝑌, 𝑋}))
4733, 46mtand 689 . . . . 5 (𝜑 → ¬ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋}))
485, 16, 17, 22, 18, 2, 29, 47lspexchn2 18952 . . . 4 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, ((invg𝑊)‘𝑍)}))
495, 7, 16lspsnneg 18827 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → (𝑁‘{((invg𝑊)‘𝑍)}) = (𝑁‘{𝑍}))
5020, 4, 49syl2anc 691 . . . . 5 (𝜑 → (𝑁‘{((invg𝑊)‘𝑍)}) = (𝑁‘{𝑍}))
5130, 50neeqtrrd 2856 . . . 4 (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{((invg𝑊)‘𝑍)}))
525, 14, 7grpinvnzcl 17310 . . . . 5 ((𝑊 ∈ Grp ∧ 𝑍 ∈ (𝑉 ∖ { 0 })) → ((invg𝑊)‘𝑍) ∈ (𝑉 ∖ { 0 }))
5335, 3, 52syl2anc 691 . . . 4 (𝜑 → ((invg𝑊)‘𝑍) ∈ (𝑉 ∖ { 0 }))
545, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6baerlem5a 36021 . . 3 (𝜑 → (𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{((invg𝑊)‘𝑍)})) ∩ ((𝑁‘{(𝑋 ((invg𝑊)‘𝑍))}) (𝑁‘{𝑌}))))
5550oveq2d 6565 . . . 4 (𝜑 → ((𝑁‘{(𝑋 𝑌)}) (𝑁‘{((invg𝑊)‘𝑍)})) = ((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})))
565, 6, 8, 7, 35, 18, 4grpsubinv 17311 . . . . . . 7 (𝜑 → (𝑋 ((invg𝑊)‘𝑍)) = (𝑋 + 𝑍))
5756sneqd 4137 . . . . . 6 (𝜑 → {(𝑋 ((invg𝑊)‘𝑍))} = {(𝑋 + 𝑍)})
5857fveq2d 6107 . . . . 5 (𝜑 → (𝑁‘{(𝑋 ((invg𝑊)‘𝑍))}) = (𝑁‘{(𝑋 + 𝑍)}))
5958oveq1d 6564 . . . 4 (𝜑 → ((𝑁‘{(𝑋 ((invg𝑊)‘𝑍))}) (𝑁‘{𝑌})) = ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌})))
6055, 59ineq12d 3777 . . 3 (𝜑 → (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{((invg𝑊)‘𝑍)})) ∩ ((𝑁‘{(𝑋 ((invg𝑊)‘𝑍))}) (𝑁‘{𝑌}))) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌}))))
6113, 54, 603eqtrd 2648 . 2 (𝜑 → (𝑁‘{(𝑋 (𝑌 𝑍))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌}))))
6210sneqd 4137 . . . 4 (𝜑 → {(𝑌 𝑍)} = {(𝑌 + ((invg𝑊)‘𝑍))})
6362fveq2d 6107 . . 3 (𝜑 → (𝑁‘{(𝑌 𝑍)}) = (𝑁‘{(𝑌 + ((invg𝑊)‘𝑍))}))
645, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6baerlem5b 36022 . . 3 (𝜑 → (𝑁‘{(𝑌 + ((invg𝑊)‘𝑍))}) = (((𝑁‘{𝑌}) (𝑁‘{((invg𝑊)‘𝑍)})) ∩ ((𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) (𝑁‘{𝑋}))))
6550oveq2d 6565 . . . 4 (𝜑 → ((𝑁‘{𝑌}) (𝑁‘{((invg𝑊)‘𝑍)})) = ((𝑁‘{𝑌}) (𝑁‘{𝑍})))
6610eqcomd 2616 . . . . . . . 8 (𝜑 → (𝑌 + ((invg𝑊)‘𝑍)) = (𝑌 𝑍))
6766oveq2d 6565 . . . . . . 7 (𝜑 → (𝑋 (𝑌 + ((invg𝑊)‘𝑍))) = (𝑋 (𝑌 𝑍)))
6867sneqd 4137 . . . . . 6 (𝜑 → {(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))} = {(𝑋 (𝑌 𝑍))})
6968fveq2d 6107 . . . . 5 (𝜑 → (𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) = (𝑁‘{(𝑋 (𝑌 𝑍))}))
7069oveq1d 6564 . . . 4 (𝜑 → ((𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) (𝑁‘{𝑋})) = ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋})))
7165, 70ineq12d 3777 . . 3 (𝜑 → (((𝑁‘{𝑌}) (𝑁‘{((invg𝑊)‘𝑍)})) ∩ ((𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) (𝑁‘{𝑋}))) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋}))))
7263, 64, 713eqtrd 2648 . 2 (𝜑 → (𝑁‘{(𝑌 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋}))))
7361, 72jca 553 1 (𝜑 → ((𝑁‘{(𝑋 (𝑌 𝑍))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌}))) ∧ (𝑁‘{(𝑌 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋})))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∖ cdif 3537   ∩ cin 3539  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  invgcminusg 17246  -gcsg 17247  LSSumclsm 17872  LModclmod 18686  LSubSpclss 18753  LSpanclspn 18792  LVecclvec 18923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-cntz 17573  df-lsm 17874  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446  df-dvdsr 18464  df-unit 18465  df-invr 18495  df-drng 18572  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lvec 18924 This theorem is referenced by: (None)
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