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Theorem baerlem5abmN 39007
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 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 3896 . . . . . . 7 (𝜑𝑌𝑉)
3 baerlem3.z . . . . . . . 8 (𝜑𝑍 ∈ (𝑉 ∖ { 0 }))
43eldifad 3896 . . . . . . 7 (𝜑𝑍𝑉)
5 baerlem3.v . . . . . . . 8 𝑉 = (Base‘𝑊)
6 baerlem5a.p . . . . . . . 8 + = (+g𝑊)
7 eqid 2801 . . . . . . . 8 (invg𝑊) = (invg𝑊)
8 baerlem3.m . . . . . . . 8 = (-g𝑊)
95, 6, 7, 8grpsubval 18144 . . . . . . 7 ((𝑌𝑉𝑍𝑉) → (𝑌 𝑍) = (𝑌 + ((invg𝑊)‘𝑍)))
102, 4, 9syl2anc 587 . . . . . 6 (𝜑 → (𝑌 𝑍) = (𝑌 + ((invg𝑊)‘𝑍)))
1110oveq2d 7155 . . . . 5 (𝜑 → (𝑋 (𝑌 𝑍)) = (𝑋 (𝑌 + ((invg𝑊)‘𝑍))))
1211sneqd 4540 . . . 4 (𝜑 → {(𝑋 (𝑌 𝑍))} = {(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))})
1312fveq2d 6653 . . 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 19874 . . . . . . 7 (𝑊 ∈ LVec → 𝑊 ∈ LMod)
2017, 19syl 17 . . . . . 6 (𝜑𝑊 ∈ LMod)
215, 7lmodvnegcl 19671 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → ((invg𝑊)‘𝑍) ∈ 𝑉)
2220, 4, 21syl2anc 587 . . . . 5 (𝜑 → ((invg𝑊)‘𝑍) ∈ 𝑉)
23 eqid 2801 . . . . . . 7 (LSubSp‘𝑊) = (LSubSp‘𝑊)
245, 23, 16, 20, 2, 4lspprcl 19746 . . . . . . 7 (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑊))
25 baerlem3.c . . . . . . 7 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
2614, 23, 20, 24, 18, 25lssneln0 19720 . . . . . 6 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
275, 16, 17, 18, 2, 4, 25lspindpi 19900 . . . . . . 7 (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})))
2827simpld 498 . . . . . 6 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
295, 14, 16, 17, 26, 2, 28lspsnne1 19885 . . . . 5 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌}))
30 baerlem3.d . . . . . . . . 9 (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
3130necomd 3045 . . . . . . . 8 (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{𝑌}))
325, 14, 16, 17, 3, 2, 31lspsnne1 19885 . . . . . . 7 (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑌}))
335, 16, 17, 18, 4, 2, 32, 25lspexchn2 19899 . . . . . 6 (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑌, 𝑋}))
34 lmodgrp 19637 . . . . . . . . . 10 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
3517, 19, 343syl 18 . . . . . . . . 9 (𝜑𝑊 ∈ Grp)
3635adantr 484 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → 𝑊 ∈ Grp)
374adantr 484 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → 𝑍𝑉)
385, 7grpinvinv 18161 . . . . . . . 8 ((𝑊 ∈ Grp ∧ 𝑍𝑉) → ((invg𝑊)‘((invg𝑊)‘𝑍)) = 𝑍)
3936, 37, 38syl2anc 587 . . . . . . 7 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → ((invg𝑊)‘((invg𝑊)‘𝑍)) = 𝑍)
4020adantr 484 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → 𝑊 ∈ LMod)
415, 23, 16, 20, 2, 18lspprcl 19746 . . . . . . . . 9 (𝜑 → (𝑁‘{𝑌, 𝑋}) ∈ (LSubSp‘𝑊))
4241adantr 484 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → (𝑁‘{𝑌, 𝑋}) ∈ (LSubSp‘𝑊))
43 simpr 488 . . . . . . . 8 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋}))
4423, 7lssvnegcl 19724 . . . . . . . 8 ((𝑊 ∈ LMod ∧ (𝑁‘{𝑌, 𝑋}) ∈ (LSubSp‘𝑊) ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → ((invg𝑊)‘((invg𝑊)‘𝑍)) ∈ (𝑁‘{𝑌, 𝑋}))
4540, 42, 43, 44syl3anc 1368 . . . . . . 7 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → ((invg𝑊)‘((invg𝑊)‘𝑍)) ∈ (𝑁‘{𝑌, 𝑋}))
4639, 45eqeltrrd 2894 . . . . . 6 ((𝜑 ∧ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋})) → 𝑍 ∈ (𝑁‘{𝑌, 𝑋}))
4733, 46mtand 815 . . . . 5 (𝜑 → ¬ ((invg𝑊)‘𝑍) ∈ (𝑁‘{𝑌, 𝑋}))
485, 16, 17, 22, 18, 2, 29, 47lspexchn2 19899 . . . 4 (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, ((invg𝑊)‘𝑍)}))
495, 7, 16lspsnneg 19774 . . . . . 6 ((𝑊 ∈ LMod ∧ 𝑍𝑉) → (𝑁‘{((invg𝑊)‘𝑍)}) = (𝑁‘{𝑍}))
5020, 4, 49syl2anc 587 . . . . 5 (𝜑 → (𝑁‘{((invg𝑊)‘𝑍)}) = (𝑁‘{𝑍}))
5130, 50neeqtrrd 3064 . . . 4 (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{((invg𝑊)‘𝑍)}))
525, 14, 7grpinvnzcl 18166 . . . . 5 ((𝑊 ∈ Grp ∧ 𝑍 ∈ (𝑉 ∖ { 0 })) → ((invg𝑊)‘𝑍) ∈ (𝑉 ∖ { 0 }))
5335, 3, 52syl2anc 587 . . . 4 (𝜑 → ((invg𝑊)‘𝑍) ∈ (𝑉 ∖ { 0 }))
545, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6baerlem5a 39003 . . 3 (𝜑 → (𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{((invg𝑊)‘𝑍)})) ∩ ((𝑁‘{(𝑋 ((invg𝑊)‘𝑍))}) (𝑁‘{𝑌}))))
5550oveq2d 7155 . . . 4 (𝜑 → ((𝑁‘{(𝑋 𝑌)}) (𝑁‘{((invg𝑊)‘𝑍)})) = ((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})))
565, 6, 8, 7, 35, 18, 4grpsubinv 18167 . . . . . . 7 (𝜑 → (𝑋 ((invg𝑊)‘𝑍)) = (𝑋 + 𝑍))
5756sneqd 4540 . . . . . 6 (𝜑 → {(𝑋 ((invg𝑊)‘𝑍))} = {(𝑋 + 𝑍)})
5857fveq2d 6653 . . . . 5 (𝜑 → (𝑁‘{(𝑋 ((invg𝑊)‘𝑍))}) = (𝑁‘{(𝑋 + 𝑍)}))
5958oveq1d 7154 . . . 4 (𝜑 → ((𝑁‘{(𝑋 ((invg𝑊)‘𝑍))}) (𝑁‘{𝑌})) = ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌})))
6055, 59ineq12d 4143 . . 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 2840 . 2 (𝜑 → (𝑁‘{(𝑋 (𝑌 𝑍))}) = (((𝑁‘{(𝑋 𝑌)}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 + 𝑍)}) (𝑁‘{𝑌}))))
6210sneqd 4540 . . . 4 (𝜑 → {(𝑌 𝑍)} = {(𝑌 + ((invg𝑊)‘𝑍))})
6362fveq2d 6653 . . 3 (𝜑 → (𝑁‘{(𝑌 𝑍)}) = (𝑁‘{(𝑌 + ((invg𝑊)‘𝑍))}))
645, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6baerlem5b 39004 . . 3 (𝜑 → (𝑁‘{(𝑌 + ((invg𝑊)‘𝑍))}) = (((𝑁‘{𝑌}) (𝑁‘{((invg𝑊)‘𝑍)})) ∩ ((𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) (𝑁‘{𝑋}))))
6550oveq2d 7155 . . . 4 (𝜑 → ((𝑁‘{𝑌}) (𝑁‘{((invg𝑊)‘𝑍)})) = ((𝑁‘{𝑌}) (𝑁‘{𝑍})))
6610eqcomd 2807 . . . . . . . 8 (𝜑 → (𝑌 + ((invg𝑊)‘𝑍)) = (𝑌 𝑍))
6766oveq2d 7155 . . . . . . 7 (𝜑 → (𝑋 (𝑌 + ((invg𝑊)‘𝑍))) = (𝑋 (𝑌 𝑍)))
6867sneqd 4540 . . . . . 6 (𝜑 → {(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))} = {(𝑋 (𝑌 𝑍))})
6968fveq2d 6653 . . . . 5 (𝜑 → (𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) = (𝑁‘{(𝑋 (𝑌 𝑍))}))
7069oveq1d 7154 . . . 4 (𝜑 → ((𝑁‘{(𝑋 (𝑌 + ((invg𝑊)‘𝑍)))}) (𝑁‘{𝑋})) = ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋})))
7165, 70ineq12d 4143 . . 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 2840 . 2 (𝜑 → (𝑁‘{(𝑌 𝑍)}) = (((𝑁‘{𝑌}) (𝑁‘{𝑍})) ∩ ((𝑁‘{(𝑋 (𝑌 𝑍))}) (𝑁‘{𝑋}))))
7361, 72jca 515 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 399   = wceq 1538  wcel 2112  wne 2990  cdif 3881  cin 3883  {csn 4528  {cpr 4530  cfv 6328  (class class class)co 7139  Basecbs 16478  +gcplusg 16560  0gc0g 16708  Grpcgrp 18098  invgcminusg 18099  -gcsg 18100  LSSumclsm 18754  LModclmod 19630  LSubSpclss 19699  LSpanclspn 19739  LVecclvec 19870
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-cnex 10586  ax-resscn 10587  ax-1cn 10588  ax-icn 10589  ax-addcl 10590  ax-addrcl 10591  ax-mulcl 10592  ax-mulrcl 10593  ax-mulcom 10594  ax-addass 10595  ax-mulass 10596  ax-distr 10597  ax-i2m1 10598  ax-1ne0 10599  ax-1rid 10600  ax-rnegex 10601  ax-rrecex 10602  ax-cnre 10603  ax-pre-lttri 10604  ax-pre-lttrn 10605  ax-pre-ltadd 10606  ax-pre-mulgt0 10607
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-nel 3095  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-tpos 7879  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10670  df-mnf 10671  df-xr 10672  df-ltxr 10673  df-le 10674  df-sub 10865  df-neg 10866  df-nn 11630  df-2 11692  df-3 11693  df-ndx 16481  df-slot 16482  df-base 16484  df-sets 16485  df-ress 16486  df-plusg 16573  df-mulr 16574  df-0g 16710  df-mgm 17847  df-sgrp 17896  df-mnd 17907  df-submnd 17952  df-grp 18101  df-minusg 18102  df-sbg 18103  df-subg 18271  df-cntz 18442  df-lsm 18756  df-cmn 18903  df-abl 18904  df-mgp 19236  df-ur 19248  df-ring 19295  df-oppr 19372  df-dvdsr 19390  df-unit 19391  df-invr 19421  df-drng 19500  df-lmod 19632  df-lss 19700  df-lsp 19740  df-lvec 19871
This theorem is referenced by: (None)
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