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Theorem dva1dim 38125
Description: Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) 𝐹 whose trace is 𝑃 rather than 𝑃 itself; 𝐹 exists by cdlemf 37703. 𝐸 is the division ring base by erngdv 38133, and 𝑠𝐹 is the scalar product by dvavsca 38157. 𝐹 must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.)
Hypotheses
Ref Expression
dva1dim.l = (le‘𝐾)
dva1dim.h 𝐻 = (LHyp‘𝐾)
dva1dim.t 𝑇 = ((LTrn‘𝐾)‘𝑊)
dva1dim.r 𝑅 = ((trL‘𝐾)‘𝑊)
dva1dim.e 𝐸 = ((TEndo‘𝐾)‘𝑊)
Assertion
Ref Expression
dva1dim (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)})
Distinct variable groups:   ,𝑠   𝐸,𝑠   𝑔,𝑠,𝐹   𝑔,𝐻,𝑠   𝑔,𝐾,𝑠   𝑅,𝑠   𝑇,𝑔,𝑠   𝑔,𝑊,𝑠
Allowed substitution hints:   𝑅(𝑔)   𝐸(𝑔)   (𝑔)

Proof of Theorem dva1dim
StepHypRef Expression
1 dva1dim.h . . . . . . . . . 10 𝐻 = (LHyp‘𝐾)
2 dva1dim.t . . . . . . . . . 10 𝑇 = ((LTrn‘𝐾)‘𝑊)
3 dva1dim.e . . . . . . . . . 10 𝐸 = ((TEndo‘𝐾)‘𝑊)
41, 2, 3tendocl 37907 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → (𝑠𝐹) ∈ 𝑇)
5 dva1dim.l . . . . . . . . . 10 = (le‘𝐾)
6 dva1dim.r . . . . . . . . . 10 𝑅 = ((trL‘𝐾)‘𝑊)
75, 1, 2, 6, 3tendotp 37901 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → (𝑅‘(𝑠𝐹)) (𝑅𝐹))
84, 7jca 514 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
983expb 1116 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝐹𝑇)) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
109anass1rs 653 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
11 eleq1 2903 . . . . . . 7 (𝑔 = (𝑠𝐹) → (𝑔𝑇 ↔ (𝑠𝐹) ∈ 𝑇))
12 fveq2 6673 . . . . . . . 8 (𝑔 = (𝑠𝐹) → (𝑅𝑔) = (𝑅‘(𝑠𝐹)))
1312breq1d 5079 . . . . . . 7 (𝑔 = (𝑠𝐹) → ((𝑅𝑔) (𝑅𝐹) ↔ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
1411, 13anbi12d 632 . . . . . 6 (𝑔 = (𝑠𝐹) → ((𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹)) ↔ ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹))))
1510, 14syl5ibrcom 249 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → (𝑔 = (𝑠𝐹) → (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
1615rexlimdva 3287 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 𝑔 = (𝑠𝐹) → (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
17 simpll 765 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
18 simplr 767 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → 𝐹𝑇)
19 simprl 769 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → 𝑔𝑇)
20 simprr 771 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → (𝑅𝑔) (𝑅𝐹))
215, 1, 2, 6, 3tendoex 38115 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑔𝑇) ∧ (𝑅𝑔) (𝑅𝐹)) → ∃𝑠𝐸 (𝑠𝐹) = 𝑔)
2217, 18, 19, 20, 21syl121anc 1371 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → ∃𝑠𝐸 (𝑠𝐹) = 𝑔)
23 eqcom 2831 . . . . . . 7 ((𝑠𝐹) = 𝑔𝑔 = (𝑠𝐹))
2423rexbii 3250 . . . . . 6 (∃𝑠𝐸 (𝑠𝐹) = 𝑔 ↔ ∃𝑠𝐸 𝑔 = (𝑠𝐹))
2522, 24sylib 220 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → ∃𝑠𝐸 𝑔 = (𝑠𝐹))
2625ex 415 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹)) → ∃𝑠𝐸 𝑔 = (𝑠𝐹)))
2716, 26impbid 214 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 𝑔 = (𝑠𝐹) ↔ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
2827abbidv 2888 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔 ∣ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))})
29 df-rab 3150 . 2 {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)} = {𝑔 ∣ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))}
3028, 29syl6eqr 2877 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1083   = wceq 1536  wcel 2113  {cab 2802  wrex 3142  {crab 3145   class class class wbr 5069  cfv 6358  lecple 16575  HLchlt 36490  LHypclh 37124  LTrncltrn 37241  trLctrl 37298  TEndoctendo 37892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-riotaBAD 36093
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-fal 1549  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-1st 7692  df-2nd 7693  df-undef 7942  df-map 8411  df-proset 17541  df-poset 17559  df-plt 17571  df-lub 17587  df-glb 17588  df-join 17589  df-meet 17590  df-p0 17652  df-p1 17653  df-lat 17659  df-clat 17721  df-oposet 36316  df-ol 36318  df-oml 36319  df-covers 36406  df-ats 36407  df-atl 36438  df-cvlat 36462  df-hlat 36491  df-llines 36638  df-lplanes 36639  df-lvols 36640  df-lines 36641  df-psubsp 36643  df-pmap 36644  df-padd 36936  df-lhyp 37128  df-laut 37129  df-ldil 37244  df-ltrn 37245  df-trl 37299  df-tendo 37895
This theorem is referenced by:  dvhb1dimN  38126  dia1dim  38201
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