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Theorem dva1dim 37059
 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 36637. 𝐸 is the division ring base by erngdv 37067, and 𝑠‘𝐹 is the scalar product by dvavsca 37091. 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 36841 . . . . . . . . 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 36835 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → (𝑅‘(𝑠𝐹)) (𝑅𝐹))
84, 7jca 507 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
983expb 1153 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝐹𝑇)) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
109anass1rs 645 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
11 eleq1 2894 . . . . . . 7 (𝑔 = (𝑠𝐹) → (𝑔𝑇 ↔ (𝑠𝐹) ∈ 𝑇))
12 fveq2 6437 . . . . . . . 8 (𝑔 = (𝑠𝐹) → (𝑅𝑔) = (𝑅‘(𝑠𝐹)))
1312breq1d 4885 . . . . . . 7 (𝑔 = (𝑠𝐹) → ((𝑅𝑔) (𝑅𝐹) ↔ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
1411, 13anbi12d 624 . . . . . 6 (𝑔 = (𝑠𝐹) → ((𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹)) ↔ ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹))))
1510, 14syl5ibrcom 239 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → (𝑔 = (𝑠𝐹) → (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
1615rexlimdva 3240 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 𝑔 = (𝑠𝐹) → (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
17 simpll 783 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
18 simplr 785 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → 𝐹𝑇)
19 simprl 787 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → 𝑔𝑇)
20 simprr 789 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → (𝑅𝑔) (𝑅𝐹))
215, 1, 2, 6, 3tendoex 37049 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑔𝑇) ∧ (𝑅𝑔) (𝑅𝐹)) → ∃𝑠𝐸 (𝑠𝐹) = 𝑔)
2217, 18, 19, 20, 21syl121anc 1498 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → ∃𝑠𝐸 (𝑠𝐹) = 𝑔)
23 eqcom 2832 . . . . . . 7 ((𝑠𝐹) = 𝑔𝑔 = (𝑠𝐹))
2423rexbii 3251 . . . . . 6 (∃𝑠𝐸 (𝑠𝐹) = 𝑔 ↔ ∃𝑠𝐸 𝑔 = (𝑠𝐹))
2522, 24sylib 210 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → ∃𝑠𝐸 𝑔 = (𝑠𝐹))
2625ex 403 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹)) → ∃𝑠𝐸 𝑔 = (𝑠𝐹)))
2716, 26impbid 204 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 𝑔 = (𝑠𝐹) ↔ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
2827abbidv 2946 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔 ∣ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))})
29 df-rab 3126 . 2 {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)} = {𝑔 ∣ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))}
3028, 29syl6eqr 2879 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  {cab 2811  ∃wrex 3118  {crab 3121   class class class wbr 4875  ‘cfv 6127  lecple 16319  HLchlt 35424  LHypclh 36058  LTrncltrn 36175  trLctrl 36232  TEndoctendo 36826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4996  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-riotaBAD 35027 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-fal 1670  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-iin 4745  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-undef 7669  df-map 8129  df-proset 17288  df-poset 17306  df-plt 17318  df-lub 17334  df-glb 17335  df-join 17336  df-meet 17337  df-p0 17399  df-p1 17400  df-lat 17406  df-clat 17468  df-oposet 35250  df-ol 35252  df-oml 35253  df-covers 35340  df-ats 35341  df-atl 35372  df-cvlat 35396  df-hlat 35425  df-llines 35572  df-lplanes 35573  df-lvols 35574  df-lines 35575  df-psubsp 35577  df-pmap 35578  df-padd 35870  df-lhyp 36062  df-laut 36063  df-ldil 36178  df-ltrn 36179  df-trl 36233  df-tendo 36829 This theorem is referenced by:  dvhb1dimN  37060  dia1dim  37135
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