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Theorem dva1dim 40590
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 40168. 𝐸 is the division ring base by erngdv 40598, and 𝑠𝐹 is the scalar product by dvavsca 40622. 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 40372 . . . . . . . . 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 40366 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → (𝑅‘(𝑠𝐹)) (𝑅𝐹))
84, 7jca 510 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
983expb 1117 . . . . . . 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 2813 . . . . . . 7 (𝑔 = (𝑠𝐹) → (𝑔𝑇 ↔ (𝑠𝐹) ∈ 𝑇))
12 fveq2 6896 . . . . . . . 8 (𝑔 = (𝑠𝐹) → (𝑅𝑔) = (𝑅‘(𝑠𝐹)))
1312breq1d 5159 . . . . . . 7 (𝑔 = (𝑠𝐹) → ((𝑅𝑔) (𝑅𝐹) ↔ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
1411, 13anbi12d 630 . . . . . 6 (𝑔 = (𝑠𝐹) → ((𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹)) ↔ ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹))))
1510, 14syl5ibrcom 246 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → (𝑔 = (𝑠𝐹) → (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
1615rexlimdva 3144 . . . 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 40580 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑔𝑇) ∧ (𝑅𝑔) (𝑅𝐹)) → ∃𝑠𝐸 (𝑠𝐹) = 𝑔)
2217, 18, 19, 20, 21syl121anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → ∃𝑠𝐸 (𝑠𝐹) = 𝑔)
23 eqcom 2732 . . . . . . 7 ((𝑠𝐹) = 𝑔𝑔 = (𝑠𝐹))
2423rexbii 3083 . . . . . 6 (∃𝑠𝐸 (𝑠𝐹) = 𝑔 ↔ ∃𝑠𝐸 𝑔 = (𝑠𝐹))
2522, 24sylib 217 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → ∃𝑠𝐸 𝑔 = (𝑠𝐹))
2625ex 411 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹)) → ∃𝑠𝐸 𝑔 = (𝑠𝐹)))
2716, 26impbid 211 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 𝑔 = (𝑠𝐹) ↔ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
2827abbidv 2794 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔 ∣ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))})
29 df-rab 3419 . 2 {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)} = {𝑔 ∣ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))}
3028, 29eqtr4di 2783 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  {cab 2702  wrex 3059  {crab 3418   class class class wbr 5149  cfv 6549  lecple 17248  HLchlt 38954  LHypclh 39589  LTrncltrn 39706  trLctrl 39763  TEndoctendo 40357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-riotaBAD 38557
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-undef 8279  df-map 8847  df-proset 18295  df-poset 18313  df-plt 18330  df-lub 18346  df-glb 18347  df-join 18348  df-meet 18349  df-p0 18425  df-p1 18426  df-lat 18432  df-clat 18499  df-oposet 38780  df-ol 38782  df-oml 38783  df-covers 38870  df-ats 38871  df-atl 38902  df-cvlat 38926  df-hlat 38955  df-llines 39103  df-lplanes 39104  df-lvols 39105  df-lines 39106  df-psubsp 39108  df-pmap 39109  df-padd 39401  df-lhyp 39593  df-laut 39594  df-ldil 39709  df-ltrn 39710  df-trl 39764  df-tendo 40360
This theorem is referenced by:  dvhb1dimN  40591  dia1dim  40666
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