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Theorem dva1dim 40987
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 40565. 𝐸 is the division ring base by erngdv 40995, and 𝑠𝐹 is the scalar product by dvavsca 41019. 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 40769 . . . . . . . . 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 40763 . . . . . . . . 9 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → (𝑅‘(𝑠𝐹)) (𝑅𝐹))
84, 7jca 511 . . . . . . . 8 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑠𝐸𝐹𝑇) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
983expb 1121 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑠𝐸𝐹𝑇)) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
109anass1rs 655 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
11 eleq1 2829 . . . . . . 7 (𝑔 = (𝑠𝐹) → (𝑔𝑇 ↔ (𝑠𝐹) ∈ 𝑇))
12 fveq2 6906 . . . . . . . 8 (𝑔 = (𝑠𝐹) → (𝑅𝑔) = (𝑅‘(𝑠𝐹)))
1312breq1d 5153 . . . . . . 7 (𝑔 = (𝑠𝐹) → ((𝑅𝑔) (𝑅𝐹) ↔ (𝑅‘(𝑠𝐹)) (𝑅𝐹)))
1411, 13anbi12d 632 . . . . . 6 (𝑔 = (𝑠𝐹) → ((𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹)) ↔ ((𝑠𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠𝐹)) (𝑅𝐹))))
1510, 14syl5ibrcom 247 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ 𝑠𝐸) → (𝑔 = (𝑠𝐹) → (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
1615rexlimdva 3155 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 𝑔 = (𝑠𝐹) → (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
17 simpll 767 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → (𝐾 ∈ HL ∧ 𝑊𝐻))
18 simplr 769 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → 𝐹𝑇)
19 simprl 771 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → 𝑔𝑇)
20 simprr 773 . . . . . . 7 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → (𝑅𝑔) (𝑅𝐹))
215, 1, 2, 6, 3tendoex 40977 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑔𝑇) ∧ (𝑅𝑔) (𝑅𝐹)) → ∃𝑠𝐸 (𝑠𝐹) = 𝑔)
2217, 18, 19, 20, 21syl121anc 1377 . . . . . 6 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → ∃𝑠𝐸 (𝑠𝐹) = 𝑔)
23 eqcom 2744 . . . . . . 7 ((𝑠𝐹) = 𝑔𝑔 = (𝑠𝐹))
2423rexbii 3094 . . . . . 6 (∃𝑠𝐸 (𝑠𝐹) = 𝑔 ↔ ∃𝑠𝐸 𝑔 = (𝑠𝐹))
2522, 24sylib 218 . . . . 5 ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) ∧ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))) → ∃𝑠𝐸 𝑔 = (𝑠𝐹))
2625ex 412 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹)) → ∃𝑠𝐸 𝑔 = (𝑠𝐹)))
2716, 26impbid 212 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∃𝑠𝐸 𝑔 = (𝑠𝐹) ↔ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))))
2827abbidv 2808 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔 ∣ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))})
29 df-rab 3437 . 2 {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)} = {𝑔 ∣ (𝑔𝑇 ∧ (𝑅𝑔) (𝑅𝐹))}
3028, 29eqtr4di 2795 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  {crab 3436   class class class wbr 5143  cfv 6561  lecple 17304  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  trLctrl 40160  TEndoctendo 40754
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-undef 8298  df-map 8868  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161  df-tendo 40757
This theorem is referenced by:  dvhb1dimN  40988  dia1dim  41063
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