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Theorem dva1dim 40367
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 39945. 𝐸 is the division ring base by erngdv 40375, and π‘ β€˜πΉ is the scalar product by dvavsca 40399. 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 ≀ = (leβ€˜πΎ)
dva1dim.h 𝐻 = (LHypβ€˜πΎ)
dva1dim.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dva1dim.r 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
dva1dim.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dva1dim (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ {𝑔 ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ)} = {𝑔 ∈ 𝑇 ∣ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)})
Distinct variable groups:   ≀ ,𝑠   𝐸,𝑠   𝑔,𝑠,𝐹   𝑔,𝐻,𝑠   𝑔,𝐾,𝑠   𝑅,𝑠   𝑇,𝑔,𝑠   𝑔,π‘Š,𝑠
Allowed substitution hints:   𝑅(𝑔)   𝐸(𝑔)   ≀ (𝑔)

Proof of Theorem dva1dim
StepHypRef Expression
1 dva1dim.h . . . . . . . . . 10 𝐻 = (LHypβ€˜πΎ)
2 dva1dim.t . . . . . . . . . 10 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
3 dva1dim.e . . . . . . . . . 10 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
41, 2, 3tendocl 40149 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘ β€˜πΉ) ∈ 𝑇)
5 dva1dim.l . . . . . . . . . 10 ≀ = (leβ€˜πΎ)
6 dva1dim.r . . . . . . . . . 10 𝑅 = ((trLβ€˜πΎ)β€˜π‘Š)
75, 1, 2, 6, 3tendotp 40143 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘ β€˜πΉ)) ≀ (π‘…β€˜πΉ))
84, 7jca 511 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ ((π‘ β€˜πΉ) ∈ 𝑇 ∧ (π‘…β€˜(π‘ β€˜πΉ)) ≀ (π‘…β€˜πΉ)))
983expb 1117 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) β†’ ((π‘ β€˜πΉ) ∈ 𝑇 ∧ (π‘…β€˜(π‘ β€˜πΉ)) ≀ (π‘…β€˜πΉ)))
109anass1rs 652 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) β†’ ((π‘ β€˜πΉ) ∈ 𝑇 ∧ (π‘…β€˜(π‘ β€˜πΉ)) ≀ (π‘…β€˜πΉ)))
11 eleq1 2815 . . . . . . 7 (𝑔 = (π‘ β€˜πΉ) β†’ (𝑔 ∈ 𝑇 ↔ (π‘ β€˜πΉ) ∈ 𝑇))
12 fveq2 6884 . . . . . . . 8 (𝑔 = (π‘ β€˜πΉ) β†’ (π‘…β€˜π‘”) = (π‘…β€˜(π‘ β€˜πΉ)))
1312breq1d 5151 . . . . . . 7 (𝑔 = (π‘ β€˜πΉ) β†’ ((π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ) ↔ (π‘…β€˜(π‘ β€˜πΉ)) ≀ (π‘…β€˜πΉ)))
1411, 13anbi12d 630 . . . . . 6 (𝑔 = (π‘ β€˜πΉ) β†’ ((𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)) ↔ ((π‘ β€˜πΉ) ∈ 𝑇 ∧ (π‘…β€˜(π‘ β€˜πΉ)) ≀ (π‘…β€˜πΉ))))
1510, 14syl5ibrcom 246 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) β†’ (𝑔 = (π‘ β€˜πΉ) β†’ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))))
1615rexlimdva 3149 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ) β†’ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))))
17 simpll 764 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ (𝐾 ∈ HL ∧ π‘Š ∈ 𝐻))
18 simplr 766 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ 𝐹 ∈ 𝑇)
19 simprl 768 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ 𝑔 ∈ 𝑇)
20 simprr 770 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))
215, 1, 2, 6, 3tendoex 40357 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)) β†’ βˆƒπ‘  ∈ 𝐸 (π‘ β€˜πΉ) = 𝑔)
2217, 18, 19, 20, 21syl121anc 1372 . . . . . 6 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ βˆƒπ‘  ∈ 𝐸 (π‘ β€˜πΉ) = 𝑔)
23 eqcom 2733 . . . . . . 7 ((π‘ β€˜πΉ) = 𝑔 ↔ 𝑔 = (π‘ β€˜πΉ))
2423rexbii 3088 . . . . . 6 (βˆƒπ‘  ∈ 𝐸 (π‘ β€˜πΉ) = 𝑔 ↔ βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ))
2522, 24sylib 217 . . . . 5 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ))
2625ex 412 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ ((𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)) β†’ βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ)))
2716, 26impbid 211 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ) ↔ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))))
2827abbidv 2795 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ {𝑔 ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))})
29 df-rab 3427 . 2 {𝑔 ∈ 𝑇 ∣ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))}
3028, 29eqtr4di 2784 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ {𝑔 ∣ βˆƒπ‘  ∈ 𝐸 𝑔 = (π‘ β€˜πΉ)} = {𝑔 ∈ 𝑇 ∣ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2703  βˆƒwrex 3064  {crab 3426   class class class wbr 5141  β€˜cfv 6536  lecple 17211  HLchlt 38731  LHypclh 39366  LTrncltrn 39483  trLctrl 39540  TEndoctendo 40134
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-riotaBAD 38334
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-undef 8256  df-map 8821  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-p1 18389  df-lat 18395  df-clat 18462  df-oposet 38557  df-ol 38559  df-oml 38560  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732  df-llines 38880  df-lplanes 38881  df-lvols 38882  df-lines 38883  df-psubsp 38885  df-pmap 38886  df-padd 39178  df-lhyp 39370  df-laut 39371  df-ldil 39486  df-ltrn 39487  df-trl 39541  df-tendo 40137
This theorem is referenced by:  dvhb1dimN  40368  dia1dim  40443
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