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Theorem dva1dim 40462
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 40040. 𝐸 is the division ring base by erngdv 40470, and π‘ β€˜πΉ is the scalar product by dvavsca 40494. 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 40244 . . . . . . . . 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 40238 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘ β€˜πΉ)) ≀ (π‘…β€˜πΉ))
84, 7jca 510 . . . . . . . 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 653 . . . . . 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 2816 . . . . . . 7 (𝑔 = (π‘ β€˜πΉ) β†’ (𝑔 ∈ 𝑇 ↔ (π‘ β€˜πΉ) ∈ 𝑇))
12 fveq2 6900 . . . . . . . 8 (𝑔 = (π‘ β€˜πΉ) β†’ (π‘…β€˜π‘”) = (π‘…β€˜(π‘ β€˜πΉ)))
1312breq1d 5160 . . . . . . 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 3151 . . . 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 765 . . . . . . 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 767 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ 𝐹 ∈ 𝑇)
19 simprl 769 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ 𝑔 ∈ 𝑇)
20 simprr 771 . . . . . . 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 40452 . . . . . . 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 2734 . . . . . . 7 ((π‘ β€˜πΉ) = 𝑔 ↔ 𝑔 = (π‘ β€˜πΉ))
2423rexbii 3090 . . . . . 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 411 . . . 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 2796 . 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 3429 . 2 {𝑔 ∈ 𝑇 ∣ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))}
3028, 29eqtr4di 2785 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  {cab 2704  βˆƒwrex 3066  {crab 3428   class class class wbr 5150  β€˜cfv 6551  lecple 17245  HLchlt 38826  LHypclh 39461  LTrncltrn 39578  trLctrl 39635  TEndoctendo 40229
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744  ax-riotaBAD 38429
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 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-iin 5001  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-undef 8283  df-map 8851  df-proset 18292  df-poset 18310  df-plt 18327  df-lub 18343  df-glb 18344  df-join 18345  df-meet 18346  df-p0 18422  df-p1 18423  df-lat 18429  df-clat 18496  df-oposet 38652  df-ol 38654  df-oml 38655  df-covers 38742  df-ats 38743  df-atl 38774  df-cvlat 38798  df-hlat 38827  df-llines 38975  df-lplanes 38976  df-lvols 38977  df-lines 38978  df-psubsp 38980  df-pmap 38981  df-padd 39273  df-lhyp 39465  df-laut 39466  df-ldil 39581  df-ltrn 39582  df-trl 39636  df-tendo 40232
This theorem is referenced by:  dvhb1dimN  40463  dia1dim  40538
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