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Theorem dva1dim 39498
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 39076. 𝐸 is the division ring base by erngdv 39506, and π‘ β€˜πΉ is the scalar product by dvavsca 39530. 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 39280 . . . . . . . . 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 39274 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘ β€˜πΉ)) ≀ (π‘…β€˜πΉ))
84, 7jca 513 . . . . . . . 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 1121 . . . . . . 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 654 . . . . . 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 2822 . . . . . . 7 (𝑔 = (π‘ β€˜πΉ) β†’ (𝑔 ∈ 𝑇 ↔ (π‘ β€˜πΉ) ∈ 𝑇))
12 fveq2 6846 . . . . . . . 8 (𝑔 = (π‘ β€˜πΉ) β†’ (π‘…β€˜π‘”) = (π‘…β€˜(π‘ β€˜πΉ)))
1312breq1d 5119 . . . . . . 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 632 . . . . . 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 247 . . . . 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 766 . . . . . . 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 768 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ 𝐹 ∈ 𝑇)
19 simprl 770 . . . . . . 7 ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))) β†’ 𝑔 ∈ 𝑇)
20 simprr 772 . . . . . . 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 39488 . . . . . . 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 1376 . . . . . 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 2740 . . . . . . 7 ((π‘ β€˜πΉ) = 𝑔 ↔ 𝑔 = (π‘ β€˜πΉ))
2423rexbii 3094 . . . . . 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 414 . . . 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 2802 . 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 3407 . 2 {𝑔 ∈ 𝑇 ∣ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (π‘…β€˜π‘”) ≀ (π‘…β€˜πΉ))}
3028, 29eqtr4di 2791 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆƒwrex 3070  {crab 3406   class class class wbr 5109  β€˜cfv 6500  lecple 17148  HLchlt 37862  LHypclh 38497  LTrncltrn 38614  trLctrl 38671  TEndoctendo 39265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-riotaBAD 37465
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-iin 4961  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7925  df-2nd 7926  df-undef 8208  df-map 8773  df-proset 18192  df-poset 18210  df-plt 18227  df-lub 18243  df-glb 18244  df-join 18245  df-meet 18246  df-p0 18322  df-p1 18323  df-lat 18329  df-clat 18396  df-oposet 37688  df-ol 37690  df-oml 37691  df-covers 37778  df-ats 37779  df-atl 37810  df-cvlat 37834  df-hlat 37863  df-llines 38011  df-lplanes 38012  df-lvols 38013  df-lines 38014  df-psubsp 38016  df-pmap 38017  df-padd 38309  df-lhyp 38501  df-laut 38502  df-ldil 38617  df-ltrn 38618  df-trl 38672  df-tendo 39268
This theorem is referenced by:  dvhb1dimN  39499  dia1dim  39574
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