Theorem dva1dim 41355

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 40933. 𝐸 is the division ring base by erngdv 41363, and 𝑠‘𝐹 is the scalar product by dvavsca 41387. 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)} = {𝑔 ∈ 𝑇 ∣ (𝑅‘𝑔) ≤ (𝑅‘𝐹)})

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

Step	Hyp	Ref	Expression
1		dva1dim.h	. . . . . . . . . 10 ⊢ 𝐻 = (LHyp‘𝐾)
2		dva1dim.t	. . . . . . . . . 10 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		dva1dim.e	. . . . . . . . . 10 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4	1, 2, 3	tendocl 41137	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇)
5		dva1dim.l	. . . . . . . . . 10 ⊢ ≤ = (le‘𝐾)
6		dva1dim.r	. . . . . . . . . 10 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
7	5, 1, 2, 6, 3	tendotp 41131	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹))
8	4, 7	jca 511	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))
9	8	3expb 1121	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))
10	9	anass1rs 656	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))
11		eleq1 2825	. . . . . . 7 ⊢ (𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ↔ (𝑠‘𝐹) ∈ 𝑇))
12		fveq2 6842	. . . . . . . 8 ⊢ (𝑔 = (𝑠‘𝐹) → (𝑅‘𝑔) = (𝑅‘(𝑠‘𝐹)))
13	12	breq1d 5110	. . . . . . 7 ⊢ (𝑔 = (𝑠‘𝐹) → ((𝑅‘𝑔) ≤ (𝑅‘𝐹) ↔ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))
14	11, 13	anbi12d 633	. . . . . 6 ⊢ (𝑔 = (𝑠‘𝐹) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)) ↔ ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹))))
15	10, 14	syl5ibrcom 247	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → (𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))))
16	15	rexlimdva 3139	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))))
17		simpll 767	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18		simplr 769	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → 𝐹 ∈ 𝑇)
19		simprl 771	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → 𝑔 ∈ 𝑇)
20		simprr 773	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → (𝑅‘𝑔) ≤ (𝑅‘𝐹))
21	5, 1, 2, 6, 3	tendoex 41345	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)) → ∃𝑠 ∈ 𝐸 (𝑠‘𝐹) = 𝑔)
22	17, 18, 19, 20, 21	syl121anc 1378	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → ∃𝑠 ∈ 𝐸 (𝑠‘𝐹) = 𝑔)
23		eqcom 2744	. . . . . . 7 ⊢ ((𝑠‘𝐹) = 𝑔 ↔ 𝑔 = (𝑠‘𝐹))
24	23	rexbii 3085	. . . . . 6 ⊢ (∃𝑠 ∈ 𝐸 (𝑠‘𝐹) = 𝑔 ↔ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹))
25	22, 24	sylib 218	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹))
26	25	ex 412	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)) → ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)))
27	16, 26	impbid 212	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹) ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))))
28	27	abbidv 2803	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))})
29		df-rab 3402	. 2 ⊢ {𝑔 ∈ 𝑇 ∣ (𝑅‘𝑔) ≤ (𝑅‘𝐹)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))}
30	28, 29	eqtr4di 2790	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)} = {𝑔 ∈ 𝑇 ∣ (𝑅‘𝑔) ≤ (𝑅‘𝐹)})

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {cab 2715  ∃wrex 3062  {crab 3401   class class class wbr 5100  ‘cfv 6500  lecple 17196  HLchlt 39720  LHypclh 40354  LTrncltrn 40471  trLctrl 40528  TEndoctendo 41122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-riotaBAD 39323

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-iin 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-undef 8225  df-map 8777  df-proset 18229  df-poset 18248  df-plt 18263  df-lub 18279  df-glb 18280  df-join 18281  df-meet 18282  df-p0 18358  df-p1 18359  df-lat 18367  df-clat 18434  df-oposet 39546  df-ol 39548  df-oml 39549  df-covers 39636  df-ats 39637  df-atl 39668  df-cvlat 39692  df-hlat 39721  df-llines 39868  df-lplanes 39869  df-lvols 39870  df-lines 39871  df-psubsp 39873  df-pmap 39874  df-padd 40166  df-lhyp 40358  df-laut 40359  df-ldil 40474  df-ltrn 40475  df-trl 40529  df-tendo 41125

This theorem is referenced by:  dvhb1dimN  41356  dia1dim  41431