Theorem dva1dim 41791

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 41369. 𝐸 is the division ring base by erngdv 41799, and 𝑠‘𝐹 is the scalar product by dvavsca 41823. 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 41573	. . . . . . . . 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 41567	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹))
8	4, 7	jca 521	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))
9	8	3expb 1138	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))
10	9	anass1rs 668	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))
11		eleq1 2854	. . . . . . 7 ⊢ (𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ↔ (𝑠‘𝐹) ∈ 𝑇))
12		fveq2 6885	. . . . . . . 8 ⊢ (𝑔 = (𝑠‘𝐹) → (𝑅‘𝑔) = (𝑅‘(𝑠‘𝐹)))
13	12	breq1d 5122	. . . . . . 7 ⊢ (𝑔 = (𝑠‘𝐹) → ((𝑅‘𝑔) ≤ (𝑅‘𝐹) ↔ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹)))
14	11, 13	anbi12d 644	. . . . . 6 ⊢ (𝑔 = (𝑠‘𝐹) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)) ↔ ((𝑠‘𝐹) ∈ 𝑇 ∧ (𝑅‘(𝑠‘𝐹)) ≤ (𝑅‘𝐹))))
15	10, 14	syl5ibrcom 250	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ 𝑠 ∈ 𝐸) → (𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))))
16	15	rexlimdva 3169	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹) → (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))))
17		simpll 779	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
18		simplr 781	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → 𝐹 ∈ 𝑇)
19		simprl 783	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → 𝑔 ∈ 𝑇)
20		simprr 785	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → (𝑅‘𝑔) ≤ (𝑅‘𝐹))
21	5, 1, 2, 6, 3	tendoex 41781	. . . . . . 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 1402	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → ∃𝑠 ∈ 𝐸 (𝑠‘𝐹) = 𝑔)
23		eqcom 2773	. . . . . . 7 ⊢ ((𝑠‘𝐹) = 𝑔 ↔ 𝑔 = (𝑠‘𝐹))
24	23	rexbii 3115	. . . . . 6 ⊢ (∃𝑠 ∈ 𝐸 (𝑠‘𝐹) = 𝑔 ↔ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹))
25	22, 24	sylib 221	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) ∧ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))) → ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹))
26	25	ex 418	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → ((𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹)) → ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)))
27	16, 26	impbid 215	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹) ↔ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))))
28	27	abbidv 2832	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))})
29		df-rab 3420	. 2 ⊢ {𝑔 ∈ 𝑇 ∣ (𝑅‘𝑔) ≤ (𝑅‘𝐹)} = {𝑔 ∣ (𝑔 ∈ 𝑇 ∧ (𝑅‘𝑔) ≤ (𝑅‘𝐹))}
30	28, 29	eqtr4di 2819	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)} = {𝑔 ∈ 𝑇 ∣ (𝑅‘𝑔) ≤ (𝑅‘𝐹)})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  {cab 2744  ∃wrex 3092  {crab 3419   class class class wbr 5112  ‘cfv 6540  lecple 17327  HLchlt 40156  LHypclh 40790  LTrncltrn 40907  trLctrl 40964  TEndoctendo 41558

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-rep 5241  ax-sep 5260  ax-nul 5272  ax-pow 5339  ax-pr 5407  ax-un 7738  ax-riotaBAD 39759

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-reu 3373  df-rab 3420  df-v 3460  df-sbc 3748  df-csb 3857  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4876  df-iun 4961  df-iin 4962  df-br 5113  df-opab 5177  df-mpt 5196  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7988  df-2nd 7989  df-undef 8271  df-map 8828  df-proset 18360  df-poset 18379  df-plt 18394  df-lub 18410  df-glb 18411  df-join 18412  df-meet 18413  df-p0 18489  df-p1 18490  df-lat 18498  df-clat 18565  df-oposet 39982  df-ol 39984  df-oml 39985  df-covers 40072  df-ats 40073  df-atl 40104  df-cvlat 40128  df-hlat 40157  df-llines 40304  df-lplanes 40305  df-lvols 40306  df-lines 40307  df-psubsp 40309  df-pmap 40310  df-padd 40602  df-lhyp 40794  df-laut 40795  df-ldil 40910  df-ltrn 40911  df-trl 40965  df-tendo 41561

This theorem is used by:  dvhb1dimN  41792  dia1dim  41867