Theorem dvhopN 41110

Description: Decompose a DVecH vector expressed as an ordered pair into the sum of two components, the first from the translation group vector base of DVecA and the other from the one-dimensional vector subspace 𝐸. Part of Lemma M of [Crawley] p. 121, line 18. We represent their e, sigma, f by ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩, 𝑈, ⟨𝐹, 𝑂⟩. We swapped the order of vector sum (their juxtaposition i.e. composition) to show ⟨𝐹, 𝑂⟩ first. Note that 𝑂 and ( I ↾ 𝑇) are the zero and one of the division ring 𝐸, and ( I ↾ 𝐵) is the zero of the translation group. 𝑆 is the scalar product. (Contributed by NM, 21-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dvhop.b	⊢ 𝐵 = (Base‘𝐾)
dvhop.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhop.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhop.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhop.p	⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))
dvhop.a	⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))⟩)
dvhop.s	⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
dvhop.o	⊢ 𝑂 = (𝑐 ∈ 𝑇 ↦ ( I ↾ 𝐵))

Assertion

Ref	Expression
dvhopN	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂⟩𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)))

Distinct variable groups:   𝐵,𝑐   𝑎,𝑏,𝑓,𝑔,𝑠,𝐸   𝐻,𝑐   𝐾,𝑐   𝑃,𝑓,𝑔   𝑎,𝑐,𝑇,𝑏,𝑓,𝑔,𝑠   𝑊,𝑎,𝑏,𝑐

Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠,𝑎,𝑏,𝑐)   𝐵(𝑓,𝑔,𝑠,𝑎,𝑏)   𝑃(𝑠,𝑎,𝑏,𝑐)   𝑆(𝑓,𝑔,𝑠,𝑎,𝑏,𝑐)   𝑈(𝑓,𝑔,𝑠,𝑎,𝑏,𝑐)   𝐸(𝑐)   𝐹(𝑓,𝑔,𝑠,𝑎,𝑏,𝑐)   𝐻(𝑓,𝑔,𝑠,𝑎,𝑏)   𝐾(𝑓,𝑔,𝑠,𝑎,𝑏)   𝑂(𝑓,𝑔,𝑠,𝑎,𝑏,𝑐)   𝑊(𝑓,𝑔,𝑠)

Proof of Theorem dvhopN

Step	Hyp	Ref	Expression
1		simprr 772	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝑈 ∈ 𝐸)
2		dvhop.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐾)
3		dvhop.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
4		dvhop.t	. . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5	2, 3, 4	idltrn 40144	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
6	5	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
7		dvhop.e	. . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
8	3, 4, 7	tendoidcl 40763	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
9	8	adantr 480	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ( I ↾ 𝑇) ∈ 𝐸)
10		dvhop.s	. . . . . 6 ⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)
11	10	dvhopspN 41109	. . . . 5 ⊢ ((𝑈 ∈ 𝐸 ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩)
12	1, 6, 9, 11	syl12anc 836	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩)
13	2, 3, 7	tendoid 40767	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
14	13	adantrl 716	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
15	3, 4, 7	tendo1mulr 40765	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
16	15	adantrl 716	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
17	14, 16	opeq12d 4845	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩ = ⟨( I ↾ 𝐵), 𝑈⟩)
18	12, 17	eqtrd 2764	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨( I ↾ 𝐵), 𝑈⟩)
19	18	oveq2d 7403	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (⟨𝐹, 𝑂⟩𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)) = (⟨𝐹, 𝑂⟩𝐴⟨( I ↾ 𝐵), 𝑈⟩))
20		simprl 770	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝐹 ∈ 𝑇)
21		dvhop.o	. . . . 5 ⊢ 𝑂 = (𝑐 ∈ 𝑇 ↦ ( I ↾ 𝐵))
22	2, 3, 4, 7, 21	tendo0cl 40784	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
23	22	adantr 480	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝑂 ∈ 𝐸)
24		dvhop.a	. . . 4 ⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))⟩)
25	24	dvhopaddN 41108	. . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑂 ∈ 𝐸) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (⟨𝐹, 𝑂⟩𝐴⟨( I ↾ 𝐵), 𝑈⟩) = ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩)
26	20, 23, 6, 1, 25	syl22anc 838	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (⟨𝐹, 𝑂⟩𝐴⟨( I ↾ 𝐵), 𝑈⟩) = ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩)
27	2, 3, 4	ltrn1o 40118	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
28	27	adantrr 717	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝐹:𝐵–1-1-onto→𝐵)
29		f1of 6800	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
30		fcoi1 6734	. . . 4 ⊢ (𝐹:𝐵⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
31	28, 29, 30	3syl 18	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
32		dvhop.p	. . . . 5 ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))
33	2, 3, 4, 7, 21, 32	tendo0pl 40785	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑂𝑃𝑈) = 𝑈)
34	33	adantrl 716	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑂𝑃𝑈) = 𝑈)
35	31, 34	opeq12d 4845	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩ = ⟨𝐹, 𝑈⟩)
36	19, 26, 35	3eqtrrd 2769	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂⟩𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   ↦ cmpt 5188   I cid 5532   × cxp 5636   ↾ cres 5640   ∘ ccom 5642  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387   ∈ cmpo 7389  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179  HLchlt 39343  LHypclh 39978  LTrncltrn 40095  TEndoctendo 40746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-riotaBAD 38946

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-undef 8252  df-map 8801  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-llines 39492  df-lplanes 39493  df-lvols 39494  df-lines 39495  df-psubsp 39497  df-pmap 39498  df-padd 39790  df-lhyp 39982  df-laut 39983  df-ldil 40098  df-ltrn 40099  df-trl 40153  df-tendo 40749

This theorem is referenced by: (None)