Theorem dvhopN 35423

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 792	. . . . 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 34454	. . . . . 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 35075	. . . . . 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 35422	. . . . 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 1316	. . . 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 35079	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
14	13	adantrl 748	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
15	3, 4, 7	tendo1mulr 35077	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
16	15	adantrl 748	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
17	14, 16	opeq12d 4348	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩ = ⟨( I ↾ 𝐵), 𝑈⟩)
18	12, 17	eqtrd 2644	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨( I ↾ 𝐵), 𝑈⟩)
19	18	oveq2d 6565	. 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 790	. . 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 35096	. . . 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 35421	. . 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 1319	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (⟨𝐹, 𝑂⟩𝐴⟨( I ↾ 𝐵), 𝑈⟩) = ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩)
27	2, 3, 4	ltrn1o 34428	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
28	27	adantrr 749	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝐹:𝐵–1-1-onto→𝐵)
29		f1of 6050	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
30		fcoi1 5991	. . . 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 35097	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑂𝑃𝑈) = 𝑈)
34	33	adantrl 748	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑂𝑃𝑈) = 𝑈)
35	31, 34	opeq12d 4348	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩ = ⟨𝐹, 𝑈⟩)
36	19, 26, 35	3eqtrrd 2649	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂⟩𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ⟨cop 4131   ↦ cmpt 4643   I cid 4948   × cxp 5036   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1^st c1st 7057  2^nd c2nd 7058  Basecbs 15695  HLchlt 33655  LHypclh 34288  LTrncltrn 34405  TEndoctendo 35058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-riotaBAD 33257

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-undef 7286  df-map 7746  df-preset 16751  df-poset 16769  df-plt 16781  df-lub 16797  df-glb 16798  df-join 16799  df-meet 16800  df-p0 16862  df-p1 16863  df-lat 16869  df-clat 16931  df-oposet 33481  df-ol 33483  df-oml 33484  df-covers 33571  df-ats 33572  df-atl 33603  df-cvlat 33627  df-hlat 33656  df-llines 33802  df-lplanes 33803  df-lvols 33804  df-lines 33805  df-psubsp 33807  df-pmap 33808  df-padd 34100  df-lhyp 34292  df-laut 34293  df-ldil 34408  df-ltrn 34409  df-trl 34464  df-tendo 35061

This theorem is referenced by: (None)