Theorem dvhopN 40719

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 771	. . . . 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 39753	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
6	5	adantr 479	. . . . 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 40372	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
9	8	adantr 479	. . . . 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 40718	. . . . 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 835	. . . 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 40376	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
14	13	adantrl 714	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
15	3, 4, 7	tendo1mulr 40374	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
16	15	adantrl 714	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
17	14, 16	opeq12d 4883	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩ = ⟨( I ↾ 𝐵), 𝑈⟩)
18	12, 17	eqtrd 2765	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨( I ↾ 𝐵), 𝑈⟩)
19	18	oveq2d 7435	. 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 769	. . 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 40393	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸)
23	22	adantr 479	. . 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 40717	. . 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 837	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (⟨𝐹, 𝑂⟩𝐴⟨( I ↾ 𝐵), 𝑈⟩) = ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩)
27	2, 3, 4	ltrn1o 39727	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵)
28	27	adantrr 715	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝐹:𝐵–1-1-onto→𝐵)
29		f1of 6838	. . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵)
30		fcoi1 6771	. . . 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 40394	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑂𝑃𝑈) = 𝑈)
34	33	adantrl 714	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑂𝑃𝑈) = 𝑈)
35	31, 34	opeq12d 4883	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩ = ⟨𝐹, 𝑈⟩)
36	19, 26, 35	3eqtrrd 2770	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂⟩𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ⟨cop 4636   ↦ cmpt 5232   I cid 5575   × cxp 5676   ↾ cres 5680   ∘ ccom 5682  ⟶wf 6545  –1-1-onto→wf1o 6548  ‘cfv 6549  (class class class)co 7419   ∈ cmpo 7421  1^st c1st 7992  2^nd c2nd 7993  Basecbs 17183  HLchlt 38952  LHypclh 39587  LTrncltrn 39704  TEndoctendo 40355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-riotaBAD 38555

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-iin 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7994  df-2nd 7995  df-undef 8279  df-map 8847  df-proset 18290  df-poset 18308  df-plt 18325  df-lub 18341  df-glb 18342  df-join 18343  df-meet 18344  df-p0 18420  df-p1 18421  df-lat 18427  df-clat 18494  df-oposet 38778  df-ol 38780  df-oml 38781  df-covers 38868  df-ats 38869  df-atl 38900  df-cvlat 38924  df-hlat 38953  df-llines 39101  df-lplanes 39102  df-lvols 39103  df-lines 39104  df-psubsp 39106  df-pmap 39107  df-padd 39399  df-lhyp 39591  df-laut 39592  df-ldil 39707  df-ltrn 39708  df-trl 39762  df-tendo 40358

This theorem is referenced by: (None)