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Theorem dvhopN 41118
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 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑐𝑇 ↦ ((𝑎𝑐) ∘ (𝑏𝑐))))
dvhop.a 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)
dvhop.s 𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
dvhop.o 𝑂 = (𝑐𝑇 ↦ ( I ↾ 𝐵))
Assertion
Ref Expression
dvhopN (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂𝐴(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
StepHypRef Expression
1 simprr 773 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → 𝑈𝐸)
2 dvhop.b . . . . . . 7 𝐵 = (Base‘𝐾)
3 dvhop.h . . . . . . 7 𝐻 = (LHyp‘𝐾)
4 dvhop.t . . . . . . 7 𝑇 = ((LTrn‘𝐾)‘𝑊)
52, 3, 4idltrn 40152 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
65adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
7 dvhop.e . . . . . . 7 𝐸 = ((TEndo‘𝐾)‘𝑊)
83, 4, 7tendoidcl 40771 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
98adantr 480 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ( I ↾ 𝑇) ∈ 𝐸)
10 dvhop.s . . . . . 6 𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
1110dvhopspN 41117 . . . . 5 ((𝑈𝐸 ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩)
121, 6, 9, 11syl12anc 837 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩)
132, 3, 7tendoid 40775 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
1413adantrl 716 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
153, 4, 7tendo1mulr 40773 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
1615adantrl 716 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
1714, 16opeq12d 4881 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩ = ⟨( I ↾ 𝐵), 𝑈⟩)
1812, 17eqtrd 2777 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨( I ↾ 𝐵), 𝑈⟩)
1918oveq2d 7447 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (⟨𝐹, 𝑂𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)) = (⟨𝐹, 𝑂𝐴⟨( I ↾ 𝐵), 𝑈⟩))
20 simprl 771 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → 𝐹𝑇)
21 dvhop.o . . . . 5 𝑂 = (𝑐𝑇 ↦ ( I ↾ 𝐵))
222, 3, 4, 7, 21tendo0cl 40792 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑂𝐸)
2322adantr 480 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → 𝑂𝐸)
24 dvhop.a . . . 4 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)
2524dvhopaddN 41116 . . 3 (((𝐹𝑇𝑂𝐸) ∧ (( I ↾ 𝐵) ∈ 𝑇𝑈𝐸)) → (⟨𝐹, 𝑂𝐴⟨( I ↾ 𝐵), 𝑈⟩) = ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩)
2620, 23, 6, 1, 25syl22anc 839 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (⟨𝐹, 𝑂𝐴⟨( I ↾ 𝐵), 𝑈⟩) = ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩)
272, 3, 4ltrn1o 40126 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:𝐵1-1-onto𝐵)
2827adantrr 717 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → 𝐹:𝐵1-1-onto𝐵)
29 f1of 6848 . . . 4 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
30 fcoi1 6782 . . . 4 (𝐹:𝐵𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
3128, 29, 303syl 18 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹)
32 dvhop.p . . . . 5 𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑐𝑇 ↦ ((𝑎𝑐) ∘ (𝑏𝑐))))
332, 3, 4, 7, 21, 32tendo0pl 40793 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑂𝑃𝑈) = 𝑈)
3433adantrl 716 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑂𝑃𝑈) = 𝑈)
3531, 34opeq12d 4881 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩ = ⟨𝐹, 𝑈⟩)
3619, 26, 353eqtrrd 2782 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cop 4632  cmpt 5225   I cid 5577   × cxp 5683  cres 5687  ccom 5689  wf 6557  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cmpo 7433  1st c1st 8012  2nd c2nd 8013  Basecbs 17247  HLchlt 39351  LHypclh 39986  LTrncltrn 40103  TEndoctendo 40754
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-riotaBAD 38954
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-undef 8298  df-map 8868  df-proset 18340  df-poset 18359  df-plt 18375  df-lub 18391  df-glb 18392  df-join 18393  df-meet 18394  df-p0 18470  df-p1 18471  df-lat 18477  df-clat 18544  df-oposet 39177  df-ol 39179  df-oml 39180  df-covers 39267  df-ats 39268  df-atl 39299  df-cvlat 39323  df-hlat 39352  df-llines 39500  df-lplanes 39501  df-lvols 39502  df-lines 39503  df-psubsp 39505  df-pmap 39506  df-padd 39798  df-lhyp 39990  df-laut 39991  df-ldil 40106  df-ltrn 40107  df-trl 40161  df-tendo 40757
This theorem is referenced by: (None)
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