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Theorem dvhopN 41099
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 40133 . . . . . 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 40752 . . . . . 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 41098 . . . . 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 40756 . . . . . 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 40754 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
1615adantrl 716 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
1714, 16opeq12d 4886 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩ = ⟨( I ↾ 𝐵), 𝑈⟩)
1812, 17eqtrd 2775 . . 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 40773 . . . 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 41097 . . 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 40107 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:𝐵1-1-onto𝐵)
2827adantrr 717 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → 𝐹:𝐵1-1-onto𝐵)
29 f1of 6849 . . . 4 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
30 fcoi1 6783 . . . 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 40774 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑂𝑃𝑈) = 𝑈)
3433adantrl 716 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑂𝑃𝑈) = 𝑈)
3531, 34opeq12d 4886 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩ = ⟨𝐹, 𝑈⟩)
3619, 26, 353eqtrrd 2780 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 1537  wcel 2106  cop 4637  cmpt 5231   I cid 5582   × cxp 5687  cres 5691  ccom 5693  wf 6559  1-1-ontowf1o 6562  cfv 6563  (class class class)co 7431  cmpo 7433  1st c1st 8011  2nd c2nd 8012  Basecbs 17245  HLchlt 39332  LHypclh 39967  LTrncltrn 40084  TEndoctendo 40735
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-riotaBAD 38935
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-undef 8297  df-map 8867  df-proset 18352  df-poset 18371  df-plt 18388  df-lub 18404  df-glb 18405  df-join 18406  df-meet 18407  df-p0 18483  df-p1 18484  df-lat 18490  df-clat 18557  df-oposet 39158  df-ol 39160  df-oml 39161  df-covers 39248  df-ats 39249  df-atl 39280  df-cvlat 39304  df-hlat 39333  df-llines 39481  df-lplanes 39482  df-lvols 39483  df-lines 39484  df-psubsp 39486  df-pmap 39487  df-padd 39779  df-lhyp 39971  df-laut 39972  df-ldil 40087  df-ltrn 40088  df-trl 40142  df-tendo 40738
This theorem is referenced by: (None)
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