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Theorem dvhopN 38374
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 772 . . . . 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 37408 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)
65adantr 484 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ( I ↾ 𝐵) ∈ 𝑇)
7 dvhop.e . . . . . . 7 𝐸 = ((TEndo‘𝐾)‘𝑊)
83, 4, 7tendoidcl 38027 . . . . . 6 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
98adantr 484 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ( I ↾ 𝑇) ∈ 𝐸)
10 dvhop.s . . . . . 6 𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)
1110dvhopspN 38373 . . . . 5 ((𝑈𝐸 ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩)
121, 6, 9, 11syl12anc 835 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩)
132, 3, 7tendoid 38031 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
1413adantrl 715 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
153, 4, 7tendo1mulr 38029 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
1615adantrl 715 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
1714, 16opeq12d 4786 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))⟩ = ⟨( I ↾ 𝐵), 𝑈⟩)
1812, 17eqtrd 2857 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩) = ⟨( I ↾ 𝐵), 𝑈⟩)
1918oveq2d 7156 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (⟨𝐹, 𝑂𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)) = (⟨𝐹, 𝑂𝐴⟨( I ↾ 𝐵), 𝑈⟩))
20 simprl 770 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → 𝐹𝑇)
21 dvhop.o . . . . 5 𝑂 = (𝑐𝑇 ↦ ( I ↾ 𝐵))
222, 3, 4, 7, 21tendo0cl 38048 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑂𝐸)
2322adantr 484 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → 𝑂𝐸)
24 dvhop.a . . . 4 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)
2524dvhopaddN 38372 . . 3 (((𝐹𝑇𝑂𝐸) ∧ (( I ↾ 𝐵) ∈ 𝑇𝑈𝐸)) → (⟨𝐹, 𝑂𝐴⟨( I ↾ 𝐵), 𝑈⟩) = ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩)
2620, 23, 6, 1, 25syl22anc 837 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (⟨𝐹, 𝑂𝐴⟨( I ↾ 𝐵), 𝑈⟩) = ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩)
272, 3, 4ltrn1o 37382 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹:𝐵1-1-onto𝐵)
2827adantrr 716 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → 𝐹:𝐵1-1-onto𝐵)
29 f1of 6597 . . . 4 (𝐹:𝐵1-1-onto𝐵𝐹:𝐵𝐵)
30 fcoi1 6533 . . . 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 38049 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑈𝐸) → (𝑂𝑃𝑈) = 𝑈)
3433adantrl 715 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → (𝑂𝑃𝑈) = 𝑈)
3531, 34opeq12d 4786 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)⟩ = ⟨𝐹, 𝑈⟩)
3619, 26, 353eqtrrd 2862 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 399   = wceq 1538  wcel 2114  cop 4545  cmpt 5122   I cid 5436   × cxp 5530  cres 5534  ccom 5536  wf 6330  1-1-ontowf1o 6333  cfv 6334  (class class class)co 7140  cmpo 7142  1st c1st 7673  2nd c2nd 7674  Basecbs 16474  HLchlt 36608  LHypclh 37242  LTrncltrn 37359  TEndoctendo 38010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-riotaBAD 36211
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-iin 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-undef 7926  df-map 8395  df-proset 17529  df-poset 17547  df-plt 17559  df-lub 17575  df-glb 17576  df-join 17577  df-meet 17578  df-p0 17640  df-p1 17641  df-lat 17647  df-clat 17709  df-oposet 36434  df-ol 36436  df-oml 36437  df-covers 36524  df-ats 36525  df-atl 36556  df-cvlat 36580  df-hlat 36609  df-llines 36756  df-lplanes 36757  df-lvols 36758  df-lines 36759  df-psubsp 36761  df-pmap 36762  df-padd 37054  df-lhyp 37246  df-laut 37247  df-ldil 37362  df-ltrn 37363  df-trl 37417  df-tendo 38013
This theorem is referenced by: (None)
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