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Theorem dvhopN 39582
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 β†Ύ 𝐡), ( I β†Ύ 𝑇)⟩, π‘ˆ, ⟨𝐹, π‘‚βŸ©. We swapped the order of vector sum (their juxtaposition i.e. composition) to show ⟨𝐹, π‘‚βŸ© first. Note that 𝑂 and ( I β†Ύ 𝑇) are the zero and one of the division ring 𝐸, and ( I β†Ύ 𝐡) 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 𝐡 = (Baseβ€˜πΎ)
dvhop.h 𝐻 = (LHypβ€˜πΎ)
dvhop.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dvhop.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dvhop.p 𝑃 = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘) ∘ (π‘β€˜π‘))))
dvhop.a 𝐴 = (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), ((2nd β€˜π‘“)𝑃(2nd β€˜π‘”))⟩)
dvhop.s 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)
dvhop.o 𝑂 = (𝑐 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
Assertion
Ref Expression
dvhopN (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ ⟨𝐹, π‘ˆβŸ© = (⟨𝐹, π‘‚βŸ©π΄(π‘ˆπ‘†βŸ¨( I β†Ύ 𝐡), ( I β†Ύ 𝑇)⟩)))
Distinct variable groups:   𝐡,𝑐   π‘Ž,𝑏,𝑓,𝑔,𝑠,𝐸   𝐻,𝑐   𝐾,𝑐   𝑃,𝑓,𝑔   π‘Ž,𝑐,𝑇,𝑏,𝑓,𝑔,𝑠   π‘Š,π‘Ž,𝑏,𝑐
Allowed substitution hints:   𝐴(𝑓,𝑔,𝑠,π‘Ž,𝑏,𝑐)   𝐡(𝑓,𝑔,𝑠,π‘Ž,𝑏)   𝑃(𝑠,π‘Ž,𝑏,𝑐)   𝑆(𝑓,𝑔,𝑠,π‘Ž,𝑏,𝑐)   π‘ˆ(𝑓,𝑔,𝑠,π‘Ž,𝑏,𝑐)   𝐸(𝑐)   𝐹(𝑓,𝑔,𝑠,π‘Ž,𝑏,𝑐)   𝐻(𝑓,𝑔,𝑠,π‘Ž,𝑏)   𝐾(𝑓,𝑔,𝑠,π‘Ž,𝑏)   𝑂(𝑓,𝑔,𝑠,π‘Ž,𝑏,𝑐)   π‘Š(𝑓,𝑔,𝑠)

Proof of Theorem dvhopN
StepHypRef Expression
1 simprr 772 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ π‘ˆ ∈ 𝐸)
2 dvhop.b . . . . . . 7 𝐡 = (Baseβ€˜πΎ)
3 dvhop.h . . . . . . 7 𝐻 = (LHypβ€˜πΎ)
4 dvhop.t . . . . . . 7 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
52, 3, 4idltrn 38616 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
65adantr 482 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
7 dvhop.e . . . . . . 7 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
83, 4, 7tendoidcl 39235 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝑇) ∈ 𝐸)
98adantr 482 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ ( I β†Ύ 𝑇) ∈ 𝐸)
10 dvhop.s . . . . . 6 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)
1110dvhopspN 39581 . . . . 5 ((π‘ˆ ∈ 𝐸 ∧ (( I β†Ύ 𝐡) ∈ 𝑇 ∧ ( I β†Ύ 𝑇) ∈ 𝐸)) β†’ (π‘ˆπ‘†βŸ¨( I β†Ύ 𝐡), ( I β†Ύ 𝑇)⟩) = ⟨(π‘ˆβ€˜( I β†Ύ 𝐡)), (π‘ˆ ∘ ( I β†Ύ 𝑇))⟩)
121, 6, 9, 11syl12anc 836 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (π‘ˆπ‘†βŸ¨( I β†Ύ 𝐡), ( I β†Ύ 𝑇)⟩) = ⟨(π‘ˆβ€˜( I β†Ύ 𝐡)), (π‘ˆ ∘ ( I β†Ύ 𝑇))⟩)
132, 3, 7tendoid 39239 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘ˆβ€˜( I β†Ύ 𝐡)) = ( I β†Ύ 𝐡))
1413adantrl 715 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (π‘ˆβ€˜( I β†Ύ 𝐡)) = ( I β†Ύ 𝐡))
153, 4, 7tendo1mulr 39237 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘ˆ ∘ ( I β†Ύ 𝑇)) = π‘ˆ)
1615adantrl 715 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (π‘ˆ ∘ ( I β†Ύ 𝑇)) = π‘ˆ)
1714, 16opeq12d 4839 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ ⟨(π‘ˆβ€˜( I β†Ύ 𝐡)), (π‘ˆ ∘ ( I β†Ύ 𝑇))⟩ = ⟨( I β†Ύ 𝐡), π‘ˆβŸ©)
1812, 17eqtrd 2777 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (π‘ˆπ‘†βŸ¨( I β†Ύ 𝐡), ( I β†Ύ 𝑇)⟩) = ⟨( I β†Ύ 𝐡), π‘ˆβŸ©)
1918oveq2d 7374 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (⟨𝐹, π‘‚βŸ©π΄(π‘ˆπ‘†βŸ¨( I β†Ύ 𝐡), ( I β†Ύ 𝑇)⟩)) = (⟨𝐹, π‘‚βŸ©π΄βŸ¨( I β†Ύ 𝐡), π‘ˆβŸ©))
20 simprl 770 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ 𝐹 ∈ 𝑇)
21 dvhop.o . . . . 5 𝑂 = (𝑐 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))
222, 3, 4, 7, 21tendo0cl 39256 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑂 ∈ 𝐸)
2322adantr 482 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ 𝑂 ∈ 𝐸)
24 dvhop.a . . . 4 𝐴 = (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), ((2nd β€˜π‘“)𝑃(2nd β€˜π‘”))⟩)
2524dvhopaddN 39580 . . 3 (((𝐹 ∈ 𝑇 ∧ 𝑂 ∈ 𝐸) ∧ (( I β†Ύ 𝐡) ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (⟨𝐹, π‘‚βŸ©π΄βŸ¨( I β†Ύ 𝐡), π‘ˆβŸ©) = ⟨(𝐹 ∘ ( I β†Ύ 𝐡)), (π‘‚π‘ƒπ‘ˆ)⟩)
2620, 23, 6, 1, 25syl22anc 838 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (⟨𝐹, π‘‚βŸ©π΄βŸ¨( I β†Ύ 𝐡), π‘ˆβŸ©) = ⟨(𝐹 ∘ ( I β†Ύ 𝐡)), (π‘‚π‘ƒπ‘ˆ)⟩)
272, 3, 4ltrn1o 38590 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹:𝐡–1-1-onto→𝐡)
2827adantrr 716 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ 𝐹:𝐡–1-1-onto→𝐡)
29 f1of 6785 . . . 4 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹:𝐡⟢𝐡)
30 fcoi1 6717 . . . 4 (𝐹:𝐡⟢𝐡 β†’ (𝐹 ∘ ( I β†Ύ 𝐡)) = 𝐹)
3128, 29, 303syl 18 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (𝐹 ∘ ( I β†Ύ 𝐡)) = 𝐹)
32 dvhop.p . . . . 5 𝑃 = (π‘Ž ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((π‘Žβ€˜π‘) ∘ (π‘β€˜π‘))))
332, 3, 4, 7, 21, 32tendo0pl 39257 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘‚π‘ƒπ‘ˆ) = π‘ˆ)
3433adantrl 715 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (π‘‚π‘ƒπ‘ˆ) = π‘ˆ)
3531, 34opeq12d 4839 . 2 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ ⟨(𝐹 ∘ ( I β†Ύ 𝐡)), (π‘‚π‘ƒπ‘ˆ)⟩ = ⟨𝐹, π‘ˆβŸ©)
3619, 26, 353eqtrrd 2782 1 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ ⟨𝐹, π‘ˆβŸ© = (⟨𝐹, π‘‚βŸ©π΄(π‘ˆπ‘†βŸ¨( I β†Ύ 𝐡), ( I β†Ύ 𝑇)⟩)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βŸ¨cop 4593   ↦ cmpt 5189   I cid 5531   Γ— cxp 5632   β†Ύ cres 5636   ∘ ccom 5638  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  Basecbs 17084  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  TEndoctendo 39218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-riotaBAD 37418
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-undef 8205  df-map 8768  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-clat 18389  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-llines 37964  df-lplanes 37965  df-lvols 37966  df-lines 37967  df-psubsp 37969  df-pmap 37970  df-padd 38262  df-lhyp 38454  df-laut 38455  df-ldil 38570  df-ltrn 38571  df-trl 38625  df-tendo 39221
This theorem is referenced by: (None)
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