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Theorem dvhopN 39987
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 39021 . . . . . 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 39640 . . . . . 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 39986 . . . . 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 39644 . . . . . 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 39642 . . . . . 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 4882 . . . 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 2773 . . 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 7425 . 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 39661 . . . 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 39985 . . 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 38995 . . . . 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 6834 . . . 4 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹:𝐡⟢𝐡)
30 fcoi1 6766 . . . 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 39662 . . . 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 4882 . 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 2778 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 4635   ↦ cmpt 5232   I cid 5574   Γ— cxp 5675   β†Ύ cres 5679   ∘ ccom 5681  βŸΆwf 6540  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1st c1st 7973  2nd c2nd 7974  Basecbs 17144  HLchlt 38220  LHypclh 38855  LTrncltrn 38972  TEndoctendo 39623
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-riotaBAD 37823
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-1st 7975  df-2nd 7976  df-undef 8258  df-map 8822  df-proset 18248  df-poset 18266  df-plt 18283  df-lub 18299  df-glb 18300  df-join 18301  df-meet 18302  df-p0 18378  df-p1 18379  df-lat 18385  df-clat 18452  df-oposet 38046  df-ol 38048  df-oml 38049  df-covers 38136  df-ats 38137  df-atl 38168  df-cvlat 38192  df-hlat 38221  df-llines 38369  df-lplanes 38370  df-lvols 38371  df-lines 38372  df-psubsp 38374  df-pmap 38375  df-padd 38667  df-lhyp 38859  df-laut 38860  df-ldil 38975  df-ltrn 38976  df-trl 39030  df-tendo 39626
This theorem is referenced by: (None)
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