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Theorem dvhopN 40498
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 770 . . . . 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 39532 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝐡) ∈ 𝑇)
65adantr 480 . . . . 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 40151 . . . . . 6 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝑇) ∈ 𝐸)
98adantr 480 . . . . 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 40497 . . . . 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 834 . . . 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 40155 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘ˆβ€˜( I β†Ύ 𝐡)) = ( I β†Ύ 𝐡))
1413adantrl 713 . . . . 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 40153 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘ˆ ∘ ( I β†Ύ 𝑇)) = π‘ˆ)
1615adantrl 713 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (π‘ˆ ∘ ( I β†Ύ 𝑇)) = π‘ˆ)
1714, 16opeq12d 4876 . . . 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 2766 . . 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 7420 . 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 768 . . 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 40172 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝑂 ∈ 𝐸)
2322adantr 480 . . 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 40496 . . 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 836 . 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 39506 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ 𝐹:𝐡–1-1-onto→𝐡)
2827adantrr 714 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ 𝐹:𝐡–1-1-onto→𝐡)
29 f1of 6826 . . . 4 (𝐹:𝐡–1-1-onto→𝐡 β†’ 𝐹:𝐡⟢𝐡)
30 fcoi1 6758 . . . 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 40173 . . . 4 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘‚π‘ƒπ‘ˆ) = π‘ˆ)
3433adantrl 713 . . 3 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ π‘ˆ ∈ 𝐸)) β†’ (π‘‚π‘ƒπ‘ˆ) = π‘ˆ)
3531, 34opeq12d 4876 . 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 2771 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 395   = wceq 1533   ∈ wcel 2098  βŸ¨cop 4629   ↦ cmpt 5224   I cid 5566   Γ— cxp 5667   β†Ύ cres 5671   ∘ ccom 5673  βŸΆwf 6532  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  (class class class)co 7404   ∈ cmpo 7406  1st c1st 7969  2nd c2nd 7970  Basecbs 17151  HLchlt 38731  LHypclh 39366  LTrncltrn 39483  TEndoctendo 40134
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-riotaBAD 38334
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-undef 8256  df-map 8821  df-proset 18258  df-poset 18276  df-plt 18293  df-lub 18309  df-glb 18310  df-join 18311  df-meet 18312  df-p0 18388  df-p1 18389  df-lat 18395  df-clat 18462  df-oposet 38557  df-ol 38559  df-oml 38560  df-covers 38647  df-ats 38648  df-atl 38679  df-cvlat 38703  df-hlat 38732  df-llines 38880  df-lplanes 38881  df-lvols 38882  df-lines 38883  df-psubsp 38885  df-pmap 38886  df-padd 39178  df-lhyp 39370  df-laut 39371  df-ldil 39486  df-ltrn 39487  df-trl 39541  df-tendo 40137
This theorem is referenced by: (None)
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