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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhopN | Structured version Visualization version GIF version |
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.) |
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 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐)))) |
dvhop.a | ⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))〉) |
dvhop.s | ⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) |
dvhop.o | ⊢ 𝑂 = (𝑐 ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
Ref | Expression |
---|---|
dvhopN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈𝐹, 𝑈〉 = (〈𝐹, 𝑂〉𝐴(𝑈𝑆〈( I ↾ 𝐵), ( I ↾ 𝑇)〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 772 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝑈 ∈ 𝐸) | |
2 | dvhop.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐾) | |
3 | dvhop.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dvhop.t | . . . . . . 7 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | 2, 3, 4 | idltrn 37446 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) |
6 | 5 | adantr 484 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ( I ↾ 𝐵) ∈ 𝑇) |
7 | dvhop.e | . . . . . . 7 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
8 | 3, 4, 7 | tendoidcl 38065 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) |
9 | 8 | adantr 484 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ( I ↾ 𝑇) ∈ 𝐸) |
10 | dvhop.s | . . . . . 6 ⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) | |
11 | 10 | dvhopspN 38411 | . . . . 5 ⊢ ((𝑈 ∈ 𝐸 ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ ( I ↾ 𝑇) ∈ 𝐸)) → (𝑈𝑆〈( I ↾ 𝐵), ( I ↾ 𝑇)〉) = 〈(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))〉) |
12 | 1, 6, 9, 11 | syl12anc 835 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈𝑆〈( I ↾ 𝐵), ( I ↾ 𝑇)〉) = 〈(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))〉) |
13 | 2, 3, 7 | tendoid 38069 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
14 | 13 | adantrl 715 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) |
15 | 3, 4, 7 | tendo1mulr 38067 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈) |
16 | 15 | adantrl 715 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈) |
17 | 14, 16 | opeq12d 4773 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))〉 = 〈( I ↾ 𝐵), 𝑈〉) |
18 | 12, 17 | eqtrd 2833 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈𝑆〈( I ↾ 𝐵), ( I ↾ 𝑇)〉) = 〈( I ↾ 𝐵), 𝑈〉) |
19 | 18 | oveq2d 7151 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (〈𝐹, 𝑂〉𝐴(𝑈𝑆〈( I ↾ 𝐵), ( I ↾ 𝑇)〉)) = (〈𝐹, 𝑂〉𝐴〈( I ↾ 𝐵), 𝑈〉)) |
20 | simprl 770 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝐹 ∈ 𝑇) | |
21 | dvhop.o | . . . . 5 ⊢ 𝑂 = (𝑐 ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
22 | 2, 3, 4, 7, 21 | tendo0cl 38086 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
23 | 22 | adantr 484 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝑂 ∈ 𝐸) |
24 | dvhop.a | . . . 4 ⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))〉) | |
25 | 24 | dvhopaddN 38410 | . . 3 ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑂 ∈ 𝐸) ∧ (( I ↾ 𝐵) ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (〈𝐹, 𝑂〉𝐴〈( I ↾ 𝐵), 𝑈〉) = 〈(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)〉) |
26 | 20, 23, 6, 1, 25 | syl22anc 837 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (〈𝐹, 𝑂〉𝐴〈( I ↾ 𝐵), 𝑈〉) = 〈(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)〉) |
27 | 2, 3, 4 | ltrn1o 37420 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) |
28 | 27 | adantrr 716 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝐹:𝐵–1-1-onto→𝐵) |
29 | f1of 6590 | . . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵) | |
30 | fcoi1 6526 | . . . 4 ⊢ (𝐹:𝐵⟶𝐵 → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹) | |
31 | 28, 29, 30 | 3syl 18 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝐹 ∘ ( I ↾ 𝐵)) = 𝐹) |
32 | dvhop.p | . . . . 5 ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐)))) | |
33 | 2, 3, 4, 7, 21, 32 | tendo0pl 38087 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑂𝑃𝑈) = 𝑈) |
34 | 33 | adantrl 715 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑂𝑃𝑈) = 𝑈) |
35 | 31, 34 | opeq12d 4773 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)〉 = 〈𝐹, 𝑈〉) |
36 | 19, 26, 35 | 3eqtrrd 2838 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈𝐹, 𝑈〉 = (〈𝐹, 𝑂〉𝐴(𝑈𝑆〈( I ↾ 𝐵), ( I ↾ 𝑇)〉))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 〈cop 4531 ↦ cmpt 5110 I cid 5424 × cxp 5517 ↾ cres 5521 ∘ ccom 5523 ⟶wf 6320 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 1st c1st 7669 2nd c2nd 7670 Basecbs 16475 HLchlt 36646 LHypclh 37280 LTrncltrn 37397 TEndoctendo 38048 |
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 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-riotaBAD 36249 |
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 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-undef 7922 df-map 8391 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tendo 38051 |
This theorem is referenced by: (None) |
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