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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 773 | . . . . 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 40152 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝐵) ∈ 𝑇) | 
| 6 | 5 | adantr 480 | . . . . 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 40771 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸) | 
| 9 | 8 | adantr 480 | . . . . 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 41117 | . . . . 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 837 | . . . 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 40775 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) | 
| 14 | 13 | adantrl 716 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈‘( I ↾ 𝐵)) = ( I ↾ 𝐵)) | 
| 15 | 3, 4, 7 | tendo1mulr 40773 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈) | 
| 16 | 15 | adantrl 716 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈) | 
| 17 | 14, 16 | opeq12d 4881 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈(𝑈‘( I ↾ 𝐵)), (𝑈 ∘ ( I ↾ 𝑇))〉 = 〈( I ↾ 𝐵), 𝑈〉) | 
| 18 | 12, 17 | eqtrd 2777 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑈𝑆〈( I ↾ 𝐵), ( I ↾ 𝑇)〉) = 〈( I ↾ 𝐵), 𝑈〉) | 
| 19 | 18 | oveq2d 7447 | . 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 771 | . . 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 40792 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) | 
| 23 | 22 | adantr 480 | . . 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 41116 | . . 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 839 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (〈𝐹, 𝑂〉𝐴〈( I ↾ 𝐵), 𝑈〉) = 〈(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)〉) | 
| 27 | 2, 3, 4 | ltrn1o 40126 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹:𝐵–1-1-onto→𝐵) | 
| 28 | 27 | adantrr 717 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 𝐹:𝐵–1-1-onto→𝐵) | 
| 29 | f1of 6848 | . . . 4 ⊢ (𝐹:𝐵–1-1-onto→𝐵 → 𝐹:𝐵⟶𝐵) | |
| 30 | fcoi1 6782 | . . . 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 40793 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑂𝑃𝑈) = 𝑈) | 
| 34 | 33 | adantrl 716 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑂𝑃𝑈) = 𝑈) | 
| 35 | 31, 34 | opeq12d 4881 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈(𝐹 ∘ ( I ↾ 𝐵)), (𝑂𝑃𝑈)〉 = 〈𝐹, 𝑈〉) | 
| 36 | 19, 26, 35 | 3eqtrrd 2782 | 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 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 ↦ cmpt 5225 I cid 5577 × cxp 5683 ↾ cres 5687 ∘ ccom 5689 ⟶wf 6557 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 1st c1st 8012 2nd c2nd 8013 Basecbs 17247 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 TEndoctendo 40754 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-riotaBAD 38954 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-undef 8298 df-map 8868 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 df-llines 39500 df-lplanes 39501 df-lvols 39502 df-lines 39503 df-psubsp 39505 df-pmap 39506 df-padd 39798 df-lhyp 39990 df-laut 39991 df-ldil 40106 df-ltrn 40107 df-trl 40161 df-tendo 40757 | 
| This theorem is referenced by: (None) | 
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