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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhvaddcl | Structured version Visualization version GIF version |
Description: Closure of the vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) |
Ref | Expression |
---|---|
dvhvaddcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvhvaddcl.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dvhvaddcl.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dvhvaddcl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dvhvaddcl.d | ⊢ 𝐷 = (Scalar‘𝑈) |
dvhvaddcl.p | ⊢ ⨣ = (+g‘𝐷) |
dvhvaddcl.a | ⊢ + = (+g‘𝑈) |
Ref | Expression |
---|---|
dvhvaddcl | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvhvaddcl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | dvhvaddcl.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
3 | dvhvaddcl.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
4 | dvhvaddcl.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | dvhvaddcl.d | . . 3 ⊢ 𝐷 = (Scalar‘𝑈) | |
6 | dvhvaddcl.a | . . 3 ⊢ + = (+g‘𝑈) | |
7 | dvhvaddcl.p | . . 3 ⊢ ⨣ = (+g‘𝐷) | |
8 | 1, 2, 3, 4, 5, 6, 7 | dvhvadd 38869 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) |
9 | simpl 486 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | xp1st 7811 | . . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (1st ‘𝐹) ∈ 𝑇) | |
11 | 10 | ad2antrl 728 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (1st ‘𝐹) ∈ 𝑇) |
12 | xp1st 7811 | . . . . 5 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (1st ‘𝐺) ∈ 𝑇) | |
13 | 12 | ad2antll 729 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (1st ‘𝐺) ∈ 𝑇) |
14 | 1, 2 | ltrnco 38496 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝐹) ∈ 𝑇 ∧ (1st ‘𝐺) ∈ 𝑇) → ((1st ‘𝐹) ∘ (1st ‘𝐺)) ∈ 𝑇) |
15 | 9, 11, 13, 14 | syl3anc 1373 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((1st ‘𝐹) ∘ (1st ‘𝐺)) ∈ 𝑇) |
16 | eqid 2738 | . . . . . . 7 ⊢ (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐)))) = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐)))) | |
17 | 1, 2, 3, 4, 5, 16, 7 | dvhfplusr 38861 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))) |
18 | 17 | adantr 484 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ⨣ = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))) |
19 | 18 | oveqd 7248 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝐹) ⨣ (2nd ‘𝐺)) = ((2nd ‘𝐹)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2nd ‘𝐺))) |
20 | xp2nd 7812 | . . . . . 6 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2nd ‘𝐹) ∈ 𝐸) | |
21 | 20 | ad2antrl 728 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (2nd ‘𝐹) ∈ 𝐸) |
22 | xp2nd 7812 | . . . . . 6 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (2nd ‘𝐺) ∈ 𝐸) | |
23 | 22 | ad2antll 729 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (2nd ‘𝐺) ∈ 𝐸) |
24 | 1, 2, 3, 16 | tendoplcl 38558 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (2nd ‘𝐹) ∈ 𝐸 ∧ (2nd ‘𝐺) ∈ 𝐸) → ((2nd ‘𝐹)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2nd ‘𝐺)) ∈ 𝐸) |
25 | 9, 21, 23, 24 | syl3anc 1373 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝐹)(𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))(2nd ‘𝐺)) ∈ 𝐸) |
26 | 19, 25 | eqeltrd 2839 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝐹) ⨣ (2nd ‘𝐺)) ∈ 𝐸) |
27 | opelxpi 5602 | . . 3 ⊢ ((((1st ‘𝐹) ∘ (1st ‘𝐺)) ∈ 𝑇 ∧ ((2nd ‘𝐹) ⨣ (2nd ‘𝐺)) ∈ 𝐸) → 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉 ∈ (𝑇 × 𝐸)) | |
28 | 15, 26, 27 | syl2anc 587 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉 ∈ (𝑇 × 𝐸)) |
29 | 8, 28 | eqeltrd 2839 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2111 〈cop 4561 ↦ cmpt 5149 × cxp 5563 ∘ ccom 5569 ‘cfv 6397 (class class class)co 7231 ∈ cmpo 7233 1st c1st 7777 2nd c2nd 7778 +gcplusg 16826 Scalarcsca 16829 HLchlt 37127 LHypclh 37761 LTrncltrn 37878 TEndoctendo 38529 DVecHcdvh 38855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5272 ax-pr 5336 ax-un 7541 ax-cnex 10809 ax-resscn 10810 ax-1cn 10811 ax-icn 10812 ax-addcl 10813 ax-addrcl 10814 ax-mulcl 10815 ax-mulrcl 10816 ax-mulcom 10817 ax-addass 10818 ax-mulass 10819 ax-distr 10820 ax-i2m1 10821 ax-1ne0 10822 ax-1rid 10823 ax-rnegex 10824 ax-rrecex 10825 ax-cnre 10826 ax-pre-lttri 10827 ax-pre-lttrn 10828 ax-pre-ltadd 10829 ax-pre-mulgt0 10830 ax-riotaBAD 36730 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3067 df-rex 3068 df-reu 3069 df-rmo 3070 df-rab 3071 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-pss 3899 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4834 df-iun 4920 df-iin 4921 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-eprel 5474 df-po 5482 df-so 5483 df-fr 5523 df-we 5525 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-res 5577 df-ima 5578 df-pred 6175 df-ord 6233 df-on 6234 df-lim 6235 df-suc 6236 df-iota 6355 df-fun 6399 df-fn 6400 df-f 6401 df-f1 6402 df-fo 6403 df-f1o 6404 df-fv 6405 df-riota 7188 df-ov 7234 df-oprab 7235 df-mpo 7236 df-om 7663 df-1st 7779 df-2nd 7780 df-undef 8035 df-wrecs 8067 df-recs 8128 df-rdg 8166 df-1o 8222 df-er 8411 df-map 8530 df-en 8647 df-dom 8648 df-sdom 8649 df-fin 8650 df-pnf 10893 df-mnf 10894 df-xr 10895 df-ltxr 10896 df-le 10897 df-sub 11088 df-neg 11089 df-nn 11855 df-2 11917 df-3 11918 df-4 11919 df-5 11920 df-6 11921 df-n0 12115 df-z 12201 df-uz 12463 df-fz 13120 df-struct 16724 df-slot 16759 df-ndx 16769 df-base 16785 df-plusg 16839 df-mulr 16840 df-sca 16842 df-vsca 16843 df-proset 17826 df-poset 17844 df-plt 17860 df-lub 17876 df-glb 17877 df-join 17878 df-meet 17879 df-p0 17955 df-p1 17956 df-lat 17962 df-clat 18029 df-oposet 36953 df-ol 36955 df-oml 36956 df-covers 37043 df-ats 37044 df-atl 37075 df-cvlat 37099 df-hlat 37128 df-llines 37275 df-lplanes 37276 df-lvols 37277 df-lines 37278 df-psubsp 37280 df-pmap 37281 df-padd 37573 df-lhyp 37765 df-laut 37766 df-ldil 37881 df-ltrn 37882 df-trl 37936 df-tendo 38532 df-edring 38534 df-dvech 38856 |
This theorem is referenced by: dvhvaddass 38874 dvhgrp 38884 dvhlveclem 38885 |
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