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Mathbox for Norm Megill |
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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 39555 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) |
9 | simpl 483 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
10 | xp1st 7953 | . . . . 5 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (1st ‘𝐹) ∈ 𝑇) | |
11 | 10 | ad2antrl 726 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (1st ‘𝐹) ∈ 𝑇) |
12 | xp1st 7953 | . . . . 5 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (1st ‘𝐺) ∈ 𝑇) | |
13 | 12 | ad2antll 727 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (1st ‘𝐺) ∈ 𝑇) |
14 | 1, 2 | ltrnco 39182 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (1st ‘𝐹) ∈ 𝑇 ∧ (1st ‘𝐺) ∈ 𝑇) → ((1st ‘𝐹) ∘ (1st ‘𝐺)) ∈ 𝑇) |
15 | 9, 11, 13, 14 | syl3anc 1371 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((1st ‘𝐹) ∘ (1st ‘𝐺)) ∈ 𝑇) |
16 | eqid 2736 | . . . . . . 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 39547 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))) |
18 | 17 | adantr 481 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ⨣ = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))) |
19 | 18 | oveqd 7374 | . . . 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 7954 | . . . . . 6 ⊢ (𝐹 ∈ (𝑇 × 𝐸) → (2nd ‘𝐹) ∈ 𝐸) | |
21 | 20 | ad2antrl 726 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (2nd ‘𝐹) ∈ 𝐸) |
22 | xp2nd 7954 | . . . . . 6 ⊢ (𝐺 ∈ (𝑇 × 𝐸) → (2nd ‘𝐺) ∈ 𝐸) | |
23 | 22 | ad2antll 727 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (2nd ‘𝐺) ∈ 𝐸) |
24 | 1, 2, 3, 16 | tendoplcl 39244 | . . . . 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 1371 | . . . 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 2838 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝐹) ⨣ (2nd ‘𝐺)) ∈ 𝐸) |
27 | opelxpi 5670 | . . 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 584 | . 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 2838 | 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 396 = wceq 1541 ∈ wcel 2106 〈cop 4592 ↦ cmpt 5188 × cxp 5631 ∘ ccom 5637 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 1st c1st 7919 2nd c2nd 7920 +gcplusg 17133 Scalarcsca 17136 HLchlt 37812 LHypclh 38447 LTrncltrn 38564 TEndoctendo 39215 DVecHcdvh 39541 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-slot 17054 df-ndx 17066 df-base 17084 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tendo 39218 df-edring 39220 df-dvech 39542 |
This theorem is referenced by: dvhvaddass 39560 dvhgrp 39570 dvhlveclem 39571 |
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