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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dvhopvadd | Structured version Visualization version GIF version |
Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
dvhvadd.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dvhvadd.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dvhvadd.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dvhvadd.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dvhvadd.f | ⊢ 𝐷 = (Scalar‘𝑈) |
dvhvadd.s | ⊢ + = (+g‘𝑈) |
dvhvadd.p | ⊢ ⨣ = (+g‘𝐷) |
Ref | Expression |
---|---|
dvhopvadd | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (〈𝐹, 𝑄〉 + 〈𝐺, 𝑅〉) = 〈(𝐹 ∘ 𝐺), (𝑄 ⨣ 𝑅)〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1135 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | opelxpi 5725 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → 〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸)) | |
3 | 2 | 3ad2ant2 1133 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → 〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸)) |
4 | opelxpi 5725 | . . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → 〈𝐺, 𝑅〉 ∈ (𝑇 × 𝐸)) | |
5 | 4 | 3ad2ant3 1134 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → 〈𝐺, 𝑅〉 ∈ (𝑇 × 𝐸)) |
6 | dvhvadd.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | dvhvadd.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | dvhvadd.e | . . . 4 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
9 | dvhvadd.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | dvhvadd.f | . . . 4 ⊢ 𝐷 = (Scalar‘𝑈) | |
11 | dvhvadd.s | . . . 4 ⊢ + = (+g‘𝑈) | |
12 | dvhvadd.p | . . . 4 ⊢ ⨣ = (+g‘𝐷) | |
13 | 6, 7, 8, 9, 10, 11, 12 | dvhvadd 41074 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸) ∧ 〈𝐺, 𝑅〉 ∈ (𝑇 × 𝐸))) → (〈𝐹, 𝑄〉 + 〈𝐺, 𝑅〉) = 〈((1st ‘〈𝐹, 𝑄〉) ∘ (1st ‘〈𝐺, 𝑅〉)), ((2nd ‘〈𝐹, 𝑄〉) ⨣ (2nd ‘〈𝐺, 𝑅〉))〉) |
14 | 1, 3, 5, 13 | syl12anc 837 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (〈𝐹, 𝑄〉 + 〈𝐺, 𝑅〉) = 〈((1st ‘〈𝐹, 𝑄〉) ∘ (1st ‘〈𝐺, 𝑅〉)), ((2nd ‘〈𝐹, 𝑄〉) ⨣ (2nd ‘〈𝐺, 𝑅〉))〉) |
15 | op1stg 8024 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (1st ‘〈𝐹, 𝑄〉) = 𝐹) | |
16 | 15 | 3ad2ant2 1133 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑄〉) = 𝐹) |
17 | op1stg 8024 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (1st ‘〈𝐺, 𝑅〉) = 𝐺) | |
18 | 17 | 3ad2ant3 1134 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘〈𝐺, 𝑅〉) = 𝐺) |
19 | 16, 18 | coeq12d 5877 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ((1st ‘〈𝐹, 𝑄〉) ∘ (1st ‘〈𝐺, 𝑅〉)) = (𝐹 ∘ 𝐺)) |
20 | op2ndg 8025 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑄〉) = 𝑄) | |
21 | 20 | 3ad2ant2 1133 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘〈𝐹, 𝑄〉) = 𝑄) |
22 | op2ndg 8025 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (2nd ‘〈𝐺, 𝑅〉) = 𝑅) | |
23 | 22 | 3ad2ant3 1134 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘〈𝐺, 𝑅〉) = 𝑅) |
24 | 21, 23 | oveq12d 7448 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ((2nd ‘〈𝐹, 𝑄〉) ⨣ (2nd ‘〈𝐺, 𝑅〉)) = (𝑄 ⨣ 𝑅)) |
25 | 19, 24 | opeq12d 4885 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → 〈((1st ‘〈𝐹, 𝑄〉) ∘ (1st ‘〈𝐺, 𝑅〉)), ((2nd ‘〈𝐹, 𝑄〉) ⨣ (2nd ‘〈𝐺, 𝑅〉))〉 = 〈(𝐹 ∘ 𝐺), (𝑄 ⨣ 𝑅)〉) |
26 | 14, 25 | eqtrd 2774 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (〈𝐹, 𝑄〉 + 〈𝐺, 𝑅〉) = 〈(𝐹 ∘ 𝐺), (𝑄 ⨣ 𝑅)〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 〈cop 4636 × cxp 5686 ∘ ccom 5692 ‘cfv 6562 (class class class)co 7430 1st c1st 8010 2nd c2nd 8011 +gcplusg 17297 Scalarcsca 17300 HLchlt 39331 LHypclh 39966 LTrncltrn 40083 TEndoctendo 40734 DVecHcdvh 41060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-n0 12524 df-z 12611 df-uz 12876 df-fz 13544 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-edring 40739 df-dvech 41061 |
This theorem is referenced by: dvhopvadd2 41076 dvhgrp 41089 dvh0g 41093 diblsmopel 41153 cdlemn4 41180 cdlemn6 41184 dihopelvalcpre 41230 |
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