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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 1137 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
| 2 | opelxpi 5722 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → 〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸)) | |
| 3 | 2 | 3ad2ant2 1135 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → 〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸)) |
| 4 | opelxpi 5722 | . . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → 〈𝐺, 𝑅〉 ∈ (𝑇 × 𝐸)) | |
| 5 | 4 | 3ad2ant3 1136 | . . 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 41094 | . . 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 8026 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (1st ‘〈𝐹, 𝑄〉) = 𝐹) | |
| 16 | 15 | 3ad2ant2 1135 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑄〉) = 𝐹) |
| 17 | op1stg 8026 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (1st ‘〈𝐺, 𝑅〉) = 𝐺) | |
| 18 | 17 | 3ad2ant3 1136 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘〈𝐺, 𝑅〉) = 𝐺) |
| 19 | 16, 18 | coeq12d 5875 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ((1st ‘〈𝐹, 𝑄〉) ∘ (1st ‘〈𝐺, 𝑅〉)) = (𝐹 ∘ 𝐺)) |
| 20 | op2ndg 8027 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑄〉) = 𝑄) | |
| 21 | 20 | 3ad2ant2 1135 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘〈𝐹, 𝑄〉) = 𝑄) |
| 22 | op2ndg 8027 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (2nd ‘〈𝐺, 𝑅〉) = 𝑅) | |
| 23 | 22 | 3ad2ant3 1136 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘〈𝐺, 𝑅〉) = 𝑅) |
| 24 | 21, 23 | oveq12d 7449 | . . 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 4881 | . 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 2777 | 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 1087 = wceq 1540 ∈ wcel 2108 〈cop 4632 × cxp 5683 ∘ ccom 5689 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 +gcplusg 17297 Scalarcsca 17300 HLchlt 39351 LHypclh 39986 LTrncltrn 40103 TEndoctendo 40754 DVecHcdvh 41080 |
| 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-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| 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-nel 3047 df-ral 3062 df-rex 3071 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-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 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-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 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-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-4 12331 df-5 12332 df-6 12333 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-struct 17184 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-sca 17313 df-vsca 17314 df-edring 40759 df-dvech 41081 |
| This theorem is referenced by: dvhopvadd2 41096 dvhgrp 41109 dvh0g 41113 diblsmopel 41173 cdlemn4 41200 cdlemn6 41204 dihopelvalcpre 41250 |
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