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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 1138 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | opelxpi 5603 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → 〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸)) | |
3 | 2 | 3ad2ant2 1136 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → 〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸)) |
4 | opelxpi 5603 | . . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → 〈𝐺, 𝑅〉 ∈ (𝑇 × 𝐸)) | |
5 | 4 | 3ad2ant3 1137 | . . 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 38870 | . . 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 7792 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (1st ‘〈𝐹, 𝑄〉) = 𝐹) | |
16 | 15 | 3ad2ant2 1136 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑄〉) = 𝐹) |
17 | op1stg 7792 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (1st ‘〈𝐺, 𝑅〉) = 𝐺) | |
18 | 17 | 3ad2ant3 1137 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘〈𝐺, 𝑅〉) = 𝐺) |
19 | 16, 18 | coeq12d 5748 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ((1st ‘〈𝐹, 𝑄〉) ∘ (1st ‘〈𝐺, 𝑅〉)) = (𝐹 ∘ 𝐺)) |
20 | op2ndg 7793 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑄〉) = 𝑄) | |
21 | 20 | 3ad2ant2 1136 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘〈𝐹, 𝑄〉) = 𝑄) |
22 | op2ndg 7793 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (2nd ‘〈𝐺, 𝑅〉) = 𝑅) | |
23 | 22 | 3ad2ant3 1137 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘〈𝐺, 𝑅〉) = 𝑅) |
24 | 21, 23 | oveq12d 7250 | . . 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 4807 | . 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 2778 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 〈cop 4562 × cxp 5564 ∘ ccom 5570 ‘cfv 6398 (class class class)co 7232 1st c1st 7778 2nd c2nd 7779 +gcplusg 16827 Scalarcsca 16830 HLchlt 37128 LHypclh 37762 LTrncltrn 37879 TEndoctendo 38530 DVecHcdvh 38856 |
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 5194 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 ax-cnex 10810 ax-resscn 10811 ax-1cn 10812 ax-icn 10813 ax-addcl 10814 ax-addrcl 10815 ax-mulcl 10816 ax-mulrcl 10817 ax-mulcom 10818 ax-addass 10819 ax-mulass 10820 ax-distr 10821 ax-i2m1 10822 ax-1ne0 10823 ax-1rid 10824 ax-rnegex 10825 ax-rrecex 10826 ax-cnre 10827 ax-pre-lttri 10828 ax-pre-lttrn 10829 ax-pre-ltadd 10830 ax-pre-mulgt0 10831 |
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-rab 3071 df-v 3423 df-sbc 3710 df-csb 3827 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-tp 4561 df-op 4563 df-uni 4835 df-iun 4921 df-br 5069 df-opab 5131 df-mpt 5151 df-tr 5177 df-id 5470 df-eprel 5475 df-po 5483 df-so 5484 df-fr 5524 df-we 5526 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-rn 5577 df-res 5578 df-ima 5579 df-pred 6176 df-ord 6234 df-on 6235 df-lim 6236 df-suc 6237 df-iota 6356 df-fun 6400 df-fn 6401 df-f 6402 df-f1 6403 df-fo 6404 df-f1o 6405 df-fv 6406 df-riota 7189 df-ov 7235 df-oprab 7236 df-mpo 7237 df-om 7664 df-1st 7780 df-2nd 7781 df-wrecs 8068 df-recs 8129 df-rdg 8167 df-1o 8223 df-er 8412 df-en 8648 df-dom 8649 df-sdom 8650 df-fin 8651 df-pnf 10894 df-mnf 10895 df-xr 10896 df-ltxr 10897 df-le 10898 df-sub 11089 df-neg 11090 df-nn 11856 df-2 11918 df-3 11919 df-4 11920 df-5 11921 df-6 11922 df-n0 12116 df-z 12202 df-uz 12464 df-fz 13121 df-struct 16725 df-slot 16760 df-ndx 16770 df-base 16786 df-plusg 16840 df-mulr 16841 df-sca 16843 df-vsca 16844 df-edring 38535 df-dvech 38857 |
This theorem is referenced by: dvhopvadd2 38872 dvhgrp 38885 dvh0g 38889 diblsmopel 38949 cdlemn4 38976 cdlemn6 38980 dihopelvalcpre 39026 |
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