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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 1133 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | opelxpi 5719 | . . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → 〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸)) | |
3 | 2 | 3ad2ant2 1131 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → 〈𝐹, 𝑄〉 ∈ (𝑇 × 𝐸)) |
4 | opelxpi 5719 | . . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → 〈𝐺, 𝑅〉 ∈ (𝑇 × 𝐸)) | |
5 | 4 | 3ad2ant3 1132 | . . 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 40791 | . . 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 835 | . 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 8015 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (1st ‘〈𝐹, 𝑄〉) = 𝐹) | |
16 | 15 | 3ad2ant2 1131 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘〈𝐹, 𝑄〉) = 𝐹) |
17 | op1stg 8015 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (1st ‘〈𝐺, 𝑅〉) = 𝐺) | |
18 | 17 | 3ad2ant3 1132 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘〈𝐺, 𝑅〉) = 𝐺) |
19 | 16, 18 | coeq12d 5871 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ((1st ‘〈𝐹, 𝑄〉) ∘ (1st ‘〈𝐺, 𝑅〉)) = (𝐹 ∘ 𝐺)) |
20 | op2ndg 8016 | . . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (2nd ‘〈𝐹, 𝑄〉) = 𝑄) | |
21 | 20 | 3ad2ant2 1131 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘〈𝐹, 𝑄〉) = 𝑄) |
22 | op2ndg 8016 | . . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (2nd ‘〈𝐺, 𝑅〉) = 𝑅) | |
23 | 22 | 3ad2ant3 1132 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘〈𝐺, 𝑅〉) = 𝑅) |
24 | 21, 23 | oveq12d 7442 | . . 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 4887 | . 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 2766 | 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 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 〈cop 4639 × cxp 5680 ∘ ccom 5686 ‘cfv 6554 (class class class)co 7424 1st c1st 8001 2nd c2nd 8002 +gcplusg 17266 Scalarcsca 17269 HLchlt 39048 LHypclh 39683 LTrncltrn 39800 TEndoctendo 40451 DVecHcdvh 40777 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-edring 40456 df-dvech 40778 |
This theorem is referenced by: dvhopvadd2 40793 dvhgrp 40806 dvh0g 40810 diblsmopel 40870 cdlemn4 40897 cdlemn6 40901 dihopelvalcpre 40947 |
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