Theorem dvhopvadd 41191

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.)

Hypotheses

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)

Assertion

Ref	Expression
dvhopvadd	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (⟨𝐹, 𝑄⟩ + ⟨𝐺, 𝑅⟩) = ⟨(𝐹 ∘ 𝐺), (𝑄 ⨣ 𝑅)⟩)

Proof of Theorem dvhopvadd

Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		opelxpi 5651	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → ⟨𝐹, 𝑄⟩ ∈ (𝑇 × 𝐸))
3	2	3ad2ant2 1134	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ⟨𝐹, 𝑄⟩ ∈ (𝑇 × 𝐸))
4		opelxpi 5651	. . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → ⟨𝐺, 𝑅⟩ ∈ (𝑇 × 𝐸))
5	4	3ad2ant3 1135	. . 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 41190	. . 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 836	. 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 7933	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (1st ‘⟨𝐹, 𝑄⟩) = 𝐹)
16	15	3ad2ant2 1134	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘⟨𝐹, 𝑄⟩) = 𝐹)
17		op1stg 7933	. . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (1st ‘⟨𝐺, 𝑅⟩) = 𝐺)
18	17	3ad2ant3 1135	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘⟨𝐺, 𝑅⟩) = 𝐺)
19	16, 18	coeq12d 5803	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ((1st ‘⟨𝐹, 𝑄⟩) ∘ (1st ‘⟨𝐺, 𝑅⟩)) = (𝐹 ∘ 𝐺))
20		op2ndg 7934	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (2nd ‘⟨𝐹, 𝑄⟩) = 𝑄)
21	20	3ad2ant2 1134	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘⟨𝐹, 𝑄⟩) = 𝑄)
22		op2ndg 7934	. . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (2nd ‘⟨𝐺, 𝑅⟩) = 𝑅)
23	22	3ad2ant3 1135	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘⟨𝐺, 𝑅⟩) = 𝑅)
24	21, 23	oveq12d 7364	. . 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 4830	. 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)⟩)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ⟨cop 4579   × cxp 5612   ∘ ccom 5618  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  +_gcplusg 17161  Scalarcsca 17164  HLchlt 39448  LHypclh 40082  LTrncltrn 40199  TEndoctendo 40850  DVecHcdvh 41176

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-n0 12382  df-z 12469  df-uz 12733  df-fz 13408  df-struct 17058  df-slot 17093  df-ndx 17105  df-base 17121  df-plusg 17174  df-mulr 17175  df-sca 17177  df-vsca 17178  df-edring 40855  df-dvech 41177

This theorem is referenced by:  dvhopvadd2  41192  dvhgrp  41205  dvh0g  41209  diblsmopel  41269  cdlemn4  41296  cdlemn6  41300  dihopelvalcpre  41346