Theorem dvhopvadd 38223

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 1132	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		opelxpi 5587	. . . 4 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → ⟨𝐹, 𝑄⟩ ∈ (𝑇 × 𝐸))
3	2	3ad2ant2 1130	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ⟨𝐹, 𝑄⟩ ∈ (𝑇 × 𝐸))
4		opelxpi 5587	. . . 4 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → ⟨𝐺, 𝑅⟩ ∈ (𝑇 × 𝐸))
5	4	3ad2ant3 1131	. . 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 38222	. . 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 834	. 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 7695	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (1st ‘⟨𝐹, 𝑄⟩) = 𝐹)
16	15	3ad2ant2 1130	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘⟨𝐹, 𝑄⟩) = 𝐹)
17		op1stg 7695	. . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (1st ‘⟨𝐺, 𝑅⟩) = 𝐺)
18	17	3ad2ant3 1131	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (1st ‘⟨𝐺, 𝑅⟩) = 𝐺)
19	16, 18	coeq12d 5730	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → ((1st ‘⟨𝐹, 𝑄⟩) ∘ (1st ‘⟨𝐺, 𝑅⟩)) = (𝐹 ∘ 𝐺))
20		op2ndg 7696	. . . . 5 ⊢ ((𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) → (2nd ‘⟨𝐹, 𝑄⟩) = 𝑄)
21	20	3ad2ant2 1130	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘⟨𝐹, 𝑄⟩) = 𝑄)
22		op2ndg 7696	. . . . 5 ⊢ ((𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸) → (2nd ‘⟨𝐺, 𝑅⟩) = 𝑅)
23	22	3ad2ant3 1131	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (2nd ‘⟨𝐺, 𝑅⟩) = 𝑅)
24	21, 23	oveq12d 7168	. . 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 4805	. 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 2856	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 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ⟨cop 4567   × cxp 5548   ∘ ccom 5554  ‘cfv 6350  (class class class)co 7150  1^st c1st 7681  2^nd c2nd 7682  +_gcplusg 16559  Scalarcsca 16562  HLchlt 36480  LHypclh 37114  LTrncltrn 37231  TEndoctendo 37882  DVecHcdvh 38208

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-oadd 8100  df-er 8283  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-2 11694  df-3 11695  df-4 11696  df-5 11697  df-6 11698  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-struct 16479  df-ndx 16480  df-slot 16481  df-base 16483  df-plusg 16572  df-mulr 16573  df-sca 16575  df-vsca 16576  df-edring 37887  df-dvech 38209

This theorem is referenced by:  dvhopvadd2  38224  dvhgrp  38237  dvh0g  38241  diblsmopel  38301  cdlemn4  38328  cdlemn6  38332  dihopelvalcpre  38378