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Theorem dvhopvadd 35883
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 + = (+g𝑈)
dvhvadd.p = (+g𝐷)
Assertion
Ref Expression
dvhopvadd (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (⟨𝐹, 𝑄+𝐺, 𝑅⟩) = ⟨(𝐹𝐺), (𝑄 𝑅)⟩)

Proof of Theorem dvhopvadd
StepHypRef Expression
1 simp1 1059 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (𝐾 ∈ HL ∧ 𝑊𝐻))
2 opelxpi 5110 . . . 4 ((𝐹𝑇𝑄𝐸) → ⟨𝐹, 𝑄⟩ ∈ (𝑇 × 𝐸))
323ad2ant2 1081 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → ⟨𝐹, 𝑄⟩ ∈ (𝑇 × 𝐸))
4 opelxpi 5110 . . . 4 ((𝐺𝑇𝑅𝐸) → ⟨𝐺, 𝑅⟩ ∈ (𝑇 × 𝐸))
543ad2ant3 1082 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → ⟨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 + = (+g𝑈)
12 dvhvadd.p . . . 4 = (+g𝐷)
136, 7, 8, 9, 10, 11, 12dvhvadd 35882 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (⟨𝐹, 𝑄⟩ ∈ (𝑇 × 𝐸) ∧ ⟨𝐺, 𝑅⟩ ∈ (𝑇 × 𝐸))) → (⟨𝐹, 𝑄+𝐺, 𝑅⟩) = ⟨((1st ‘⟨𝐹, 𝑄⟩) ∘ (1st ‘⟨𝐺, 𝑅⟩)), ((2nd ‘⟨𝐹, 𝑄⟩) (2nd ‘⟨𝐺, 𝑅⟩))⟩)
141, 3, 5, 13syl12anc 1321 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (⟨𝐹, 𝑄+𝐺, 𝑅⟩) = ⟨((1st ‘⟨𝐹, 𝑄⟩) ∘ (1st ‘⟨𝐺, 𝑅⟩)), ((2nd ‘⟨𝐹, 𝑄⟩) (2nd ‘⟨𝐺, 𝑅⟩))⟩)
15 op1stg 7128 . . . . 5 ((𝐹𝑇𝑄𝐸) → (1st ‘⟨𝐹, 𝑄⟩) = 𝐹)
16153ad2ant2 1081 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (1st ‘⟨𝐹, 𝑄⟩) = 𝐹)
17 op1stg 7128 . . . . 5 ((𝐺𝑇𝑅𝐸) → (1st ‘⟨𝐺, 𝑅⟩) = 𝐺)
18173ad2ant3 1082 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (1st ‘⟨𝐺, 𝑅⟩) = 𝐺)
1916, 18coeq12d 5248 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → ((1st ‘⟨𝐹, 𝑄⟩) ∘ (1st ‘⟨𝐺, 𝑅⟩)) = (𝐹𝐺))
20 op2ndg 7129 . . . . 5 ((𝐹𝑇𝑄𝐸) → (2nd ‘⟨𝐹, 𝑄⟩) = 𝑄)
21203ad2ant2 1081 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (2nd ‘⟨𝐹, 𝑄⟩) = 𝑄)
22 op2ndg 7129 . . . . 5 ((𝐺𝑇𝑅𝐸) → (2nd ‘⟨𝐺, 𝑅⟩) = 𝑅)
23223ad2ant3 1082 . . . 4 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (2nd ‘⟨𝐺, 𝑅⟩) = 𝑅)
2421, 23oveq12d 6625 . . 3 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → ((2nd ‘⟨𝐹, 𝑄⟩) (2nd ‘⟨𝐺, 𝑅⟩)) = (𝑄 𝑅))
2519, 24opeq12d 4380 . 2 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → ⟨((1st ‘⟨𝐹, 𝑄⟩) ∘ (1st ‘⟨𝐺, 𝑅⟩)), ((2nd ‘⟨𝐹, 𝑄⟩) (2nd ‘⟨𝐺, 𝑅⟩))⟩ = ⟨(𝐹𝐺), (𝑄 𝑅)⟩)
2614, 25eqtrd 2655 1 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (⟨𝐹, 𝑄+𝐺, 𝑅⟩) = ⟨(𝐹𝐺), (𝑄 𝑅)⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  cop 4156   × cxp 5074  ccom 5080  cfv 5849  (class class class)co 6607  1st c1st 7114  2nd c2nd 7115  +gcplusg 15865  Scalarcsca 15868  HLchlt 34138  LHypclh 34771  LTrncltrn 34888  TEndoctendo 35541  DVecHcdvh 35868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4733  ax-sep 4743  ax-nul 4751  ax-pow 4805  ax-pr 4869  ax-un 6905  ax-cnex 9939  ax-resscn 9940  ax-1cn 9941  ax-icn 9942  ax-addcl 9943  ax-addrcl 9944  ax-mulcl 9945  ax-mulrcl 9946  ax-mulcom 9947  ax-addass 9948  ax-mulass 9949  ax-distr 9950  ax-i2m1 9951  ax-1ne0 9952  ax-1rid 9953  ax-rnegex 9954  ax-rrecex 9955  ax-cnre 9956  ax-pre-lttri 9957  ax-pre-lttrn 9958  ax-pre-ltadd 9959  ax-pre-mulgt0 9960
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3419  df-csb 3516  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-int 4443  df-iun 4489  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-f1 5854  df-fo 5855  df-f1o 5856  df-fv 5857  df-riota 6568  df-ov 6610  df-oprab 6611  df-mpt2 6612  df-om 7016  df-1st 7116  df-2nd 7117  df-wrecs 7355  df-recs 7416  df-rdg 7454  df-1o 7508  df-oadd 7512  df-er 7690  df-en 7903  df-dom 7904  df-sdom 7905  df-fin 7906  df-pnf 10023  df-mnf 10024  df-xr 10025  df-ltxr 10026  df-le 10027  df-sub 10215  df-neg 10216  df-nn 10968  df-2 11026  df-3 11027  df-4 11028  df-5 11029  df-6 11030  df-n0 11240  df-z 11325  df-uz 11635  df-fz 12272  df-struct 15786  df-ndx 15787  df-slot 15788  df-base 15789  df-plusg 15878  df-mulr 15879  df-sca 15881  df-vsca 15882  df-edring 35546  df-dvech 35869
This theorem is referenced by:  dvhopvadd2  35884  dvhgrp  35897  dvh0g  35901  diblsmopel  35961  cdlemn4  35988  cdlemn6  35992  dihopelvalcpre  36038
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