Theorem dvhfvadd 39554

Description: The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)

Hypotheses

Ref	Expression
dvhfvadd.h	⊢ 𝐻 = (LHyp‘𝐾)
dvhfvadd.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dvhfvadd.e	⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
dvhfvadd.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dvhfvadd.f	⊢ 𝐷 = (Scalar‘𝑈)
dvhfvadd.p	⊢ ⨣ = (+g‘𝐷)
dvhfvadd.a	⊢ ✚ = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩)
dvhfvadd.s	⊢ + = (+g‘𝑈)

Assertion

Ref	Expression
dvhfvadd	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = ✚ )

Distinct variable groups:   𝑓,𝑔,𝐸   𝑓,𝐻,𝑔   𝑓,𝐾,𝑔   𝑇,𝑓,𝑔   𝑓,𝑊,𝑔

Allowed substitution hints:   𝐷(𝑓,𝑔)   + (𝑓,𝑔)   ⨣ (𝑓,𝑔)   ✚ (𝑓,𝑔)   𝑈(𝑓,𝑔)

Proof of Theorem dvhfvadd

Dummy variables ℎ 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvhfvadd.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		dvhfvadd.t	. . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
3		dvhfvadd.e	. . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)
4		eqid 2736	. . . . 5 ⊢ ((EDRing‘𝐾)‘𝑊) = ((EDRing‘𝐾)‘𝑊)
5		dvhfvadd.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	dvhset 39544	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩}))
7	6	fveq2d 6846	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝑈) = (+g‘({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})))
8		dvhfvadd.p	. . . . . . . . . 10 ⊢ ⨣ = (+g‘𝐷)
9		dvhfvadd.f	. . . . . . . . . . . 12 ⊢ 𝐷 = (Scalar‘𝑈)
10	1, 4, 5, 9	dvhsca 39545	. . . . . . . . . . 11 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐷 = ((EDRing‘𝐾)‘𝑊))
11	10	fveq2d 6846	. . . . . . . . . 10 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝐷) = (+g‘((EDRing‘𝐾)‘𝑊)))
12	8, 11	eqtrid 2788	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ⨣ = (+g‘((EDRing‘𝐾)‘𝑊)))
13	12	oveqd 7374	. . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((2nd ‘𝑓) ⨣ (2nd ‘𝑔)) = ((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔)))
14	13	3ad2ant1 1133	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓) ⨣ (2nd ‘𝑔)) = ((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔)))
15		xp2nd 7954	. . . . . . . . . 10 ⊢ (𝑓 ∈ (𝑇 × 𝐸) → (2nd ‘𝑓) ∈ 𝐸)
16		xp2nd 7954	. . . . . . . . . 10 ⊢ (𝑔 ∈ (𝑇 × 𝐸) → (2nd ‘𝑔) ∈ 𝐸)
17	15, 16	anim12i 613	. . . . . . . . 9 ⊢ ((𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓) ∈ 𝐸 ∧ (2nd ‘𝑔) ∈ 𝐸))
18		eqid 2736	. . . . . . . . . 10 ⊢ (+g‘((EDRing‘𝐾)‘𝑊)) = (+g‘((EDRing‘𝐾)‘𝑊))
19	1, 2, 3, 4, 18	erngplus 39266	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((2nd ‘𝑓) ∈ 𝐸 ∧ (2nd ‘𝑔) ∈ 𝐸)) → ((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔)) = (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ))))
20	17, 19	sylan2 593	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸))) → ((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔)) = (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ))))
21	20	3impb 1115	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓)(+g‘((EDRing‘𝐾)‘𝑊))(2nd ‘𝑔)) = (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ))))
22	14, 21	eqtrd 2776	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ((2nd ‘𝑓) ⨣ (2nd ‘𝑔)) = (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ))))
23	22	opeq2d 4837	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 × 𝐸) ∧ 𝑔 ∈ (𝑇 × 𝐸)) → ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩ = ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)
24	23	mpoeq3dva 7434	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩))
25	2	fvexi 6856	. . . . . . 7 ⊢ 𝑇 ∈ V
26	3	fvexi 6856	. . . . . . 7 ⊢ 𝐸 ∈ V
27	25, 26	xpex 7687	. . . . . 6 ⊢ (𝑇 × 𝐸) ∈ V
28	27, 27	mpoex 8012	. . . . 5 ⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩) ∈ V
29		eqid 2736	. . . . . 6 ⊢ ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩}) = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})
30	29	lmodplusg 17208	. . . . 5 ⊢ ((𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩) ∈ V → (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩) = (+g‘({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})))
31	28, 30	ax-mp 5	. . . 4 ⊢ (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩) = (+g‘({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩}))
32	24, 31	eqtr2di 2793	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑊)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩))
33	7, 32	eqtrd 2776	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘𝑈) = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩))
34		dvhfvadd.s	. 2 ⊢ + = (+g‘𝑈)
35		dvhfvadd.a	. 2 ⊢ ✚ = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩)
36	33, 34, 35	3eqtr4g 2801	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = ✚ )

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  Vcvv 3445   ∪ cun 3908  {csn 4586  {ctp 4590  ⟨cop 4592   ↦ cmpt 5188   × cxp 5631   ∘ ccom 5637  ‘cfv 6496  (class class class)co 7357   ∈ cmpo 7359  1^st c1st 7919  2^nd c2nd 7920  ndxcnx 17065  Basecbs 17083  +_gcplusg 17133  Scalarcsca 17136   ·_𝑠 cvsca 17137  HLchlt 37812  LHypclh 38447  LTrncltrn 38564  TEndoctendo 39215  EDRingcedring 39216  DVecHcdvh 39541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-n0 12414  df-z 12500  df-uz 12764  df-fz 13425  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-mulr 17147  df-sca 17149  df-vsca 17150  df-edring 39220  df-dvech 39542

This theorem is referenced by:  dvhvadd  39555