dvhfvadd

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)

	Ref	Expression
Hypotheses	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 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ )
	dvhfvadd.s	⊢ + = ( +_g ‘ 𝑈 )
Assertion	dvhfvadd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → + = ✚ )

Proof

Step	Hyp	Ref	Expression
1		dvhfvadd.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
2		dvhfvadd.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
3		dvhfvadd.e	⊢ 𝐸 = ( ( TEndo ‘ 𝐾 ) ‘ 𝑊 )
4		dvhfvadd.u	⊢ 𝑈 = ( ( DVecH ‘ 𝐾 ) ‘ 𝑊 )
5		dvhfvadd.f	⊢ 𝐷 = ( Scalar ‘ 𝑈 )
6		dvhfvadd.p	⊢ ⨣ = ( +_g ‘ 𝐷 )
7		dvhfvadd.a	⊢ ✚ = ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ )
8		dvhfvadd.s	⊢ + = ( +_g ‘ 𝑈 )
9		eqid	⊢ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 )
10	1 2 3 9 4	dvhset	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝑈 = ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
11	10	fveq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +_g ‘ 𝑈 ) = ( +_g ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
12	1 9 4 5	dvhsca	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → 𝐷 = ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
13	12	fveq2d	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +_g ‘ 𝐷 ) = ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
14	6 13	syl5eq	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ⨣ = ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) )
15	14	oveqd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) = ( ( 2^nd ‘ 𝑓 ) ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ( 2^nd ‘ 𝑔 ) ) )
16	15	3ad2ant1	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( 𝑇 × 𝐸 ) ∧ 𝑔 ∈ ( 𝑇 × 𝐸 ) ) → ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) = ( ( 2^nd ‘ 𝑓 ) ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ( 2^nd ‘ 𝑔 ) ) )
17		xp2nd	⊢ ( 𝑓 ∈ ( 𝑇 × 𝐸 ) → ( 2^nd ‘ 𝑓 ) ∈ 𝐸 )
18		xp2nd	⊢ ( 𝑔 ∈ ( 𝑇 × 𝐸 ) → ( 2^nd ‘ 𝑔 ) ∈ 𝐸 )
19	17 18	anim12i	⊢ ( ( 𝑓 ∈ ( 𝑇 × 𝐸 ) ∧ 𝑔 ∈ ( 𝑇 × 𝐸 ) ) → ( ( 2^nd ‘ 𝑓 ) ∈ 𝐸 ∧ ( 2^nd ‘ 𝑔 ) ∈ 𝐸 ) )
20		eqid	⊢ ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) = ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) )
21	1 2 3 9 20	erngplus	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 2^nd ‘ 𝑓 ) ∈ 𝐸 ∧ ( 2^nd ‘ 𝑔 ) ∈ 𝐸 ) ) → ( ( 2^nd ‘ 𝑓 ) ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ( 2^nd ‘ 𝑔 ) ) = ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) )
22	19 21	sylan2	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑓 ∈ ( 𝑇 × 𝐸 ) ∧ 𝑔 ∈ ( 𝑇 × 𝐸 ) ) ) → ( ( 2^nd ‘ 𝑓 ) ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ( 2^nd ‘ 𝑔 ) ) = ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) )
23	22	3impb	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( 𝑇 × 𝐸 ) ∧ 𝑔 ∈ ( 𝑇 × 𝐸 ) ) → ( ( 2^nd ‘ 𝑓 ) ( +_g ‘ ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ) ( 2^nd ‘ 𝑔 ) ) = ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) )
24	16 23	eqtrd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( 𝑇 × 𝐸 ) ∧ 𝑔 ∈ ( 𝑇 × 𝐸 ) ) → ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) = ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) )
25	24	opeq2d	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑓 ∈ ( 𝑇 × 𝐸 ) ∧ 𝑔 ∈ ( 𝑇 × 𝐸 ) ) → ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ = ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ )
26	25	mpoeq3dva	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ ) = ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) )
27	2	fvexi	⊢ 𝑇 ∈ V
28	3	fvexi	⊢ 𝐸 ∈ V
29	27 28	xpex	⊢ ( 𝑇 × 𝐸 ) ∈ V
30	29 29	mpoex	⊢ ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ∈ V
31		eqid	⊢ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) = ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } )
32	31	lmodplusg	⊢ ( ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ∈ V → ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) = ( +_g ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) )
33	30 32	ax-mp	⊢ ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) = ( +_g ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) )
34	26 33	eqtr2di	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +_g ‘ ( { ⟨ ( Base ‘ ndx ) , ( 𝑇 × 𝐸 ) ⟩ , ⟨ ( +_g ‘ ndx ) , ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ℎ ∈ 𝑇 ↦ ( ( ( 2^nd ‘ 𝑓 ) ‘ ℎ ) ∘ ( ( 2^nd ‘ 𝑔 ) ‘ ℎ ) ) ) ⟩ ) ⟩ , ⟨ ( Scalar ‘ ndx ) , ( ( EDRing ‘ 𝐾 ) ‘ 𝑊 ) ⟩ } ∪ { ⟨ ( ·_𝑠 ‘ ndx ) , ( 𝑠 ∈ 𝐸 , 𝑓 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( 𝑠 ‘ ( 1^st ‘ 𝑓 ) ) , ( 𝑠 ∘ ( 2^nd ‘ 𝑓 ) ) ⟩ ) ⟩ } ) ) = ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ ) )
35	11 34	eqtrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( +_g ‘ 𝑈 ) = ( 𝑓 ∈ ( 𝑇 × 𝐸 ) , 𝑔 ∈ ( 𝑇 × 𝐸 ) ↦ ⟨ ( ( 1^st ‘ 𝑓 ) ∘ ( 1^st ‘ 𝑔 ) ) , ( ( 2^nd ‘ 𝑓 ) ⨣ ( 2^nd ‘ 𝑔 ) ) ⟩ ) )
36	35 8 7	3eqtr4g	⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → + = ✚ )

Metamath Proof Explorer

Theorem dvhfvadd

Proof