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Theorem dvhfvadd 39951
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 = (LHypβ€˜πΎ)
dvhfvadd.t 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
dvhfvadd.e 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
dvhfvadd.u π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
dvhfvadd.f 𝐷 = (Scalarβ€˜π‘ˆ)
dvhfvadd.p ⨣ = (+gβ€˜π·)
dvhfvadd.a ✚ = (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”))⟩)
dvhfvadd.s + = (+gβ€˜π‘ˆ)
Assertion
Ref Expression
dvhfvadd ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ + = ✚ )
Distinct variable groups:   𝑓,𝑔,𝐸   𝑓,𝐻,𝑔   𝑓,𝐾,𝑔   𝑇,𝑓,𝑔   𝑓,π‘Š,𝑔
Allowed substitution hints:   𝐷(𝑓,𝑔)   + (𝑓,𝑔)   ⨣ (𝑓,𝑔)   ✚ (𝑓,𝑔)   π‘ˆ(𝑓,𝑔)
Proof of Theorem dvhfvadd
Dummy variables β„Ž 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dvhfvadd.h . . . . 5 𝐻 = (LHypβ€˜πΎ)
2 dvhfvadd.t . . . . 5 𝑇 = ((LTrnβ€˜πΎ)β€˜π‘Š)
3 dvhfvadd.e . . . . 5 𝐸 = ((TEndoβ€˜πΎ)β€˜π‘Š)
4 eqid 2733 . . . . 5 ((EDRingβ€˜πΎ)β€˜π‘Š) = ((EDRingβ€˜πΎ)β€˜π‘Š)
5 dvhfvadd.u . . . . 5 π‘ˆ = ((DVecHβ€˜πΎ)β€˜π‘Š)
61, 2, 3, 4, 5dvhset 39941 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ π‘ˆ = ({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))
76fveq2d 6893 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜π‘ˆ) = (+gβ€˜({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩})))
8 dvhfvadd.p . . . . . . . . . 10 ⨣ = (+gβ€˜π·)
9 dvhfvadd.f . . . . . . . . . . . 12 𝐷 = (Scalarβ€˜π‘ˆ)
101, 4, 5, 9dvhsca 39942 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
1110fveq2d 6893 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜π·) = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
128, 11eqtrid 2785 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ⨣ = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
1312oveqd 7423 . . . . . . . 8 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”)) = ((2nd β€˜π‘“)(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))(2nd β€˜π‘”)))
14133ad2ant1 1134 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”)) = ((2nd β€˜π‘“)(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))(2nd β€˜π‘”)))
15 xp2nd 8005 . . . . . . . . . 10 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘“) ∈ 𝐸)
16 xp2nd 8005 . . . . . . . . . 10 (𝑔 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘”) ∈ 𝐸)
1715, 16anim12i 614 . . . . . . . . 9 ((𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ ((2nd β€˜π‘“) ∈ 𝐸 ∧ (2nd β€˜π‘”) ∈ 𝐸))
18 eqid 2733 . . . . . . . . . 10 (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)) = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))
191, 2, 3, 4, 18erngplus 39663 . . . . . . . . 9 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((2nd β€˜π‘“) ∈ 𝐸 ∧ (2nd β€˜π‘”) ∈ 𝐸)) β†’ ((2nd β€˜π‘“)(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))(2nd β€˜π‘”)) = (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž))))
2017, 19sylan2 594 . . . . . . . 8 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸))) β†’ ((2nd β€˜π‘“)(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))(2nd β€˜π‘”)) = (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž))))
21203impb 1116 . . . . . . 7 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ ((2nd β€˜π‘“)(+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))(2nd β€˜π‘”)) = (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž))))
2214, 21eqtrd 2773 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”)) = (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž))))
2322opeq2d 4880 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑓 ∈ (𝑇 Γ— 𝐸) ∧ 𝑔 ∈ (𝑇 Γ— 𝐸)) β†’ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”))⟩ = ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)
2423mpoeq3dva 7483 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”))⟩) = (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩))
252fvexi 6903 . . . . . . 7 𝑇 ∈ V
263fvexi 6903 . . . . . . 7 𝐸 ∈ V
2725, 26xpex 7737 . . . . . 6 (𝑇 Γ— 𝐸) ∈ V
2827, 27mpoex 8063 . . . . 5 (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩) ∈ V
29 eqid 2733 . . . . . 6 ({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}) = ({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩})
3029lmodplusg 17269 . . . . 5 ((𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩) ∈ V β†’ (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩) = (+gβ€˜({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩})))
3128, 30ax-mp 5 . . . 4 (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩) = (+gβ€˜({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩}))
3224, 31eqtr2di 2790 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜({⟨(Baseβ€˜ndx), (𝑇 Γ— 𝐸)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), (β„Ž ∈ 𝑇 ↦ (((2nd β€˜π‘“)β€˜β„Ž) ∘ ((2nd β€˜π‘”)β€˜β„Ž)))⟩)⟩, ⟨(Scalarβ€˜ndx), ((EDRingβ€˜πΎ)β€˜π‘Š)⟩} βˆͺ {⟨( ·𝑠 β€˜ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 Γ— 𝐸) ↦ ⟨(π‘ β€˜(1st β€˜π‘“)), (𝑠 ∘ (2nd β€˜π‘“))⟩)⟩})) = (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”))⟩))
337, 32eqtrd 2773 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜π‘ˆ) = (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”))⟩))
34 dvhfvadd.s . 2 + = (+gβ€˜π‘ˆ)
35 dvhfvadd.a . 2 ✚ = (𝑓 ∈ (𝑇 Γ— 𝐸), 𝑔 ∈ (𝑇 Γ— 𝐸) ↦ ⟨((1st β€˜π‘“) ∘ (1st β€˜π‘”)), ((2nd β€˜π‘“) ⨣ (2nd β€˜π‘”))⟩)
3633, 34, 353eqtr4g 2798 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ + = ✚ )
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475   βˆͺ cun 3946  {csn 4628  {ctp 4632  βŸ¨cop 4634   ↦ cmpt 5231   Γ— cxp 5674   ∘ ccom 5680  β€˜cfv 6541  (class class class)co 7406   ∈ cmpo 7408  1st c1st 7970  2nd c2nd 7971  ndxcnx 17123  Basecbs 17141  +gcplusg 17194  Scalarcsca 17197   ·𝑠 cvsca 17198  HLchlt 38209  LHypclh 38844  LTrncltrn 38961  TEndoctendo 39612  EDRingcedring 39613  DVecHcdvh 39938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-5 12275  df-6 12276  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-struct 17077  df-slot 17112  df-ndx 17124  df-base 17142  df-plusg 17207  df-mulr 17208  df-sca 17210  df-vsca 17211  df-edring 39617  df-dvech 39939
This theorem is referenced by:  dvhvadd  39952
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