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Theorem dvhfvadd 39557
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 2737 . . . . 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 39547 . . . 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 6847 . . 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 39548 . . . . . . . . . . 11 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = ((EDRingβ€˜πΎ)β€˜π‘Š))
1110fveq2d 6847 . . . . . . . . . 10 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (+gβ€˜π·) = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
128, 11eqtrid 2789 . . . . . . . . 9 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ⨣ = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)))
1312oveqd 7375 . . . . . . . 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 7955 . . . . . . . . . 10 (𝑓 ∈ (𝑇 Γ— 𝐸) β†’ (2nd β€˜π‘“) ∈ 𝐸)
16 xp2nd 7955 . . . . . . . . . 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 2737 . . . . . . . . . 10 (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š)) = (+gβ€˜((EDRingβ€˜πΎ)β€˜π‘Š))
191, 2, 3, 4, 18erngplus 39269 . . . . . . . . 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 2777 . . . . . 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 4838 . . . . 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 7435 . . . 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 6857 . . . . . . 7 𝑇 ∈ V
263fvexi 6857 . . . . . . 7 𝐸 ∈ V
2725, 26xpex 7688 . . . . . 6 (𝑇 Γ— 𝐸) ∈ V
2827, 27mpoex 8013 . . . . 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 2737 . . . . . 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 17209 . . . . 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 2794 . . 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 2777 . 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 2802 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 3446   βˆͺ cun 3909  {csn 4587  {ctp 4591  βŸ¨cop 4593   ↦ cmpt 5189   Γ— cxp 5632   ∘ ccom 5638  β€˜cfv 6497  (class class class)co 7358   ∈ cmpo 7360  1st c1st 7920  2nd c2nd 7921  ndxcnx 17066  Basecbs 17084  +gcplusg 17134  Scalarcsca 17137   ·𝑠 cvsca 17138  HLchlt 37815  LHypclh 38450  LTrncltrn 38567  TEndoctendo 39218  EDRingcedring 39219  DVecHcdvh 39544
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109  ax-1cn 11110  ax-icn 11111  ax-addcl 11112  ax-addrcl 11113  ax-mulcl 11114  ax-mulrcl 11115  ax-mulcom 11116  ax-addass 11117  ax-mulass 11118  ax-distr 11119  ax-i2m1 11120  ax-1ne0 11121  ax-1rid 11122  ax-rnegex 11123  ax-rrecex 11124  ax-cnre 11125  ax-pre-lttri 11126  ax-pre-lttrn 11127  ax-pre-ltadd 11128  ax-pre-mulgt0 11129
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196  df-sub 11388  df-neg 11389  df-nn 12155  df-2 12217  df-3 12218  df-4 12219  df-5 12220  df-6 12221  df-n0 12415  df-z 12501  df-uz 12765  df-fz 13426  df-struct 17020  df-slot 17055  df-ndx 17067  df-base 17085  df-plusg 17147  df-mulr 17148  df-sca 17150  df-vsca 17151  df-edring 39223  df-dvech 39545
This theorem is referenced by:  dvhvadd  39558
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