| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | elex 3501 | . 2
⊢ (𝐾 ∈ 𝑋 → 𝐾 ∈ V) | 
| 2 |  | fveq2 6906 | . . . . 5
⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = (LHyp‘𝐾)) | 
| 3 |  | hvmapval.h | . . . . 5
⊢ 𝐻 = (LHyp‘𝐾) | 
| 4 | 2, 3 | eqtr4di 2795 | . . . 4
⊢ (𝑘 = 𝐾 → (LHyp‘𝑘) = 𝐻) | 
| 5 |  | fveq2 6906 | . . . . . . . 8
⊢ (𝑘 = 𝐾 → (DVecH‘𝑘) = (DVecH‘𝐾)) | 
| 6 | 5 | fveq1d 6908 | . . . . . . 7
⊢ (𝑘 = 𝐾 → ((DVecH‘𝑘)‘𝑤) = ((DVecH‘𝐾)‘𝑤)) | 
| 7 | 6 | fveq2d 6910 | . . . . . 6
⊢ (𝑘 = 𝐾 → (Base‘((DVecH‘𝑘)‘𝑤)) = (Base‘((DVecH‘𝐾)‘𝑤))) | 
| 8 | 6 | fveq2d 6910 | . . . . . . 7
⊢ (𝑘 = 𝐾 →
(0g‘((DVecH‘𝑘)‘𝑤)) = (0g‘((DVecH‘𝐾)‘𝑤))) | 
| 9 | 8 | sneqd 4638 | . . . . . 6
⊢ (𝑘 = 𝐾 →
{(0g‘((DVecH‘𝑘)‘𝑤))} =
{(0g‘((DVecH‘𝐾)‘𝑤))}) | 
| 10 | 7, 9 | difeq12d 4127 | . . . . 5
⊢ (𝑘 = 𝐾 → ((Base‘((DVecH‘𝑘)‘𝑤)) ∖
{(0g‘((DVecH‘𝑘)‘𝑤))}) = ((Base‘((DVecH‘𝐾)‘𝑤)) ∖
{(0g‘((DVecH‘𝐾)‘𝑤))})) | 
| 11 | 6 | fveq2d 6910 | . . . . . . . 8
⊢ (𝑘 = 𝐾 → (Scalar‘((DVecH‘𝑘)‘𝑤)) = (Scalar‘((DVecH‘𝐾)‘𝑤))) | 
| 12 | 11 | fveq2d 6910 | . . . . . . 7
⊢ (𝑘 = 𝐾 →
(Base‘(Scalar‘((DVecH‘𝑘)‘𝑤))) =
(Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))) | 
| 13 |  | fveq2 6906 | . . . . . . . . . 10
⊢ (𝑘 = 𝐾 → (ocH‘𝑘) = (ocH‘𝐾)) | 
| 14 | 13 | fveq1d 6908 | . . . . . . . . 9
⊢ (𝑘 = 𝐾 → ((ocH‘𝑘)‘𝑤) = ((ocH‘𝐾)‘𝑤)) | 
| 15 | 14 | fveq1d 6908 | . . . . . . . 8
⊢ (𝑘 = 𝐾 → (((ocH‘𝑘)‘𝑤)‘{𝑥}) = (((ocH‘𝐾)‘𝑤)‘{𝑥})) | 
| 16 | 6 | fveq2d 6910 | . . . . . . . . . 10
⊢ (𝑘 = 𝐾 →
(+g‘((DVecH‘𝑘)‘𝑤)) = (+g‘((DVecH‘𝐾)‘𝑤))) | 
| 17 |  | eqidd 2738 | . . . . . . . . . 10
⊢ (𝑘 = 𝐾 → 𝑡 = 𝑡) | 
| 18 | 6 | fveq2d 6910 | . . . . . . . . . . 11
⊢ (𝑘 = 𝐾 → (
·𝑠 ‘((DVecH‘𝑘)‘𝑤)) = ( ·𝑠
‘((DVecH‘𝐾)‘𝑤))) | 
| 19 | 18 | oveqd 7448 | . . . . . . . . . 10
⊢ (𝑘 = 𝐾 → (𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥) = (𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥)) | 
| 20 | 16, 17, 19 | oveq123d 7452 | . . . . . . . . 9
⊢ (𝑘 = 𝐾 → (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥)) = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥))) | 
| 21 | 20 | eqeq2d 2748 | . . . . . . . 8
⊢ (𝑘 = 𝐾 → (𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥)) ↔ 𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥)))) | 
| 22 | 15, 21 | rexeqbidv 3347 | . . . . . . 7
⊢ (𝑘 = 𝐾 → (∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥)) ↔ ∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥)))) | 
| 23 | 12, 22 | riotaeqbidv 7391 | . . . . . 6
⊢ (𝑘 = 𝐾 → (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥))) = (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥)))) | 
| 24 | 7, 23 | mpteq12dv 5233 | . . . . 5
⊢ (𝑘 = 𝐾 → (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥)))) = (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥))))) | 
| 25 | 10, 24 | mpteq12dv 5233 | . . . 4
⊢ (𝑘 = 𝐾 → (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖
{(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥))))) = (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖
{(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥)))))) | 
| 26 | 4, 25 | mpteq12dv 5233 | . . 3
⊢ (𝑘 = 𝐾 → (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖
{(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥)))))) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖
{(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥))))))) | 
| 27 |  | df-hvmap 41759 | . . 3
⊢ HVMap =
(𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ ((Base‘((DVecH‘𝑘)‘𝑤)) ∖
{(0g‘((DVecH‘𝑘)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝑘)‘𝑤)))∃𝑡 ∈ (((ocH‘𝑘)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝑘)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝑘)‘𝑤))𝑥))))))) | 
| 28 | 26, 27, 3 | mptfvmpt 7248 | . 2
⊢ (𝐾 ∈ V →
(HVMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖
{(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥))))))) | 
| 29 | 1, 28 | syl 17 | 1
⊢ (𝐾 ∈ 𝑋 → (HVMap‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ ((Base‘((DVecH‘𝐾)‘𝑤)) ∖
{(0g‘((DVecH‘𝐾)‘𝑤))}) ↦ (𝑣 ∈ (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (℩𝑗 ∈
(Base‘(Scalar‘((DVecH‘𝐾)‘𝑤)))∃𝑡 ∈ (((ocH‘𝐾)‘𝑤)‘{𝑥})𝑣 = (𝑡(+g‘((DVecH‘𝐾)‘𝑤))(𝑗( ·𝑠
‘((DVecH‘𝐾)‘𝑤))𝑥))))))) |