| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | xpsval.t | . 2
⊢ 𝑇 = (𝑅 ×s 𝑆) | 
| 2 |  | xpsval.1 | . . . 4
⊢ (𝜑 → 𝑅 ∈ 𝑉) | 
| 3 | 2 | elexd 3504 | . . 3
⊢ (𝜑 → 𝑅 ∈ V) | 
| 4 |  | xpsval.2 | . . . 4
⊢ (𝜑 → 𝑆 ∈ 𝑊) | 
| 5 | 4 | elexd 3504 | . . 3
⊢ (𝜑 → 𝑆 ∈ V) | 
| 6 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅)) | 
| 7 |  | xpsval.x | . . . . . . . . 9
⊢ 𝑋 = (Base‘𝑅) | 
| 8 | 6, 7 | eqtr4di 2795 | . . . . . . . 8
⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝑋) | 
| 9 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆)) | 
| 10 |  | xpsval.y | . . . . . . . . 9
⊢ 𝑌 = (Base‘𝑆) | 
| 11 | 9, 10 | eqtr4di 2795 | . . . . . . . 8
⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝑌) | 
| 12 |  | mpoeq12 7506 | . . . . . . . 8
⊢
(((Base‘𝑟) =
𝑋 ∧ (Base‘𝑠) = 𝑌) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})) | 
| 13 | 8, 11, 12 | syl2an 596 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})) | 
| 14 |  | xpsval.f | . . . . . . 7
⊢ 𝐹 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) | 
| 15 | 13, 14 | eqtr4di 2795 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = 𝐹) | 
| 16 | 15 | cnveqd 5886 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉}) = ◡𝐹) | 
| 17 |  | fveq2 6906 | . . . . . . . . 9
⊢ (𝑟 = 𝑅 → (Scalar‘𝑟) = (Scalar‘𝑅)) | 
| 18 | 17 | adantr 480 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = (Scalar‘𝑅)) | 
| 19 |  | xpsval.k | . . . . . . . 8
⊢ 𝐺 = (Scalar‘𝑅) | 
| 20 | 18, 19 | eqtr4di 2795 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (Scalar‘𝑟) = 𝐺) | 
| 21 |  | simpl 482 | . . . . . . . . 9
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑟 = 𝑅) | 
| 22 | 21 | opeq2d 4880 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 〈∅, 𝑟〉 = 〈∅, 𝑅〉) | 
| 23 |  | simpr 484 | . . . . . . . . 9
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 𝑠 = 𝑆) | 
| 24 | 23 | opeq2d 4880 | . . . . . . . 8
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → 〈1o, 𝑠〉 = 〈1o,
𝑆〉) | 
| 25 | 22, 24 | preq12d 4741 | . . . . . . 7
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → {〈∅, 𝑟〉, 〈1o, 𝑠〉} = {〈∅, 𝑅〉, 〈1o,
𝑆〉}) | 
| 26 | 20, 25 | oveq12d 7449 | . . . . . 6
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{〈∅, 𝑟〉, 〈1o, 𝑠〉}) = (𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉})) | 
| 27 |  | xpsval.u | . . . . . 6
⊢ 𝑈 = (𝐺Xs{〈∅, 𝑅〉, 〈1o, 𝑆〉}) | 
| 28 | 26, 27 | eqtr4di 2795 | . . . . 5
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → ((Scalar‘𝑟)Xs{〈∅, 𝑟〉, 〈1o, 𝑠〉}) = 𝑈) | 
| 29 | 16, 28 | oveq12d 7449 | . . . 4
⊢ ((𝑟 = 𝑅 ∧ 𝑠 = 𝑆) → (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})
“s ((Scalar‘𝑟)Xs{〈∅, 𝑟〉, 〈1o, 𝑠〉})) = (◡𝐹 “s 𝑈)) | 
| 30 |  | df-xps 17555 | . . . 4
⊢ 
×s = (𝑟 ∈ V, 𝑠 ∈ V ↦ (◡(𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Base‘𝑠) ↦ {〈∅, 𝑥〉, 〈1o, 𝑦〉})
“s ((Scalar‘𝑟)Xs{〈∅, 𝑟〉, 〈1o, 𝑠〉}))) | 
| 31 |  | ovex 7464 | . . . 4
⊢ (◡𝐹 “s 𝑈) ∈ V | 
| 32 | 29, 30, 31 | ovmpoa 7588 | . . 3
⊢ ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×s
𝑆) = (◡𝐹 “s 𝑈)) | 
| 33 | 3, 5, 32 | syl2anc 584 | . 2
⊢ (𝜑 → (𝑅 ×s 𝑆) = (◡𝐹 “s 𝑈)) | 
| 34 | 1, 33 | eqtrid 2789 | 1
⊢ (𝜑 → 𝑇 = (◡𝐹 “s 𝑈)) |