| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . 3
⊢
(Base‘(𝑆
⊕m 𝑅)) =
(Base‘(𝑆
⊕m 𝑅)) |
| 2 | 1 | dsmmval2 21756 |
. 2
⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) |
| 3 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑆Xs𝑅) = (𝑆Xs𝑅) |
| 4 | | eqid 2737 |
. . . . . . . . . . 11
⊢
(Base‘(𝑆Xs𝑅)) = (Base‘(𝑆Xs𝑅)) |
| 5 | | noel 4338 |
. . . . . . . . . . . . . 14
⊢ ¬
𝑓 ∈
∅ |
| 6 | | reldmprds 17493 |
. . . . . . . . . . . . . . . . . 18
⊢ Rel dom
Xs |
| 7 | 6 | ovprc1 7470 |
. . . . . . . . . . . . . . . . 17
⊢ (¬
𝑆 ∈ V → (𝑆Xs𝑅) = ∅) |
| 8 | 7 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (¬
𝑆 ∈ V →
(Base‘(𝑆Xs𝑅)) = (Base‘∅)) |
| 9 | | base0 17252 |
. . . . . . . . . . . . . . . 16
⊢ ∅ =
(Base‘∅) |
| 10 | 8, 9 | eqtr4di 2795 |
. . . . . . . . . . . . . . 15
⊢ (¬
𝑆 ∈ V →
(Base‘(𝑆Xs𝑅)) = ∅) |
| 11 | 10 | eleq2d 2827 |
. . . . . . . . . . . . . 14
⊢ (¬
𝑆 ∈ V → (𝑓 ∈ (Base‘(𝑆Xs𝑅)) ↔ 𝑓 ∈ ∅)) |
| 12 | 5, 11 | mtbiri 327 |
. . . . . . . . . . . . 13
⊢ (¬
𝑆 ∈ V → ¬
𝑓 ∈ (Base‘(𝑆Xs𝑅))) |
| 13 | 12 | con4i 114 |
. . . . . . . . . . . 12
⊢ (𝑓 ∈ (Base‘(𝑆Xs𝑅)) → 𝑆 ∈ V) |
| 14 | 13 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑆 ∈ V) |
| 15 | | simplr 769 |
. . . . . . . . . . 11
⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝐼 ∈ Fin) |
| 16 | | simpll 767 |
. . . . . . . . . . 11
⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑅 Fn 𝐼) |
| 17 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 ∈ (Base‘(𝑆Xs𝑅))) |
| 18 | 3, 4, 14, 15, 16, 17 | prdsbasfn 17516 |
. . . . . . . . . 10
⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → 𝑓 Fn 𝐼) |
| 19 | 18 | fndmd 6673 |
. . . . . . . . 9
⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom 𝑓 = 𝐼) |
| 20 | 19, 15 | eqeltrd 2841 |
. . . . . . . 8
⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom 𝑓 ∈ Fin) |
| 21 | | difss 4136 |
. . . . . . . . 9
⊢ (𝑓 ∖ (0g ∘
𝑅)) ⊆ 𝑓 |
| 22 | | dmss 5913 |
. . . . . . . . 9
⊢ ((𝑓 ∖ (0g ∘
𝑅)) ⊆ 𝑓 → dom (𝑓 ∖ (0g ∘ 𝑅)) ⊆ dom 𝑓) |
| 23 | 21, 22 | ax-mp 5 |
. . . . . . . 8
⊢ dom
(𝑓 ∖ (0g
∘ 𝑅)) ⊆ dom
𝑓 |
| 24 | | ssfi 9213 |
. . . . . . . 8
⊢ ((dom
𝑓 ∈ Fin ∧ dom
(𝑓 ∖ (0g
∘ 𝑅)) ⊆ dom
𝑓) → dom (𝑓 ∖ (0g ∘
𝑅)) ∈
Fin) |
| 25 | 20, 23, 24 | sylancl 586 |
. . . . . . 7
⊢ (((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) ∧ 𝑓 ∈ (Base‘(𝑆Xs𝑅))) → dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin) |
| 26 | 25 | ralrimiva 3146 |
. . . . . 6
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → ∀𝑓 ∈ (Base‘(𝑆Xs𝑅))dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin) |
| 27 | | rabid2 3470 |
. . . . . 6
⊢
((Base‘(𝑆Xs𝑅)) = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} ↔
∀𝑓 ∈
(Base‘(𝑆Xs𝑅))dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin) |
| 28 | 26, 27 | sylibr 234 |
. . . . 5
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → (Base‘(𝑆Xs𝑅)) = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}) |
| 29 | | eqid 2737 |
. . . . . 6
⊢ {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} |
| 30 | 3, 29 | dsmmbas2 21757 |
. . . . 5
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} =
(Base‘(𝑆
⊕m 𝑅))) |
| 31 | 28, 30 | eqtr2d 2778 |
. . . 4
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → (Base‘(𝑆 ⊕m 𝑅)) = (Base‘(𝑆Xs𝑅))) |
| 32 | 31 | oveq2d 7447 |
. . 3
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) = ((𝑆Xs𝑅) ↾s (Base‘(𝑆Xs𝑅)))) |
| 33 | | ovex 7464 |
. . . 4
⊢ (𝑆Xs𝑅) ∈ V |
| 34 | 4 | ressid 17290 |
. . . 4
⊢ ((𝑆Xs𝑅) ∈ V → ((𝑆Xs𝑅) ↾s (Base‘(𝑆Xs𝑅))) = (𝑆Xs𝑅)) |
| 35 | 33, 34 | ax-mp 5 |
. . 3
⊢ ((𝑆Xs𝑅) ↾s (Base‘(𝑆Xs𝑅))) = (𝑆Xs𝑅) |
| 36 | 32, 35 | eqtrdi 2793 |
. 2
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → ((𝑆Xs𝑅) ↾s (Base‘(𝑆 ⊕m 𝑅))) = (𝑆Xs𝑅)) |
| 37 | 2, 36 | eqtrid 2789 |
1
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → (𝑆 ⊕m 𝑅) = (𝑆Xs𝑅)) |