| Step | Hyp | Ref
| Expression |
| 1 | | gsum2d.w |
. . . 4
⊢ (𝜑 → 𝐹 finSupp 0 ) |
| 2 | 1 | fsuppimpd 9409 |
. . 3
⊢ (𝜑 → (𝐹 supp 0 ) ∈
Fin) |
| 3 | | dmfi 9375 |
. . 3
⊢ ((𝐹 supp 0 ) ∈ Fin → dom
(𝐹 supp 0 ) ∈
Fin) |
| 4 | 2, 3 | syl 17 |
. 2
⊢ (𝜑 → dom (𝐹 supp 0 ) ∈
Fin) |
| 5 | | reseq2 5992 |
. . . . . . . . 9
⊢ (𝑥 = ∅ → (𝐴 ↾ 𝑥) = (𝐴 ↾ ∅)) |
| 6 | | res0 6001 |
. . . . . . . . 9
⊢ (𝐴 ↾ ∅) =
∅ |
| 7 | 5, 6 | eqtrdi 2793 |
. . . . . . . 8
⊢ (𝑥 = ∅ → (𝐴 ↾ 𝑥) = ∅) |
| 8 | 7 | reseq2d 5997 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝐹 ↾ (𝐴 ↾ 𝑥)) = (𝐹 ↾ ∅)) |
| 9 | | res0 6001 |
. . . . . . 7
⊢ (𝐹 ↾ ∅) =
∅ |
| 10 | 8, 9 | eqtrdi 2793 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝐹 ↾ (𝐴 ↾ 𝑥)) = ∅) |
| 11 | 10 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = ∅ → (𝐺 Σg
(𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg
∅)) |
| 12 | | mpteq1 5235 |
. . . . . . 7
⊢ (𝑥 = ∅ → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ ∅ ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) |
| 13 | | mpt0 6710 |
. . . . . . 7
⊢ (𝑗 ∈ ∅ ↦ (𝐺 Σg
(𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅ |
| 14 | 12, 13 | eqtrdi 2793 |
. . . . . 6
⊢ (𝑥 = ∅ → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = ∅) |
| 15 | 14 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = ∅ → (𝐺 Σg
(𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg
∅)) |
| 16 | 11, 15 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = ∅ → ((𝐺 Σg
(𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg ∅) =
(𝐺
Σg ∅))) |
| 17 | 16 | imbi2d 340 |
. . 3
⊢ (𝑥 = ∅ → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg ∅) =
(𝐺
Σg ∅)))) |
| 18 | | reseq2 5992 |
. . . . . . 7
⊢ (𝑥 = 𝑦 → (𝐴 ↾ 𝑥) = (𝐴 ↾ 𝑦)) |
| 19 | 18 | reseq2d 5997 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝐹 ↾ (𝐴 ↾ 𝑥)) = (𝐹 ↾ (𝐴 ↾ 𝑦))) |
| 20 | 19 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))) |
| 21 | | mpteq1 5235 |
. . . . . 6
⊢ (𝑥 = 𝑦 → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) |
| 22 | 21 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = 𝑦 → (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |
| 23 | 20, 22 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = 𝑦 → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))) |
| 24 | 23 | imbi2d 340 |
. . 3
⊢ (𝑥 = 𝑦 → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))) |
| 25 | | reseq2 5992 |
. . . . . . 7
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐴 ↾ 𝑥) = (𝐴 ↾ (𝑦 ∪ {𝑧}))) |
| 26 | 25 | reseq2d 5997 |
. . . . . 6
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐹 ↾ (𝐴 ↾ 𝑥)) = (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) |
| 27 | 26 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))))) |
| 28 | | mpteq1 5235 |
. . . . . 6
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) |
| 29 | 28 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |
| 30 | 27, 29 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))) |
| 31 | 30 | imbi2d 340 |
. . 3
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))) |
| 32 | | reseq2 5992 |
. . . . . . 7
⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝐴 ↾ 𝑥) = (𝐴 ↾ dom (𝐹 supp 0 ))) |
| 33 | 32 | reseq2d 5997 |
. . . . . 6
⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝐹 ↾ (𝐴 ↾ 𝑥)) = (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) |
| 34 | 33 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝐺 Σg
(𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 ))))) |
| 35 | | mpteq1 5235 |
. . . . . 6
⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))) = (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg
(𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) |
| 36 | 35 | oveq2d 7447 |
. . . . 5
⊢ (𝑥 = dom (𝐹 supp 0 ) → (𝐺 Σg
(𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg
(𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |
| 37 | 34, 36 | eqeq12d 2753 |
. . . 4
⊢ (𝑥 = dom (𝐹 supp 0 ) → ((𝐺 Σg
(𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg
(𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))) |
| 38 | 37 | imbi2d 340 |
. . 3
⊢ (𝑥 = dom (𝐹 supp 0 ) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑥))) = (𝐺 Σg (𝑗 ∈ 𝑥 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) ↔ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg
(𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))) |
| 39 | | eqidd 2738 |
. . 3
⊢ (𝜑 → (𝐺 Σg ∅) =
(𝐺
Σg ∅)) |
| 40 | | oveq1 7438 |
. . . . . 6
⊢ ((𝐺 Σg
(𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))) |
| 41 | | gsum2d.b |
. . . . . . . . 9
⊢ 𝐵 = (Base‘𝐺) |
| 42 | | gsum2d.z |
. . . . . . . . 9
⊢ 0 =
(0g‘𝐺) |
| 43 | | eqid 2737 |
. . . . . . . . 9
⊢
(+g‘𝐺) = (+g‘𝐺) |
| 44 | | gsum2d.g |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ CMnd) |
| 45 | 44 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝐺 ∈ CMnd) |
| 46 | | gsum2d.a |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
| 47 | 46 | resexd 6046 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V) |
| 48 | 47 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) ∈ V) |
| 49 | | gsum2d.f |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
| 50 | | resss 6019 |
. . . . . . . . . . 11
⊢ (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴 |
| 51 | | fssres 6774 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴⟶𝐵 ∧ (𝐴 ↾ (𝑦 ∪ {𝑧})) ⊆ 𝐴) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵) |
| 52 | 49, 50, 51 | sylancl 586 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵) |
| 53 | 52 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))):(𝐴 ↾ (𝑦 ∪ {𝑧}))⟶𝐵) |
| 54 | 49 | ffund 6740 |
. . . . . . . . . . . 12
⊢ (𝜑 → Fun 𝐹) |
| 55 | 54 | funresd 6609 |
. . . . . . . . . . 11
⊢ (𝜑 → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) |
| 56 | 55 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) |
| 57 | 2 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹 supp 0 ) ∈
Fin) |
| 58 | 49, 46 | fexd 7247 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹 ∈ V) |
| 59 | 42 | fvexi 6920 |
. . . . . . . . . . . . 13
⊢ 0 ∈
V |
| 60 | | ressuppss 8208 |
. . . . . . . . . . . . 13
⊢ ((𝐹 ∈ V ∧ 0 ∈ V)
→ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 )) |
| 61 | 58, 59, 60 | sylancl 586 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 )) |
| 62 | 61 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ⊆ (𝐹 supp 0 )) |
| 63 | 57, 62 | ssfid 9301 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈
Fin) |
| 64 | 58 | resexd 6046 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V) |
| 65 | | isfsupp 9405 |
. . . . . . . . . . . 12
⊢ (((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∈ V ∧ 0 ∈ V) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈
Fin))) |
| 66 | 64, 59, 65 | sylancl 586 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈
Fin))) |
| 67 | 66 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ↔ (Fun (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ∧ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) supp 0 ) ∈
Fin))) |
| 68 | 56, 63, 67 | mpbir2and 713 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) finSupp 0 ) |
| 69 | | simprr 773 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ¬ 𝑧 ∈ 𝑦) |
| 70 | | disjsn 4711 |
. . . . . . . . . . . 12
⊢ ((𝑦 ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ 𝑦) |
| 71 | 69, 70 | sylibr 234 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝑦 ∩ {𝑧}) = ∅) |
| 72 | 71 | reseq2d 5997 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↾ (𝑦 ∩ {𝑧})) = (𝐴 ↾ ∅)) |
| 73 | | resindi 6013 |
. . . . . . . . . 10
⊢ (𝐴 ↾ (𝑦 ∩ {𝑧})) = ((𝐴 ↾ 𝑦) ∩ (𝐴 ↾ {𝑧})) |
| 74 | 72, 73, 6 | 3eqtr3g 2800 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐴 ↾ 𝑦) ∩ (𝐴 ↾ {𝑧})) = ∅) |
| 75 | | resundi 6011 |
. . . . . . . . . 10
⊢ (𝐴 ↾ (𝑦 ∪ {𝑧})) = ((𝐴 ↾ 𝑦) ∪ (𝐴 ↾ {𝑧})) |
| 76 | 75 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐴 ↾ (𝑦 ∪ {𝑧})) = ((𝐴 ↾ 𝑦) ∪ (𝐴 ↾ {𝑧}))) |
| 77 | 41, 42, 43, 45, 48, 53, 68, 74, 76 | gsumsplit 19946 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧}))))) |
| 78 | | ssun1 4178 |
. . . . . . . . . . 11
⊢ 𝑦 ⊆ (𝑦 ∪ {𝑧}) |
| 79 | | ssres2 6022 |
. . . . . . . . . . 11
⊢ (𝑦 ⊆ (𝑦 ∪ {𝑧}) → (𝐴 ↾ 𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧}))) |
| 80 | | resabs1 6024 |
. . . . . . . . . . 11
⊢ ((𝐴 ↾ 𝑦) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦)) = (𝐹 ↾ (𝐴 ↾ 𝑦))) |
| 81 | 78, 79, 80 | mp2b 10 |
. . . . . . . . . 10
⊢ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦)) = (𝐹 ↾ (𝐴 ↾ 𝑦)) |
| 82 | 81 | oveq2i 7442 |
. . . . . . . . 9
⊢ (𝐺 Σg
((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) |
| 83 | | ssun2 4179 |
. . . . . . . . . . 11
⊢ {𝑧} ⊆ (𝑦 ∪ {𝑧}) |
| 84 | | ssres2 6022 |
. . . . . . . . . . 11
⊢ ({𝑧} ⊆ (𝑦 ∪ {𝑧}) → (𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧}))) |
| 85 | | resabs1 6024 |
. . . . . . . . . . 11
⊢ ((𝐴 ↾ {𝑧}) ⊆ (𝐴 ↾ (𝑦 ∪ {𝑧})) → ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})) = (𝐹 ↾ (𝐴 ↾ {𝑧}))) |
| 86 | 83, 84, 85 | mp2b 10 |
. . . . . . . . . 10
⊢ ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})) = (𝐹 ↾ (𝐴 ↾ {𝑧})) |
| 87 | 86 | oveq2i 7442 |
. . . . . . . . 9
⊢ (𝐺 Σg
((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))) |
| 88 | 82, 87 | oveq12i 7443 |
. . . . . . . 8
⊢ ((𝐺 Σg
((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg ((𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧}))) ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) |
| 89 | 77, 88 | eqtrdi 2793 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))) |
| 90 | | simprl 771 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑦 ∈ Fin) |
| 91 | | gsum2d.r |
. . . . . . . . . . 11
⊢ (𝜑 → Rel 𝐴) |
| 92 | | gsum2d.d |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐷 ∈ 𝑊) |
| 93 | | gsum2d.s |
. . . . . . . . . . 11
⊢ (𝜑 → dom 𝐴 ⊆ 𝐷) |
| 94 | 41, 42, 44, 46, 91, 92, 93, 49, 1 | gsum2dlem1 19988 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) |
| 95 | 94 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) ∧ 𝑗 ∈ 𝑦) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) |
| 96 | | vex 3484 |
. . . . . . . . . 10
⊢ 𝑧 ∈ V |
| 97 | 96 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → 𝑧 ∈ V) |
| 98 | | sneq 4636 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑧 → {𝑗} = {𝑧}) |
| 99 | 98 | imaeq2d 6078 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑧 → (𝐴 “ {𝑗}) = (𝐴 “ {𝑧})) |
| 100 | | oveq1 7438 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑧 → (𝑗𝐹𝑘) = (𝑧𝐹𝑘)) |
| 101 | 99, 100 | mpteq12dv 5233 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑧 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) |
| 102 | 101 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑧 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))) |
| 103 | 102 | eleq1d 2826 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑧 → ((𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵 ↔ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵)) |
| 104 | 103 | imbi2d 340 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑧 → ((𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) ∈ 𝐵) ↔ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵))) |
| 105 | 104, 94 | chvarvv 1998 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵) |
| 106 | 105 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) ∈ 𝐵) |
| 107 | 41, 43, 45, 90, 95, 97, 69, 106, 102 | gsumunsn 19978 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))))) |
| 108 | 98 | reseq2d 5997 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑧 → (𝐴 ↾ {𝑗}) = (𝐴 ↾ {𝑧})) |
| 109 | 108 | reseq2d 5997 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑧 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝐹 ↾ (𝐴 ↾ {𝑧}))) |
| 110 | 109 | oveq2d 7447 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑧 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) |
| 111 | 102, 110 | eqeq12d 2753 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑧 → ((𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) ↔ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))) |
| 112 | 111 | imbi2d 340 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑧 → ((𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗})))) ↔ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))) |
| 113 | | imaexg 7935 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ 𝑉 → (𝐴 “ {𝑗}) ∈ V) |
| 114 | 46, 113 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐴 “ {𝑗}) ∈ V) |
| 115 | | vex 3484 |
. . . . . . . . . . . . . . . 16
⊢ 𝑗 ∈ V |
| 116 | | vex 3484 |
. . . . . . . . . . . . . . . 16
⊢ 𝑘 ∈ V |
| 117 | 115, 116 | elimasn 6108 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐴 “ {𝑗}) ↔ 〈𝑗, 𝑘〉 ∈ 𝐴) |
| 118 | | df-ov 7434 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗𝐹𝑘) = (𝐹‘〈𝑗, 𝑘〉) |
| 119 | 49 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝐹‘〈𝑗, 𝑘〉) ∈ 𝐵) |
| 120 | 118, 119 | eqeltrid 2845 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ 𝐴) → (𝑗𝐹𝑘) ∈ 𝐵) |
| 121 | 117, 120 | sylan2b 594 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 “ {𝑗})) → (𝑗𝐹𝑘) ∈ 𝐵) |
| 122 | 121 | fmpttd 7135 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)):(𝐴 “ {𝑗})⟶𝐵) |
| 123 | | funmpt 6604 |
. . . . . . . . . . . . . . 15
⊢ Fun
(𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) |
| 124 | 123 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) |
| 125 | | rnfi 9380 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹 supp 0 ) ∈ Fin → ran
(𝐹 supp 0 ) ∈
Fin) |
| 126 | 2, 125 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ran (𝐹 supp 0 ) ∈
Fin) |
| 127 | 117 | biimpi 216 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (𝐴 “ {𝑗}) → 〈𝑗, 𝑘〉 ∈ 𝐴) |
| 128 | 115, 116 | opelrn 5954 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ) → 𝑘 ∈ ran (𝐹 supp 0 )) |
| 129 | 128 | con3i 154 |
. . . . . . . . . . . . . . . . . . 19
⊢ (¬
𝑘 ∈ ran (𝐹 supp 0 ) → ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 )) |
| 130 | 127, 129 | anim12i 613 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 )) → (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) |
| 131 | | eldif 3961 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) ↔ (𝑘 ∈ (𝐴 “ {𝑗}) ∧ ¬ 𝑘 ∈ ran (𝐹 supp 0 ))) |
| 132 | | eldif 3961 |
. . . . . . . . . . . . . . . . . 18
⊢
(〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 )) ↔ (〈𝑗, 𝑘〉 ∈ 𝐴 ∧ ¬ 〈𝑗, 𝑘〉 ∈ (𝐹 supp 0 ))) |
| 133 | 130, 131,
132 | 3imtr4i 292 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 )) → 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) |
| 134 | | ssidd 4007 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (𝐹 supp 0 ) ⊆ (𝐹 supp 0 )) |
| 135 | 59 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 0 ∈ V) |
| 136 | 49, 134, 46, 135 | suppssr 8220 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝐹‘〈𝑗, 𝑘〉) = 0 ) |
| 137 | 118, 136 | eqtrid 2789 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 〈𝑗, 𝑘〉 ∈ (𝐴 ∖ (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
| 138 | 133, 137 | sylan2 593 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ ((𝐴 “ {𝑗}) ∖ ran (𝐹 supp 0 ))) → (𝑗𝐹𝑘) = 0 ) |
| 139 | 138, 114 | suppss2 8225 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ⊆ ran (𝐹 supp 0 )) |
| 140 | 126, 139 | ssfid 9301 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈
Fin) |
| 141 | 114 | mptexd 7244 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V) |
| 142 | | isfsupp 9405 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∈ V ∧ 0 ∈ V) → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ↔ (Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈
Fin))) |
| 143 | 141, 59, 142 | sylancl 586 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ↔ (Fun (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∧ ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) supp 0 ) ∈
Fin))) |
| 144 | 124, 140,
143 | mpbir2and 713 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) finSupp 0 ) |
| 145 | | 2ndconst 8126 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ V → (2nd
↾ ({𝑗} × (𝐴 “ {𝑗}))):({𝑗} × (𝐴 “ {𝑗}))–1-1-onto→(𝐴 “ {𝑗})) |
| 146 | 115, 145 | mp1i 13 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2nd ↾
({𝑗} × (𝐴 “ {𝑗}))):({𝑗} × (𝐴 “ {𝑗}))–1-1-onto→(𝐴 “ {𝑗})) |
| 147 | 41, 42, 44, 114, 122, 144, 146 | gsumf1o 19934 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))))) |
| 148 | | 1st2nd2 8053 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = 〈(1st ‘𝑥), (2nd ‘𝑥)〉) |
| 149 | | xp1st 8046 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st ‘𝑥) ∈ {𝑗}) |
| 150 | | elsni 4643 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((1st ‘𝑥) ∈ {𝑗} → (1st ‘𝑥) = 𝑗) |
| 151 | 149, 150 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (1st ‘𝑥) = 𝑗) |
| 152 | 151 | opeq1d 4879 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 〈(1st ‘𝑥), (2nd ‘𝑥)〉 = 〈𝑗, (2nd ‘𝑥)〉) |
| 153 | 148, 152 | eqtrd 2777 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → 𝑥 = 〈𝑗, (2nd ‘𝑥)〉) |
| 154 | 153 | fveq2d 6910 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹‘𝑥) = (𝐹‘〈𝑗, (2nd ‘𝑥)〉)) |
| 155 | | df-ov 7434 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗𝐹(2nd ‘𝑥)) = (𝐹‘〈𝑗, (2nd ‘𝑥)〉) |
| 156 | 154, 155 | eqtr4di 2795 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (𝐹‘𝑥) = (𝑗𝐹(2nd ‘𝑥))) |
| 157 | 156 | mpteq2ia 5245 |
. . . . . . . . . . . . . 14
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹‘𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd ‘𝑥))) |
| 158 | 49 | feqmptd 6977 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥))) |
| 159 | 158 | reseq1d 5996 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐴 ↾ {𝑗}))) |
| 160 | | resss 6019 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ↾ {𝑗}) ⊆ 𝐴 |
| 161 | | resmpt 6055 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ↾ {𝑗}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹‘𝑥))) |
| 162 | 160, 161 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹‘𝑥)) |
| 163 | | ressn 6305 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐴 ↾ {𝑗}) = ({𝑗} × (𝐴 “ {𝑗})) |
| 164 | 163 | mpteq1i 5238 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ (𝐴 ↾ {𝑗}) ↦ (𝐹‘𝑥)) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹‘𝑥)) |
| 165 | 162, 164 | eqtri 2765 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 ∈ 𝐴 ↦ (𝐹‘𝑥)) ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹‘𝑥)) |
| 166 | 159, 165 | eqtrdi 2793 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝐹‘𝑥))) |
| 167 | | xp2nd 8047 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) → (2nd ‘𝑥) ∈ (𝐴 “ {𝑗})) |
| 168 | 167 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗}))) → (2nd ‘𝑥) ∈ (𝐴 “ {𝑗})) |
| 169 | | fo2nd 8035 |
. . . . . . . . . . . . . . . . . . 19
⊢
2nd :V–onto→V |
| 170 | | fof 6820 |
. . . . . . . . . . . . . . . . . . 19
⊢
(2nd :V–onto→V → 2nd
:V⟶V) |
| 171 | 169, 170 | mp1i 13 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 2nd
:V⟶V) |
| 172 | 171 | feqmptd 6977 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 2nd = (𝑥 ∈ V ↦
(2nd ‘𝑥))) |
| 173 | 172 | reseq1d 5996 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (2nd ↾
({𝑗} × (𝐴 “ {𝑗}))) = ((𝑥 ∈ V ↦ (2nd
‘𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗})))) |
| 174 | | ssv 4008 |
. . . . . . . . . . . . . . . . 17
⊢ ({𝑗} × (𝐴 “ {𝑗})) ⊆ V |
| 175 | | resmpt 6055 |
. . . . . . . . . . . . . . . . 17
⊢ (({𝑗} × (𝐴 “ {𝑗})) ⊆ V → ((𝑥 ∈ V ↦ (2nd
‘𝑥)) ↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd ‘𝑥))) |
| 176 | 174, 175 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ∈ V ↦
(2nd ‘𝑥))
↾ ({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd ‘𝑥)) |
| 177 | 173, 176 | eqtrdi 2793 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (2nd ↾
({𝑗} × (𝐴 “ {𝑗}))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (2nd ‘𝑥))) |
| 178 | | eqidd 2738 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) = (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) |
| 179 | | oveq2 7439 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (2nd ‘𝑥) → (𝑗𝐹𝑘) = (𝑗𝐹(2nd ‘𝑥))) |
| 180 | 168, 177,
178, 179 | fmptco 7149 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))) = (𝑥 ∈ ({𝑗} × (𝐴 “ {𝑗})) ↦ (𝑗𝐹(2nd ‘𝑥)))) |
| 181 | 157, 166,
180 | 3eqtr4a 2803 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐹 ↾ (𝐴 ↾ {𝑗})) = ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗}))))) |
| 182 | 181 | oveq2d 7447 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗}))) = (𝐺 Σg ((𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)) ∘ (2nd ↾ ({𝑗} × (𝐴 “ {𝑗})))))) |
| 183 | 147, 182 | eqtr4d 2780 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑗})))) |
| 184 | 112, 183 | chvarvv 1998 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) |
| 185 | 184 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘))) = (𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) |
| 186 | 185 | oveq2d 7447 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑧}) ↦ (𝑧𝐹𝑘)))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))) |
| 187 | 107, 186 | eqtrd 2777 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧}))))) |
| 188 | 89, 187 | eqeq12d 2753 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) ↔ ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦)))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))) = ((𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))(+g‘𝐺)(𝐺 Σg (𝐹 ↾ (𝐴 ↾ {𝑧})))))) |
| 189 | 40, 188 | imbitrrid 246 |
. . . . 5
⊢ ((𝜑 ∧ (𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦)) → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))) |
| 190 | 189 | expcom 413 |
. . . 4
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → (𝜑 → ((𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))) → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))) |
| 191 | 190 | a2d 29 |
. . 3
⊢ ((𝑦 ∈ Fin ∧ ¬ 𝑧 ∈ 𝑦) → ((𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ 𝑦))) = (𝐺 Σg (𝑗 ∈ 𝑦 ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) → (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ (𝑦 ∪ {𝑧})))) = (𝐺 Σg (𝑗 ∈ (𝑦 ∪ {𝑧}) ↦ (𝐺 Σg (𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))))) |
| 192 | 17, 24, 31, 38, 39, 191 | findcard2s 9205 |
. 2
⊢ (dom
(𝐹 supp 0 ) ∈ Fin → (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg
(𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘))))))) |
| 193 | 4, 192 | mpcom 38 |
1
⊢ (𝜑 → (𝐺 Σg (𝐹 ↾ (𝐴 ↾ dom (𝐹 supp 0 )))) = (𝐺 Σg (𝑗 ∈ dom (𝐹 supp 0 ) ↦ (𝐺 Σg
(𝑘 ∈ (𝐴 “ {𝑗}) ↦ (𝑗𝐹𝑘)))))) |