| Step | Hyp | Ref
| Expression |
| 1 | | eqidd 2738 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
| 2 | | 0zd 12625 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ∈ ℤ) |
| 3 | | dvnprodlem1.j |
. . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈
ℕ0) |
| 4 | 3 | nn0zd 12639 |
. . . . . . . . 9
⊢ (𝜑 → 𝐽 ∈ ℤ) |
| 5 | 4 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℤ) |
| 6 | | oveq2 7439 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽)) |
| 7 | 6 | oveq1d 7446 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝐽 → ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑m (𝑅 ∪ {𝑍}))) |
| 8 | | eqeq2 2749 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽)) |
| 9 | 7, 8 | rabeqbidv 3455 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
| 10 | | dvnprodlem1.c |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})) |
| 11 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑m 𝑠) = ((0...𝑛) ↑m (𝑅 ∪ {𝑍}))) |
| 12 | | sumeq1 15725 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡)) |
| 13 | 12 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛)) |
| 14 | 11, 13 | rabeqbidv 3455 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) |
| 15 | 14 | mpteq2dv 5244 |
. . . . . . . . . . . . . . . 16
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})) |
| 16 | | dvnprodlem1.t |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑇 ∈ Fin) |
| 17 | | dvnprodlem1.rzt |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇) |
| 18 | 16, 17 | sselpwd 5328 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇) |
| 19 | | nn0ex 12532 |
. . . . . . . . . . . . . . . . . 18
⊢
ℕ0 ∈ V |
| 20 | 19 | mptex 7243 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛) ↑m
(𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V |
| 21 | 20 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V) |
| 22 | 10, 15, 18, 21 | fvmptd3 7039 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})) |
| 23 | | ovex 7464 |
. . . . . . . . . . . . . . . . 17
⊢
((0...𝐽)
↑m (𝑅 ∪
{𝑍})) ∈
V |
| 24 | 23 | rabex 5339 |
. . . . . . . . . . . . . . . 16
⊢ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V |
| 25 | 24 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V) |
| 26 | 9, 22, 3, 25 | fvmptd4 7040 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
| 27 | | ssrab2 4080 |
. . . . . . . . . . . . . 14
⊢ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) |
| 28 | 26, 27 | eqsstrdi 4028 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑m (𝑅 ∪ {𝑍}))) |
| 29 | 28 | sselda 3983 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍}))) |
| 30 | | elmapi 8889 |
. . . . . . . . . . . 12
⊢ (𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
| 31 | 29, 30 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
| 32 | | dvnprodlem1.z |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑍 ∈ 𝑇) |
| 33 | | snidg 4660 |
. . . . . . . . . . . . 13
⊢ (𝑍 ∈ 𝑇 → 𝑍 ∈ {𝑍}) |
| 34 | | elun2 4183 |
. . . . . . . . . . . . 13
⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
| 35 | 32, 33, 34 | 3syl 18 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
| 36 | 35 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
| 37 | 31, 36 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝐽)) |
| 38 | 37 | elfzelzd 13565 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℤ) |
| 39 | 5, 38 | zsubcld 12727 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ) |
| 40 | | elfzle2 13568 |
. . . . . . . . . 10
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ≤ 𝐽) |
| 41 | 37, 40 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ≤ 𝐽) |
| 42 | 5 | zred 12722 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℝ) |
| 43 | 38 | zred 12722 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℝ) |
| 44 | 42, 43 | subge0d 11853 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝐽 − (𝑐‘𝑍)) ↔ (𝑐‘𝑍) ≤ 𝐽)) |
| 45 | 41, 44 | mpbird 257 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝐽 − (𝑐‘𝑍))) |
| 46 | | elfzle1 13567 |
. . . . . . . . . 10
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑍)) |
| 47 | 37, 46 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝑐‘𝑍)) |
| 48 | 42, 43 | subge02d 11855 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝑐‘𝑍) ↔ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)) |
| 49 | 47, 48 | mpbid 232 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ≤ 𝐽) |
| 50 | 2, 5, 39, 45, 49 | elfzd 13555 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽)) |
| 51 | | eqidd 2738 |
. . . . . . . . . . 11
⊢ (𝑒 = (𝑐 ↾ 𝑅) → 𝑅 = 𝑅) |
| 52 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝑒 = (𝑐 ↾ 𝑅) ∧ 𝑡 ∈ 𝑅) → 𝑒 = (𝑐 ↾ 𝑅)) |
| 53 | 52 | fveq1d 6908 |
. . . . . . . . . . 11
⊢ ((𝑒 = (𝑐 ↾ 𝑅) ∧ 𝑡 ∈ 𝑅) → (𝑒‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
| 54 | 51, 53 | sumeq12rdv 15743 |
. . . . . . . . . 10
⊢ (𝑒 = (𝑐 ↾ 𝑅) → Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡)) |
| 55 | 54 | eqeq1d 2739 |
. . . . . . . . 9
⊢ (𝑒 = (𝑐 ↾ 𝑅) → (Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍)) ↔ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍)))) |
| 56 | | ovexd 7466 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...(𝐽 − (𝑐‘𝑍))) ∈ V) |
| 57 | 17 | unssad 4193 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 ⊆ 𝑇) |
| 58 | 16, 57 | ssfid 9301 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑅 ∈ Fin) |
| 59 | 58 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin) |
| 60 | | elmapfn 8905 |
. . . . . . . . . . . . 13
⊢ (𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) → 𝑐 Fn (𝑅 ∪ {𝑍})) |
| 61 | 29, 60 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 Fn (𝑅 ∪ {𝑍})) |
| 62 | | ssun1 4178 |
. . . . . . . . . . . . 13
⊢ 𝑅 ⊆ (𝑅 ∪ {𝑍}) |
| 63 | 62 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ⊆ (𝑅 ∪ {𝑍})) |
| 64 | 61, 63 | fnssresd 6692 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) Fn 𝑅) |
| 65 | | nfv 1914 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡𝜑 |
| 66 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑡𝒫 𝑇 |
| 67 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑡ℕ0 |
| 68 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑡𝑠 |
| 69 | 68 | nfsum1 15726 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑡Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) |
| 70 | 69 | nfeq1 2921 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 |
| 71 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡((0...𝑛) ↑m 𝑠) |
| 72 | 70, 71 | nfrabw 3475 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑡{𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} |
| 73 | 67, 72 | nfmpt 5249 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑡(𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
| 74 | 66, 73 | nfmpt 5249 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡(𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})) |
| 75 | 10, 74 | nfcxfr 2903 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡𝐶 |
| 76 | | nfcv 2905 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡(𝑅 ∪ {𝑍}) |
| 77 | 75, 76 | nffv 6916 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑡(𝐶‘(𝑅 ∪ {𝑍})) |
| 78 | | nfcv 2905 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑡𝐽 |
| 79 | 77, 78 | nffv 6916 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) |
| 80 | 79 | nfcri 2897 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) |
| 81 | 65, 80 | nfan 1899 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
| 82 | | fvres 6925 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)) |
| 83 | 82 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)) |
| 84 | | 0zd 12625 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 0 ∈ ℤ) |
| 85 | 39 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ) |
| 86 | 31 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
| 87 | 63 | sselda 3983 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍})) |
| 88 | 86, 87 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝐽)) |
| 89 | 88 | elfzelzd 13565 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℤ) |
| 90 | | elfzle1 13567 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐‘𝑡) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑡)) |
| 91 | 88, 90 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 0 ≤ (𝑐‘𝑡)) |
| 92 | 58 | ad2antrr 726 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑅 ∈ Fin) |
| 93 | | fzssre 45326 |
. . . . . . . . . . . . . . . . . 18
⊢
(0...𝐽) ⊆
ℝ |
| 94 | 31 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
| 95 | 63 | sselda 3983 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ (𝑅 ∪ {𝑍})) |
| 96 | 94, 95 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ (0...𝐽)) |
| 97 | 93, 96 | sselid 3981 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℝ) |
| 98 | 97 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℝ) |
| 99 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑐‘𝑟) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑟)) |
| 100 | 96, 99 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 0 ≤ (𝑐‘𝑟)) |
| 101 | 100 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) ∧ 𝑟 ∈ 𝑅) → 0 ≤ (𝑐‘𝑟)) |
| 102 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = 𝑡 → (𝑐‘𝑟) = (𝑐‘𝑡)) |
| 103 | | simpr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑅) |
| 104 | 92, 98, 101, 102, 103 | fsumge1 15833 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ≤ Σ𝑟 ∈ 𝑅 (𝑐‘𝑟)) |
| 105 | 97 | recnd 11289 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℂ) |
| 106 | 59, 105 | fsumcl 15769 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) ∈ ℂ) |
| 107 | 38 | zcnd 12723 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℂ) |
| 108 | 102 | cbvsumv 15732 |
. . . . . . . . . . . . . . . . . 18
⊢
Σ𝑟 ∈
(𝑅 ∪ {𝑍})(𝑐‘𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) |
| 109 | | nfv 1914 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑟(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
| 110 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑟(𝑐‘𝑍) |
| 111 | 32 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇) |
| 112 | | dvnprodlem1.zr |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅) |
| 113 | 112 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅) |
| 114 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑟 = 𝑍 → (𝑐‘𝑟) = (𝑐‘𝑍)) |
| 115 | 109, 110,
59, 111, 113, 105, 114, 107 | fsumsplitsn 15780 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍))) |
| 116 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
| 117 | 26 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
| 118 | 116, 117 | eleqtrd 2843 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
| 119 | | rabid 3458 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ↔ (𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽)) |
| 120 | 118, 119 | sylib 218 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽)) |
| 121 | 120 | simprd 495 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽) |
| 122 | 108, 115,
121 | 3eqtr3a 2801 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) = 𝐽) |
| 123 | 106, 107,
122 | mvlraddd 11673 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍))) |
| 124 | 123 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍))) |
| 125 | 104, 124 | breqtrd 5169 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍))) |
| 126 | 84, 85, 89, 91, 125 | elfzd 13555 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))) |
| 127 | 83, 126 | eqeltrd 2841 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))) |
| 128 | 81, 127 | ralrimia 3258 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))) |
| 129 | | ffnfv 7139 |
. . . . . . . . . . 11
⊢ ((𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍))) ↔ ((𝑐 ↾ 𝑅) Fn 𝑅 ∧ ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))) |
| 130 | 64, 128, 129 | sylanbrc 583 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍)))) |
| 131 | 56, 59, 130 | elmapdd 8881 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅)) |
| 132 | 82 | a1i 11 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))) |
| 133 | 81, 132 | ralrimi 3257 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)) |
| 134 | 133 | sumeq2d 15737 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡)) |
| 135 | 102 | cbvsumv 15732 |
. . . . . . . . . . . 12
⊢
Σ𝑟 ∈
𝑅 (𝑐‘𝑟) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) |
| 136 | 135 | eqcomi 2746 |
. . . . . . . . . . 11
⊢
Σ𝑡 ∈
𝑅 (𝑐‘𝑡) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) |
| 137 | 136 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟)) |
| 138 | 134, 137,
123 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍))) |
| 139 | 55, 131, 138 | elrabd 3694 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))}) |
| 140 | | fveq1 6905 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = 𝑒 → (𝑐‘𝑡) = (𝑒‘𝑡)) |
| 141 | 140 | sumeq2sdv 15739 |
. . . . . . . . . . . . 13
⊢ (𝑐 = 𝑒 → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑒‘𝑡)) |
| 142 | 141 | eqeq1d 2739 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝑒 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚)) |
| 143 | 142 | cbvrabv 3447 |
. . . . . . . . . . 11
⊢ {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} |
| 144 | 143 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚}) |
| 145 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → (0...𝑚) = (0...(𝐽 − (𝑐‘𝑍)))) |
| 146 | 145 | oveq1d 7446 |
. . . . . . . . . . 11
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → ((0...𝑚) ↑m 𝑅) = ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅)) |
| 147 | 146 | rabeqdv 3452 |
. . . . . . . . . 10
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑒 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚}) |
| 148 | | eqeq2 2749 |
. . . . . . . . . . 11
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → (Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍)))) |
| 149 | 148 | rabbidv 3444 |
. . . . . . . . . 10
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))}) |
| 150 | 144, 147,
149 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))}) |
| 151 | | oveq2 7439 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑅 → ((0...𝑛) ↑m 𝑠) = ((0...𝑛) ↑m 𝑅)) |
| 152 | | sumeq1 15725 |
. . . . . . . . . . . . . . 15
⊢ (𝑠 = 𝑅 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡)) |
| 153 | 152 | eqeq1d 2739 |
. . . . . . . . . . . . . 14
⊢ (𝑠 = 𝑅 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛)) |
| 154 | 151, 153 | rabeqbidv 3455 |
. . . . . . . . . . . . 13
⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
| 155 | 154 | mpteq2dv 5244 |
. . . . . . . . . . . 12
⊢ (𝑠 = 𝑅 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
| 156 | 16, 57 | sselpwd 5328 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑅 ∈ 𝒫 𝑇) |
| 157 | 19 | mptex 7243 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛) ↑m
𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V |
| 158 | 157 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V) |
| 159 | 10, 155, 156, 158 | fvmptd3 7039 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐶‘𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
| 160 | 159 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
| 161 | | oveq2 7439 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚)) |
| 162 | 161 | oveq1d 7446 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → ((0...𝑛) ↑m 𝑅) = ((0...𝑚) ↑m 𝑅)) |
| 163 | | eqeq2 2749 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚)) |
| 164 | 162, 163 | rabeqbidv 3455 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}) |
| 165 | 164 | cbvmptv 5255 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛) ↑m
𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}) |
| 166 | 160, 165 | eqtrdi 2793 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶‘𝑅) = (𝑚 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑚) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})) |
| 167 | | elnn0z 12626 |
. . . . . . . . . 10
⊢ ((𝐽 − (𝑐‘𝑍)) ∈ ℕ0 ↔ ((𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐‘𝑍)))) |
| 168 | 39, 45, 167 | sylanbrc 583 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈
ℕ0) |
| 169 | | ovex 7464 |
. . . . . . . . . . 11
⊢
((0...(𝐽 −
(𝑐‘𝑍))) ↑m 𝑅) ∈ V |
| 170 | 169 | rabex 5339 |
. . . . . . . . . 10
⊢ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ∈ V |
| 171 | 170 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ∈ V) |
| 172 | 150, 166,
168, 171 | fvmptd4 7040 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))) = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))}) |
| 173 | 139, 172 | eleqtrrd 2844 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))) |
| 174 | 50, 173 | jca 511 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) |
| 175 | | ovexd 7466 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ V) |
| 176 | | vex 3484 |
. . . . . . . 8
⊢ 𝑐 ∈ V |
| 177 | 176 | resex 6047 |
. . . . . . 7
⊢ (𝑐 ↾ 𝑅) ∈ V |
| 178 | | opeq12 4875 |
. . . . . . . . . 10
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → 〈𝑘, 𝑑〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
| 179 | 178 | eqeq2d 2748 |
. . . . . . . . 9
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈𝑘, 𝑑〉 ↔ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
| 180 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ (𝑘 = (𝐽 − (𝑐‘𝑍)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))) |
| 181 | 180 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))) |
| 182 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → 𝑑 = (𝑐 ↾ 𝑅)) |
| 183 | | fveq2 6906 |
. . . . . . . . . . . 12
⊢ (𝑘 = (𝐽 − (𝑐‘𝑍)) → ((𝐶‘𝑅)‘𝑘) = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))) |
| 184 | 183 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((𝐶‘𝑅)‘𝑘) = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))) |
| 185 | 182, 184 | eleq12d 2835 |
. . . . . . . . . 10
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (𝑑 ∈ ((𝐶‘𝑅)‘𝑘) ↔ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) |
| 186 | 181, 185 | anbi12d 632 |
. . . . . . . . 9
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)) ↔ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))))) |
| 187 | 179, 186 | anbi12d 632 |
. . . . . . . 8
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈𝑘, 𝑑〉 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))) ↔ (〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))))) |
| 188 | 187 | spc2egv 3599 |
. . . . . . 7
⊢ (((𝐽 − (𝑐‘𝑍)) ∈ V ∧ (𝑐 ↾ 𝑅) ∈ V) → ((〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) → ∃𝑘∃𝑑(〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈𝑘, 𝑑〉 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))))) |
| 189 | 175, 177,
188 | sylancl 586 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∧ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) → ∃𝑘∃𝑑(〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈𝑘, 𝑑〉 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘))))) |
| 190 | 1, 174, 189 | mp2and 699 |
. . . . 5
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∃𝑘∃𝑑(〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈𝑘, 𝑑〉 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)))) |
| 191 | | eliunxp 5848 |
. . . . 5
⊢
(〈(𝐽 −
(𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘∃𝑑(〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈𝑘, 𝑑〉 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)))) |
| 192 | 190, 191 | sylibr 234 |
. . . 4
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 193 | | dvnprodlem1.d |
. . . 4
⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
| 194 | 192, 193 | fmptd 7134 |
. . 3
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 195 | 79 | nfcri 2897 |
. . . . . . . . . 10
⊢
Ⅎ𝑡 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) |
| 196 | 80, 195 | nfan 1899 |
. . . . . . . . 9
⊢
Ⅎ𝑡(𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
| 197 | 65, 196 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑡(𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) |
| 198 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑡(𝐷‘𝑐) = (𝐷‘𝑒) |
| 199 | 197, 198 | nfan 1899 |
. . . . . . 7
⊢
Ⅎ𝑡((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) |
| 200 | 83 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
| 201 | 200 | adantlrr 721 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
| 202 | 201 | adantlr 715 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
| 203 | 193 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
| 204 | | opex 5469 |
. . . . . . . . . . . . . . . . 17
⊢
〈(𝐽 −
(𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ V |
| 205 | 204 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ V) |
| 206 | 203, 205 | fvmpt2d 7029 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
| 207 | 206 | fveq2d 6910 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (2nd ‘(𝐷‘𝑐)) = (2nd ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
| 208 | 207 | fveq1d 6908 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2nd ‘(𝐷‘𝑐))‘𝑡) = ((2nd ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)‘𝑡)) |
| 209 | | ovex 7464 |
. . . . . . . . . . . . . . 15
⊢ (𝐽 − (𝑐‘𝑍)) ∈ V |
| 210 | 209, 177 | op2nd 8023 |
. . . . . . . . . . . . . 14
⊢
(2nd ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) = (𝑐 ↾ 𝑅) |
| 211 | 210 | fveq1i 6907 |
. . . . . . . . . . . . 13
⊢
((2nd ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡) |
| 212 | 208, 211 | eqtr2di 2794 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2nd ‘(𝐷‘𝑐))‘𝑡)) |
| 213 | 212 | adantrr 717 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2nd ‘(𝐷‘𝑐))‘𝑡)) |
| 214 | 213 | ad2antrr 726 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2nd ‘(𝐷‘𝑐))‘𝑡)) |
| 215 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑐) = (𝐷‘𝑒)) |
| 216 | | fveq1 6905 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 = 𝑒 → (𝑐‘𝑍) = (𝑒‘𝑍)) |
| 217 | 216 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 = 𝑒 → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝑒‘𝑍))) |
| 218 | | reseq1 5991 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 = 𝑒 → (𝑐 ↾ 𝑅) = (𝑒 ↾ 𝑅)) |
| 219 | 217, 218 | opeq12d 4881 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = 𝑒 → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) |
| 220 | | simpr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
| 221 | | opex 5469 |
. . . . . . . . . . . . . . . . . . . 20
⊢
〈(𝐽 −
(𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉 ∈ V |
| 222 | 221 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉 ∈ V) |
| 223 | 193, 219,
220, 222 | fvmptd3 7039 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑒) = 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) |
| 224 | 223 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑒) = 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) |
| 225 | 215, 224 | eqtrd 2777 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑐) = 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) |
| 226 | 225 | fveq2d 6910 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (2nd ‘(𝐷‘𝑐)) = (2nd ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
| 227 | 226 | adantlrl 720 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (2nd ‘(𝐷‘𝑐)) = (2nd ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
| 228 | 227 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2nd ‘(𝐷‘𝑐)) = (2nd ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
| 229 | | ovex 7464 |
. . . . . . . . . . . . . 14
⊢ (𝐽 − (𝑒‘𝑍)) ∈ V |
| 230 | | vex 3484 |
. . . . . . . . . . . . . . 15
⊢ 𝑒 ∈ V |
| 231 | 230 | resex 6047 |
. . . . . . . . . . . . . 14
⊢ (𝑒 ↾ 𝑅) ∈ V |
| 232 | 229, 231 | op2nd 8023 |
. . . . . . . . . . . . 13
⊢
(2nd ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) = (𝑒 ↾ 𝑅) |
| 233 | 228, 232 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2nd ‘(𝐷‘𝑐)) = (𝑒 ↾ 𝑅)) |
| 234 | 233 | fveq1d 6908 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘(𝐷‘𝑐))‘𝑡) = ((𝑒 ↾ 𝑅)‘𝑡)) |
| 235 | | fvres 6925 |
. . . . . . . . . . . 12
⊢ (𝑡 ∈ 𝑅 → ((𝑒 ↾ 𝑅)‘𝑡) = (𝑒‘𝑡)) |
| 236 | 235 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((𝑒 ↾ 𝑅)‘𝑡) = (𝑒‘𝑡)) |
| 237 | 234, 236 | eqtrd 2777 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘(𝐷‘𝑐))‘𝑡) = (𝑒‘𝑡)) |
| 238 | 202, 214,
237 | 3eqtrd 2781 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
| 239 | 238 | adantlr 715 |
. . . . . . . 8
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
| 240 | | elunnel1 4154 |
. . . . . . . . . . 11
⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 ∈ {𝑍}) |
| 241 | | elsni 4643 |
. . . . . . . . . . 11
⊢ (𝑡 ∈ {𝑍} → 𝑡 = 𝑍) |
| 242 | 240, 241 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍) |
| 243 | 242 | adantll 714 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍) |
| 244 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍) |
| 245 | 244 | fveq2d 6910 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑐‘𝑍)) |
| 246 | 3 | nn0cnd 12589 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐽 ∈ ℂ) |
| 247 | 246 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ) |
| 248 | 247, 107 | nncand 11625 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝑐‘𝑍)) |
| 249 | 248 | eqcomd 2743 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍)))) |
| 250 | 249 | adantrr 717 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍)))) |
| 251 | 250 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍)))) |
| 252 | 206 | fveq2d 6910 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1st ‘(𝐷‘𝑐)) = (1st ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
| 253 | 209, 177 | op1st 8022 |
. . . . . . . . . . . . . . . . 17
⊢
(1st ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) = (𝐽 − (𝑐‘𝑍)) |
| 254 | 252, 253 | eqtr2di 2794 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) = (1st ‘(𝐷‘𝑐))) |
| 255 | 254 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1st ‘(𝐷‘𝑐)))) |
| 256 | 255 | adantrr 717 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1st ‘(𝐷‘𝑐)))) |
| 257 | 256 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1st ‘(𝐷‘𝑐)))) |
| 258 | | fveq2 6906 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐷‘𝑐) = (𝐷‘𝑒) → (1st ‘(𝐷‘𝑐)) = (1st ‘(𝐷‘𝑒))) |
| 259 | 258 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1st ‘(𝐷‘𝑐)) = (1st ‘(𝐷‘𝑒))) |
| 260 | 223 | fveq2d 6910 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1st ‘(𝐷‘𝑒)) = (1st ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
| 261 | 260 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1st ‘(𝐷‘𝑒)) = (1st ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
| 262 | 229, 231 | op1st 8022 |
. . . . . . . . . . . . . . . . . 18
⊢
(1st ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) = (𝐽 − (𝑒‘𝑍)) |
| 263 | 262 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1st ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) = (𝐽 − (𝑒‘𝑍))) |
| 264 | 259, 261,
263 | 3eqtrd 2781 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1st ‘(𝐷‘𝑐)) = (𝐽 − (𝑒‘𝑍))) |
| 265 | 264 | oveq2d 7447 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1st ‘(𝐷‘𝑐))) = (𝐽 − (𝐽 − (𝑒‘𝑍)))) |
| 266 | 246 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ) |
| 267 | | fzsscn 45323 |
. . . . . . . . . . . . . . . . . 18
⊢
(0...𝐽) ⊆
ℂ |
| 268 | | eleq1w 2824 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑐 = 𝑒 → (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↔ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) |
| 269 | 268 | anbi2d 630 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 = 𝑒 → ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ↔ (𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)))) |
| 270 | | feq1 6716 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 = 𝑒 → (𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽) ↔ 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽))) |
| 271 | 269, 270 | imbi12d 344 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 = 𝑒 → (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) ↔ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽)))) |
| 272 | 271, 31 | chvarvv 1998 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
| 273 | 35 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
| 274 | 272, 273 | ffvelcdmd 7105 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒‘𝑍) ∈ (0...𝐽)) |
| 275 | 267, 274 | sselid 3981 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒‘𝑍) ∈ ℂ) |
| 276 | 266, 275 | nncand 11625 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑒‘𝑍))) = (𝑒‘𝑍)) |
| 277 | 276 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (𝐽 − (𝑒‘𝑍))) = (𝑒‘𝑍)) |
| 278 | 265, 277 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1st ‘(𝐷‘𝑐))) = (𝑒‘𝑍)) |
| 279 | 278 | adantlrl 720 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1st ‘(𝐷‘𝑐))) = (𝑒‘𝑍)) |
| 280 | 251, 257,
279 | 3eqtrd 2781 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐‘𝑍) = (𝑒‘𝑍)) |
| 281 | 280 | adantr 480 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑍) = (𝑒‘𝑍)) |
| 282 | | fveq2 6906 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑍 → (𝑒‘𝑡) = (𝑒‘𝑍)) |
| 283 | 282 | eqcomd 2743 |
. . . . . . . . . . . 12
⊢ (𝑡 = 𝑍 → (𝑒‘𝑍) = (𝑒‘𝑡)) |
| 284 | 283 | adantl 481 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑒‘𝑍) = (𝑒‘𝑡)) |
| 285 | 245, 281,
284 | 3eqtrd 2781 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
| 286 | 285 | adantlr 715 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
| 287 | 243, 286 | syldan 591 |
. . . . . . . 8
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
| 288 | 239, 287 | pm2.61dan 813 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
| 289 | 199, 288 | ralrimia 3258 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡)) |
| 290 | 61 | adantrr 717 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑐 Fn (𝑅 ∪ {𝑍})) |
| 291 | 290 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑐 Fn (𝑅 ∪ {𝑍})) |
| 292 | 272 | ffnd 6737 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 Fn (𝑅 ∪ {𝑍})) |
| 293 | 292 | adantrl 716 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑒 Fn (𝑅 ∪ {𝑍})) |
| 294 | 293 | adantr 480 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑒 Fn (𝑅 ∪ {𝑍})) |
| 295 | | eqfnfv 7051 |
. . . . . . 7
⊢ ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑒 Fn (𝑅 ∪ {𝑍})) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡))) |
| 296 | 291, 294,
295 | syl2anc 584 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡))) |
| 297 | 289, 296 | mpbird 257 |
. . . . 5
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑐 = 𝑒) |
| 298 | 297 | ex 412 |
. . . 4
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒)) |
| 299 | 298 | ralrimivva 3202 |
. . 3
⊢ (𝜑 → ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒)) |
| 300 | | dff13 7275 |
. . 3
⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒))) |
| 301 | 194, 299,
300 | sylanbrc 583 |
. 2
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 302 | | eliun 4995 |
. . . . . . . . 9
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 303 | 302 | biimpi 216 |
. . . . . . . 8
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 304 | 303 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 305 | | nfv 1914 |
. . . . . . . . 9
⊢
Ⅎ𝑘𝜑 |
| 306 | | nfiu1 5027 |
. . . . . . . . . 10
⊢
Ⅎ𝑘∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
| 307 | 306 | nfcri 2897 |
. . . . . . . . 9
⊢
Ⅎ𝑘 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
| 308 | 305, 307 | nfan 1899 |
. . . . . . . 8
⊢
Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 309 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} |
| 310 | | eleq1w 2824 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑟 → (𝑡 ∈ 𝑅 ↔ 𝑟 ∈ 𝑅)) |
| 311 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑟 → ((2nd ‘𝑝)‘𝑡) = ((2nd ‘𝑝)‘𝑟)) |
| 312 | 310, 311 | ifbieq1d 4550 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑟 → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝)))) |
| 313 | 312 | cbvmptv 5255 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝)))) |
| 314 | 313 | eqeq2i 2750 |
. . . . . . . . . . . . 13
⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ↔ 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))) |
| 315 | | fveq1 6905 |
. . . . . . . . . . . . . 14
⊢ (𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝)))) → (𝑐‘𝑡) = ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡)) |
| 316 | 315 | sumeq2sdv 15739 |
. . . . . . . . . . . . 13
⊢ (𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝)))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡)) |
| 317 | 314, 316 | sylbi 217 |
. . . . . . . . . . . 12
⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡)) |
| 318 | 317 | eqeq1d 2739 |
. . . . . . . . . . 11
⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡) = 𝐽)) |
| 319 | | ovexd 7466 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (0...𝐽) ∈ V) |
| 320 | 16, 17 | ssexd 5324 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ V) |
| 321 | 320 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑅 ∪ {𝑍}) ∈ V) |
| 322 | | nfv 1914 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡 𝑘 ∈ (0...𝐽) |
| 323 | | nfcv 2905 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡{𝑘} |
| 324 | | nfcv 2905 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑡𝑅 |
| 325 | 75, 324 | nffv 6916 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡(𝐶‘𝑅) |
| 326 | | nfcv 2905 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡𝑘 |
| 327 | 325, 326 | nffv 6916 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡((𝐶‘𝑅)‘𝑘) |
| 328 | 323, 327 | nfxp 5718 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑡({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
| 329 | 328 | nfcri 2897 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑡 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
| 330 | 65, 322, 329 | nf3an 1901 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡(𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 331 | | 0zd 12625 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ∈
ℤ) |
| 332 | 4 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ) |
| 333 | 332 | 3ad2antl1 1186 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ) |
| 334 | | iftrue 4531 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 ∈ 𝑅 → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡)) |
| 335 | 334 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡)) |
| 336 | | xp2nd 8047 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘)) |
| 337 | 336 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘)) |
| 338 | | oveq2 7439 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘)) |
| 339 | 338 | oveq1d 7446 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑘 → ((0...𝑛) ↑m 𝑅) = ((0...𝑘) ↑m 𝑅)) |
| 340 | | eqeq2 2749 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘)) |
| 341 | 339, 340 | rabeqbidv 3455 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
| 342 | 159 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
| 343 | | elfznn0 13660 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ0) |
| 344 | 343 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ0) |
| 345 | | ovex 7464 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((0...𝑘)
↑m 𝑅)
∈ V |
| 346 | 345 | rabex 5339 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V |
| 347 | 346 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V) |
| 348 | 341, 342,
344, 347 | fvmptd4 7040 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
| 349 | 348 | 3adant3 1133 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
| 350 | 337, 349 | eleqtrd 2843 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
| 351 | | elrabi 3687 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} → (2nd ‘𝑝) ∈ ((0...𝑘) ↑m 𝑅)) |
| 352 | | elmapi 8889 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘𝑝) ∈ ((0...𝑘) ↑m 𝑅) → (2nd ‘𝑝):𝑅⟶(0...𝑘)) |
| 353 | 350, 351,
352 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝):𝑅⟶(0...𝑘)) |
| 354 | 353 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (2nd ‘𝑝):𝑅⟶(0...𝑘)) |
| 355 | 354 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘)) |
| 356 | 355 | elfzelzd 13565 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ ℤ) |
| 357 | 335, 356 | eqeltrd 2841 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
| 358 | 242 | adantll 714 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍) |
| 359 | | simpr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍) |
| 360 | 112 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → ¬ 𝑍 ∈ 𝑅) |
| 361 | 359, 360 | eqneltrd 2861 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → ¬ 𝑡 ∈ 𝑅) |
| 362 | 361 | iffalsed 4536 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
| 363 | 362 | 3ad2antl1 1186 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
| 364 | 4 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ) |
| 365 | 364 | 3ad2antl1 1186 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ) |
| 366 | | xp1st 8046 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ {𝑘}) |
| 367 | | elsni 4643 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((1st ‘𝑝) ∈ {𝑘} → (1st ‘𝑝) = 𝑘) |
| 368 | 366, 367 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) = 𝑘) |
| 369 | 368 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) = 𝑘) |
| 370 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ) |
| 371 | 370 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ ℤ) |
| 372 | 369, 371 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈
ℤ) |
| 373 | 372 | 3adant1 1131 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈
ℤ) |
| 374 | 373 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (1st ‘𝑝) ∈
ℤ) |
| 375 | 365, 374 | zsubcld 12727 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1st ‘𝑝)) ∈
ℤ) |
| 376 | 363, 375 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
| 377 | 376 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
| 378 | 358, 377 | syldan 591 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
| 379 | 357, 378 | pm2.61dan 813 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
| 380 | 353 | ffvelcdmda 7104 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘)) |
| 381 | | elfzle1 13567 |
. . . . . . . . . . . . . . . . . 18
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → 0 ≤ ((2nd ‘𝑝)‘𝑡)) |
| 382 | 380, 381 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 0 ≤ ((2nd
‘𝑝)‘𝑡)) |
| 383 | 334 | eqcomd 2743 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 ∈ 𝑅 → ((2nd ‘𝑝)‘𝑡) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 384 | 383 | adantl 481 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 385 | 382, 384 | breqtrd 5169 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 386 | 385 | adantlr 715 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 387 | | simpll 767 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))) |
| 388 | | elfzle2 13568 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ≤ 𝐽) |
| 389 | | elfzel2 13562 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ) |
| 390 | 389 | zred 12722 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ) |
| 391 | 93 | sseli 3979 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ) |
| 392 | 390, 391 | subge0d 11853 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ (𝐽 − 𝑘) ↔ 𝑘 ≤ 𝐽)) |
| 393 | 388, 392 | mpbird 257 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ (𝐽 − 𝑘)) |
| 394 | 393 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽 − 𝑘)) |
| 395 | 394 | 3ad2antl2 1187 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽 − 𝑘)) |
| 396 | 361 | 3ad2antl1 1186 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → ¬ 𝑡 ∈ 𝑅) |
| 397 | 396 | iffalsed 4536 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
| 398 | 368 | 3ad2ant3 1136 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) = 𝑘) |
| 399 | 398 | oveq2d 7447 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) = (𝐽 − 𝑘)) |
| 400 | 399 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1st ‘𝑝)) = (𝐽 − 𝑘)) |
| 401 | 397, 400 | eqtr2d 2778 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 402 | 395, 401 | breqtrd 5169 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 403 | 387, 358,
402 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 404 | 386, 403 | pm2.61dan 813 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 405 | | simpl2 1193 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑘 ∈ (0...𝐽)) |
| 406 | | elfzelz 13564 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd ‘𝑝)‘𝑡) ∈ ℤ) |
| 407 | 406 | zred 12722 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd ‘𝑝)‘𝑡) ∈ ℝ) |
| 408 | 407 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd ‘𝑝)‘𝑡) ∈ ℝ) |
| 409 | 391 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ) |
| 410 | 390 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ) |
| 411 | | elfzle2 13568 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd ‘𝑝)‘𝑡) ≤ 𝑘) |
| 412 | 411 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd ‘𝑝)‘𝑡) ≤ 𝑘) |
| 413 | 388 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝐽) |
| 414 | 408, 409,
410, 412, 413 | letrd 11418 |
. . . . . . . . . . . . . . . . . 18
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd ‘𝑝)‘𝑡) ≤ 𝐽) |
| 415 | 380, 405,
414 | syl2anc 584 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ≤ 𝐽) |
| 416 | 415 | adantlr 715 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ≤ 𝐽) |
| 417 | 335, 416 | eqbrtrd 5165 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽) |
| 418 | 344 | nn0ge0d 12590 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘) |
| 419 | 390 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ) |
| 420 | 391 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ) |
| 421 | 419, 420 | subge02d 11855 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0 ≤ 𝑘 ↔ (𝐽 − 𝑘) ≤ 𝐽)) |
| 422 | 418, 421 | mpbid 232 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽 − 𝑘) ≤ 𝐽) |
| 423 | 422 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) ≤ 𝐽) |
| 424 | 423 | 3adantl3 1169 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) ≤ 𝐽) |
| 425 | 401, 424 | eqbrtrrd 5167 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽) |
| 426 | 387, 358,
425 | syl2anc 584 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽) |
| 427 | 417, 426 | pm2.61dan 813 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽) |
| 428 | 331, 333,
379, 404, 427 | elfzd 13555 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈ (0...𝐽)) |
| 429 | 330, 428 | fmptd2f 45240 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))):(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
| 430 | 319, 321,
429 | elmapdd 8881 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍}))) |
| 431 | | eleq1w 2824 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = 𝑡 → (𝑟 ∈ 𝑅 ↔ 𝑡 ∈ 𝑅)) |
| 432 | | fveq2 6906 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = 𝑡 → ((2nd ‘𝑝)‘𝑟) = ((2nd ‘𝑝)‘𝑡)) |
| 433 | 431, 432 | ifbieq1d 4550 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 = 𝑡 → if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 434 | | eqidd 2738 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))) |
| 435 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ (𝑅 ∪ {𝑍})) |
| 436 | 433, 434,
435, 379 | fvmptd4 7040 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 437 | 330, 436 | ralrimia 3258 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 438 | 437 | sumeq2d 15737 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
| 439 | | nfcv 2905 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑡if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) |
| 440 | 58 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin) |
| 441 | 32 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑍 ∈ 𝑇) |
| 442 | 112 | 3ad2ant1 1134 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ¬ 𝑍 ∈ 𝑅) |
| 443 | 334 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡)) |
| 444 | 380 | elfzelzd 13565 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ ℤ) |
| 445 | 444 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ ℂ) |
| 446 | 443, 445 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℂ) |
| 447 | | eleq1 2829 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑍 → (𝑡 ∈ 𝑅 ↔ 𝑍 ∈ 𝑅)) |
| 448 | | fveq2 6906 |
. . . . . . . . . . . . . 14
⊢ (𝑡 = 𝑍 → ((2nd ‘𝑝)‘𝑡) = ((2nd ‘𝑝)‘𝑍)) |
| 449 | 447, 448 | ifbieq1d 4550 |
. . . . . . . . . . . . 13
⊢ (𝑡 = 𝑍 → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝)))) |
| 450 | 112 | adantr 480 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ¬ 𝑍 ∈ 𝑅) |
| 451 | 450 | iffalsed 4536 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
| 452 | 451 | 3adant2 1132 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
| 453 | 4 | 3ad2ant1 1134 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℤ) |
| 454 | 453, 373 | zsubcld 12727 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) ∈
ℤ) |
| 455 | 454 | zcnd 12723 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) ∈
ℂ) |
| 456 | 452, 455 | eqeltrd 2841 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) ∈
ℂ) |
| 457 | 330, 439,
440, 441, 442, 446, 449, 456 | fsumsplitsn 15780 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))))) |
| 458 | 334 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡))) |
| 459 | 330, 458 | ralrimi 3257 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∀𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡)) |
| 460 | 459 | sumeq2d 15737 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡)) |
| 461 | | eqidd 2738 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑐 = (2nd ‘𝑝) → 𝑅 = 𝑅) |
| 462 | | simpl 482 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑐 = (2nd ‘𝑝) ∧ 𝑡 ∈ 𝑅) → 𝑐 = (2nd ‘𝑝)) |
| 463 | 462 | fveq1d 6908 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐 = (2nd ‘𝑝) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((2nd ‘𝑝)‘𝑡)) |
| 464 | 461, 463 | sumeq12rdv 15743 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = (2nd ‘𝑝) → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡)) |
| 465 | 464 | eqeq1d 2739 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = (2nd ‘𝑝) → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘 ↔ Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡) = 𝑘)) |
| 466 | 465 | elrab 3692 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑m 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ↔ ((2nd ‘𝑝) ∈ ((0...𝑘) ↑m 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡) = 𝑘)) |
| 467 | 350, 466 | sylib 218 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((2nd ‘𝑝) ∈ ((0...𝑘) ↑m 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡) = 𝑘)) |
| 468 | 467 | simprd 495 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡) = 𝑘) |
| 469 | 460, 468 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = 𝑘) |
| 470 | 442 | iffalsed 4536 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
| 471 | 470, 399 | eqtrd 2777 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) = (𝐽 − 𝑘)) |
| 472 | 469, 471 | oveq12d 7449 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝)))) = (𝑘 + (𝐽 − 𝑘))) |
| 473 | 267 | sseli 3979 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℂ) |
| 474 | 473 | 3ad2ant2 1135 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ ℂ) |
| 475 | 246 | 3ad2ant1 1134 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℂ) |
| 476 | 474, 475 | pncan3d 11623 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 + (𝐽 − 𝑘)) = 𝐽) |
| 477 | 472, 476 | eqtrd 2777 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) + if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝)))) = 𝐽) |
| 478 | 438, 457,
477 | 3eqtrd 2781 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡) = 𝐽) |
| 479 | 318, 430,
478 | elrabd 3694 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
| 480 | 479 | 3exp 1120 |
. . . . . . . . 9
⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}))) |
| 481 | 480 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}))) |
| 482 | 308, 309,
481 | rexlimd 3266 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽})) |
| 483 | 304, 482 | mpd 15 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
| 484 | 26 | eqcomd 2743 |
. . . . . . 7
⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
| 485 | 484 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → {𝑐 ∈ ((0...𝐽) ↑m (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
| 486 | 483, 485 | eleqtrd 2843 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
| 487 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) |
| 488 | 487, 313 | eqtrdi 2793 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → 𝑐 = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))) |
| 489 | | simpr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑟 = 𝑍) → 𝑟 = 𝑍) |
| 490 | 112 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑟 = 𝑍) → ¬ 𝑍 ∈ 𝑅) |
| 491 | 489, 490 | eqneltrd 2861 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑟 = 𝑍) → ¬ 𝑟 ∈ 𝑅) |
| 492 | 491 | iffalsed 4536 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑟 = 𝑍) → if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
| 493 | 492 | adantlr 715 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) ∧ 𝑟 = 𝑍) → if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
| 494 | 35 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
| 495 | | ovexd 7466 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝐽 − (1st ‘𝑝)) ∈ V) |
| 496 | 488, 493,
494, 495 | fvmptd 7023 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝑐‘𝑍) = (𝐽 − (1st ‘𝑝))) |
| 497 | 496 | oveq2d 7447 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝐽 − (1st ‘𝑝)))) |
| 498 | 497 | adantlr 715 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝐽 − (1st ‘𝑝)))) |
| 499 | 246 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → 𝐽 ∈ ℂ) |
| 500 | | nfv 1914 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘(1st ‘𝑝) ∈ (0...𝐽) |
| 501 | | simpl 482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽)) |
| 502 | 369, 501 | eqeltrd 2841 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈ (0...𝐽)) |
| 503 | 502 | ex 412 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽))) |
| 504 | 503 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽)))) |
| 505 | 307, 500,
504 | rexlimd 3266 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽))) |
| 506 | 303, 505 | mpd 15 |
. . . . . . . . . . . . 13
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽)) |
| 507 | 506 | elfzelzd 13565 |
. . . . . . . . . . . 12
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈
ℤ) |
| 508 | 507 | zcnd 12723 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈
ℂ) |
| 509 | 508 | ad2antlr 727 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (1st
‘𝑝) ∈
ℂ) |
| 510 | 499, 509 | nncand 11625 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝐽 − (𝐽 − (1st ‘𝑝))) = (1st
‘𝑝)) |
| 511 | 498, 510 | eqtrd 2777 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (1st ‘𝑝)) |
| 512 | | reseq1 5991 |
. . . . . . . . . 10
⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) → (𝑐 ↾ 𝑅) = ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ↾ 𝑅)) |
| 513 | 512 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝑐 ↾ 𝑅) = ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ↾ 𝑅)) |
| 514 | 62 | a1i 11 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → 𝑅 ⊆ (𝑅 ∪ {𝑍})) |
| 515 | 514 | resmptd 6058 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ↾ 𝑅) = (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) |
| 516 | | nfv 1914 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘(𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝) |
| 517 | 334 | mpteq2ia 5245 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (𝑡 ∈ 𝑅 ↦ ((2nd ‘𝑝)‘𝑡)) |
| 518 | 353 | feqmptd 6977 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) = (𝑡 ∈ 𝑅 ↦ ((2nd ‘𝑝)‘𝑡))) |
| 519 | 517, 518 | eqtr4id 2796 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝)) |
| 520 | 519 | 3exp 1120 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝)))) |
| 521 | 520 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝)))) |
| 522 | 308, 516,
521 | rexlimd 3266 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝))) |
| 523 | 304, 522 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝)) |
| 524 | 523 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝)) |
| 525 | 513, 515,
524 | 3eqtrd 2781 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝑐 ↾ 𝑅) = (2nd ‘𝑝)) |
| 526 | 511, 525 | opeq12d 4881 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(1st ‘𝑝), (2nd ‘𝑝)〉) |
| 527 | | opex 5469 |
. . . . . . . 8
⊢
〈(1st ‘𝑝), (2nd ‘𝑝)〉 ∈ V |
| 528 | 527 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 ∈
V) |
| 529 | 193, 526,
486, 528 | fvmptd2 7024 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) = 〈(1st
‘𝑝), (2nd
‘𝑝)〉) |
| 530 | | nfv 1914 |
. . . . . . . 8
⊢
Ⅎ𝑘〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝 |
| 531 | | 1st2nd2 8053 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 𝑝 = 〈(1st ‘𝑝), (2nd ‘𝑝)〉) |
| 532 | 531 | eqcomd 2743 |
. . . . . . . . 9
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝) |
| 533 | 532 | 2a1i 12 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝))) |
| 534 | 308, 530,
533 | rexlimd 3266 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝)) |
| 535 | 304, 534 | mpd 15 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝) |
| 536 | 529, 535 | eqtr2d 2778 |
. . . . 5
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))))) |
| 537 | | fveq2 6906 |
. . . . . 6
⊢ (𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) → (𝐷‘𝑐) = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))))) |
| 538 | 537 | rspceeqv 3645 |
. . . . 5
⊢ (((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)) |
| 539 | 486, 536,
538 | syl2anc 584 |
. . . 4
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)) |
| 540 | 539 | ralrimiva 3146 |
. . 3
⊢ (𝜑 → ∀𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)) |
| 541 | | dffo3 7122 |
. . 3
⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ ∀𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐))) |
| 542 | 194, 540,
541 | sylanbrc 583 |
. 2
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
| 543 | | df-f1o 6568 |
. 2
⊢ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ (𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ∧ 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)))) |
| 544 | 301, 542,
543 | sylanbrc 583 |
1
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |