Step | Hyp | Ref
| Expression |
1 | | eqidd 2779 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
2 | | 0zd 11740 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ∈ ℤ) |
3 | | dvnprodlem1.j |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐽 ∈
ℕ0) |
4 | 3 | nn0zd 11832 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐽 ∈ ℤ) |
5 | 4 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℤ) |
6 | | fzssz 12660 |
. . . . . . . . . . . . . . . 16
⊢
(0...𝐽) ⊆
ℤ |
7 | 6 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℤ) |
8 | | dvnprodlem1.c |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝐶 = (𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})) |
9 | | oveq2 6930 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → ((0...𝑛) ↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍}))) |
10 | | rabeq 3389 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((0...𝑛)
↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
11 | 9, 10 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
12 | | sumeq1 14827 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡)) |
13 | 12 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛)) |
14 | 13 | rabbidv 3386 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑠 = (𝑅 ∪ {𝑍}) → {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) |
15 | 11, 14 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (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}) |
16 | 15 | mpteq2dv 4980 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (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})) |
17 | | dvnprodlem1.rzt |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ⊆ 𝑇) |
18 | | dvnprodlem1.t |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝑇 ∈ Fin) |
19 | | ssexg 5041 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑅 ∪ {𝑍}) ⊆ 𝑇 ∧ 𝑇 ∈ Fin) → (𝑅 ∪ {𝑍}) ∈ V) |
20 | 17, 18, 19 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ V) |
21 | | elpwg 4387 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑅 ∪ {𝑍}) ∈ V → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇)) |
22 | 20, 21 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → ((𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇 ↔ (𝑅 ∪ {𝑍}) ⊆ 𝑇)) |
23 | 17, 22 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑅 ∪ {𝑍}) ∈ 𝒫 𝑇) |
24 | | nn0ex 11649 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
ℕ0 ∈ V |
25 | 24 | mptex 6758 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛)
↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V |
26 | 25 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) ∈ V) |
27 | 8, 16, 23, 26 | fvmptd3 6564 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝐶‘(𝑅 ∪ {𝑍})) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛})) |
28 | | oveq2 6930 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑛 = 𝐽 → (0...𝑛) = (0...𝐽)) |
29 | 28 | oveq1d 6937 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝐽 → ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
30 | | rabeq 3389 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((0...𝑛)
↑𝑚 (𝑅 ∪ {𝑍})) = ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) → {𝑐 ∈ ((0...𝑛) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛}) |
31 | 29, 30 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (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) = 𝑛}) |
32 | | eqeq2 2789 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑛 = 𝐽 → (Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽)) |
33 | 32 | rabbidv 3386 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑛 = 𝐽 → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
34 | 31, 33 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (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}) |
35 | 34 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 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}) |
36 | | ovex 6954 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((0...𝐽)
↑𝑚 (𝑅 ∪ {𝑍})) ∈ V |
37 | 36 | rabex 5049 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V |
38 | 37 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ∈ V) |
39 | 27, 35, 3, 38 | fvmptd 6548 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
40 | | ssrab2 3908 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) |
41 | 40 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ⊆ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
42 | 39, 41 | eqsstrd 3858 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
43 | 42 | adantr 474 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ⊆ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
44 | | simpr 479 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
45 | 43, 44 | sseldd 3822 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍}))) |
46 | | elmapi 8162 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
47 | 45, 46 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
48 | | dvnprodlem1.z |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑍 ∈ 𝑇) |
49 | | snidg 4428 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑍 ∈ 𝑇 → 𝑍 ∈ {𝑍}) |
50 | 48, 49 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑍 ∈ {𝑍}) |
51 | | elun2 4004 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
52 | 50, 51 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
53 | 52 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
54 | 47, 53 | ffvelrnd 6624 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ (0...𝐽)) |
55 | 7, 54 | sseldd 3822 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℤ) |
56 | 5, 55 | zsubcld 11839 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ) |
57 | 2, 5, 56 | 3jca 1119 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ)) |
58 | | elfzle2 12662 |
. . . . . . . . . . . . . 14
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → (𝑐‘𝑍) ≤ 𝐽) |
59 | 54, 58 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ≤ 𝐽) |
60 | 5 | zred 11834 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℝ) |
61 | 55 | zred 11834 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℝ) |
62 | 60, 61 | subge0d 10965 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝐽 − (𝑐‘𝑍)) ↔ (𝑐‘𝑍) ≤ 𝐽)) |
63 | 59, 62 | mpbird 249 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝐽 − (𝑐‘𝑍))) |
64 | | elfzle1 12661 |
. . . . . . . . . . . . . 14
⊢ ((𝑐‘𝑍) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑍)) |
65 | 54, 64 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 0 ≤ (𝑐‘𝑍)) |
66 | 60, 61 | subge02d 10967 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0 ≤ (𝑐‘𝑍) ↔ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽)) |
67 | 65, 66 | mpbid 224 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ≤ 𝐽) |
68 | 57, 63, 67 | jca32 511 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))) |
69 | | elfz2 12650 |
. . . . . . . . . . 11
⊢ ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ↔ ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ) ∧ (0 ≤ (𝐽 − (𝑐‘𝑍)) ∧ (𝐽 − (𝑐‘𝑍)) ≤ 𝐽))) |
70 | 68, 69 | sylibr 226 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽)) |
71 | | elmapfn 8163 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) → 𝑐 Fn (𝑅 ∪ {𝑍})) |
72 | 45, 71 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 Fn (𝑅 ∪ {𝑍})) |
73 | | ssun1 3999 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑅 ⊆ (𝑅 ∪ {𝑍}) |
74 | 73 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ⊆ (𝑅 ∪ {𝑍})) |
75 | | fnssres 6250 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑅 ⊆ (𝑅 ∪ {𝑍})) → (𝑐 ↾ 𝑅) Fn 𝑅) |
76 | 72, 74, 75 | syl2anc 579 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) Fn 𝑅) |
77 | | nfv 1957 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑡𝜑 |
78 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑡𝑐 |
79 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑡𝒫 𝑇 |
80 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑡ℕ0 |
81 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
Ⅎ𝑡𝑠 |
82 | 81 | nfsum1 14828 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑡Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) |
83 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑡𝑛 |
84 | 82, 83 | nfeq 2945 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑡Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 |
85 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑡((0...𝑛) ↑𝑚 𝑠) |
86 | 84, 85 | nfrab 3310 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑡{𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} |
87 | 80, 86 | nfmpt 4981 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑡(𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
88 | 79, 87 | nfmpt 4981 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑡(𝑠 ∈ 𝒫 𝑇 ↦ (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛})) |
89 | 8, 88 | nfcxfr 2932 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑡𝐶 |
90 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑡(𝑅 ∪ {𝑍}) |
91 | 89, 90 | nffv 6456 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡(𝐶‘(𝑅 ∪ {𝑍})) |
92 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡𝐽 |
93 | 91, 92 | nffv 6456 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑡((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) |
94 | 78, 93 | nfel 2946 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑡 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) |
95 | 77, 94 | nfan 1946 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
96 | | fvres 6465 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)) |
97 | 96 | adantl 475 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)) |
98 | 2 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 0 ∈ ℤ) |
99 | 56 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝐽 − (𝑐‘𝑍)) ∈ ℤ) |
100 | 6 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0...𝐽) ⊆ ℤ) |
101 | 47 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
102 | 74 | sselda 3821 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ (𝑅 ∪ {𝑍})) |
103 | 101, 102 | ffvelrnd 6624 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...𝐽)) |
104 | 100, 103 | sseldd 3822 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ ℤ) |
105 | 98, 99, 104 | 3jca 1119 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ)) |
106 | | elfzle1 12661 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑐‘𝑡) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑡)) |
107 | 103, 106 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 0 ≤ (𝑐‘𝑡)) |
108 | 17 | unssad 4013 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → 𝑅 ⊆ 𝑇) |
109 | | ssfi 8468 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑇 ∈ Fin ∧ 𝑅 ⊆ 𝑇) → 𝑅 ∈ Fin) |
110 | 18, 108, 109 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑅 ∈ Fin) |
111 | 110 | ad2antrr 716 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑅 ∈ Fin) |
112 | | zssre 11735 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ℤ
⊆ ℝ |
113 | 6, 112 | sstri 3830 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(0...𝐽) ⊆
ℝ |
114 | 113 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (0...𝐽) ⊆ ℝ) |
115 | 47 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
116 | 74 | sselda 3821 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 𝑟 ∈ (𝑅 ∪ {𝑍})) |
117 | 115, 116 | ffvelrnd 6624 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ (0...𝐽)) |
118 | 114, 117 | sseldd 3822 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℝ) |
119 | 118 | adantlr 705 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℝ) |
120 | | elfzle1 12661 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑐‘𝑟) ∈ (0...𝐽) → 0 ≤ (𝑐‘𝑟)) |
121 | 117, 120 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → 0 ≤ (𝑐‘𝑟)) |
122 | 121 | adantlr 705 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) ∧ 𝑟 ∈ 𝑅) → 0 ≤ (𝑐‘𝑟)) |
123 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑟 = 𝑡 → (𝑐‘𝑟) = (𝑐‘𝑡)) |
124 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → 𝑡 ∈ 𝑅) |
125 | 111, 119,
122, 123, 124 | fsumge1 14933 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ≤ Σ𝑟 ∈ 𝑅 (𝑐‘𝑟)) |
126 | 110 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ Fin) |
127 | 118 | recnd 10405 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑟 ∈ 𝑅) → (𝑐‘𝑟) ∈ ℂ) |
128 | 126, 127 | fsumcl 14871 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) ∈ ℂ) |
129 | 61 | recnd 10405 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) ∈ ℂ) |
130 | 128, 129 | pncand 10735 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) − (𝑐‘𝑍)) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟)) |
131 | | nfv 1957 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
Ⅎ𝑟(𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
132 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
Ⅎ𝑟(𝑐‘𝑍) |
133 | 48 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ 𝑇) |
134 | | dvnprodlem1.zr |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → ¬ 𝑍 ∈ 𝑅) |
135 | 134 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ¬ 𝑍 ∈ 𝑅) |
136 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑟 = 𝑍 → (𝑐‘𝑟) = (𝑐‘𝑍)) |
137 | 131, 132,
126, 133, 135, 127, 136, 129 | fsumsplitsn 14881 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍))) |
138 | 137 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) = Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟)) |
139 | 123 | cbvsumv 14834 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
Σ𝑟 ∈
(𝑅 ∪ {𝑍})(𝑐‘𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) |
140 | 139 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡)) |
141 | 39 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) = {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
142 | 44, 141 | eleqtrd 2861 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐 ∈ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽}) |
143 | | rabid 3302 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (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)) |
144 | 142, 143 | sylib 210 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽)) |
145 | 144 | simprd 491 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽) |
146 | 140, 145 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑟) = 𝐽) |
147 | 138, 146 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) = 𝐽) |
148 | 147 | oveq1d 6937 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) + (𝑐‘𝑍)) − (𝑐‘𝑍)) = (𝐽 − (𝑐‘𝑍))) |
149 | 130, 148 | eqtr3d 2816 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍))) |
150 | 149 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍))) |
151 | 125, 150 | breqtrd 4912 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍))) |
152 | 105, 107,
151 | jca32 511 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ) ∧ (0 ≤ (𝑐‘𝑡) ∧ (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍))))) |
153 | | elfz2 12650 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑐‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))) ↔ ((0 ∈ ℤ ∧ (𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ (𝑐‘𝑡) ∈ ℤ) ∧ (0 ≤ (𝑐‘𝑡) ∧ (𝑐‘𝑡) ≤ (𝐽 − (𝑐‘𝑍))))) |
154 | 152, 153 | sylibr 226 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))) |
155 | 97, 154 | eqeltrd 2859 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))) |
156 | 155 | ex 403 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))) |
157 | 95, 156 | ralrimi 3139 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍)))) |
158 | 76, 157 | jca 507 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) Fn 𝑅 ∧ ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))) |
159 | | ffnfv 6652 |
. . . . . . . . . . . . . . 15
⊢ ((𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍))) ↔ ((𝑐 ↾ 𝑅) Fn 𝑅 ∧ ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) ∈ (0...(𝐽 − (𝑐‘𝑍))))) |
160 | 158, 159 | sylibr 226 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍)))) |
161 | | ovexd 6956 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...(𝐽 − (𝑐‘𝑍))) ∈ V) |
162 | 18, 108 | ssexd 5042 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑅 ∈ V) |
163 | 162 | adantr 474 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑅 ∈ V) |
164 | 161, 163 | elmapd 8154 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ↔ (𝑐 ↾ 𝑅):𝑅⟶(0...(𝐽 − (𝑐‘𝑍))))) |
165 | 160, 164 | mpbird 249 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅)) |
166 | 96 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑡 ∈ 𝑅 → ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡))) |
167 | 95, 166 | ralrimi 3139 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∀𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝑐‘𝑡)) |
168 | 167 | sumeq2d 14840 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡)) |
169 | 123 | cbvsumv 14834 |
. . . . . . . . . . . . . . . 16
⊢
Σ𝑟 ∈
𝑅 (𝑐‘𝑟) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) |
170 | 169 | eqcomi 2787 |
. . . . . . . . . . . . . . 15
⊢
Σ𝑡 ∈
𝑅 (𝑐‘𝑡) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) |
171 | 170 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑟 ∈ 𝑅 (𝑐‘𝑟)) |
172 | 149 | idi 2 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑟 ∈ 𝑅 (𝑐‘𝑟) = (𝐽 − (𝑐‘𝑍))) |
173 | 168, 171,
172 | 3eqtrd 2818 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍))) |
174 | 165, 173 | jca 507 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍)))) |
175 | | eqidd 2779 |
. . . . . . . . . . . . . . 15
⊢ (𝑒 = (𝑐 ↾ 𝑅) → 𝑅 = 𝑅) |
176 | | simpl 476 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑒 = (𝑐 ↾ 𝑅) ∧ 𝑡 ∈ 𝑅) → 𝑒 = (𝑐 ↾ 𝑅)) |
177 | 176 | fveq1d 6448 |
. . . . . . . . . . . . . . 15
⊢ ((𝑒 = (𝑐 ↾ 𝑅) ∧ 𝑡 ∈ 𝑅) → (𝑒‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
178 | 175, 177 | sumeq12rdv 14845 |
. . . . . . . . . . . . . 14
⊢ (𝑒 = (𝑐 ↾ 𝑅) → Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡)) |
179 | 178 | eqeq1d 2780 |
. . . . . . . . . . . . 13
⊢ (𝑒 = (𝑐 ↾ 𝑅) → (Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍)) ↔ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍)))) |
180 | 179 | elrab 3572 |
. . . . . . . . . . . 12
⊢ ((𝑐 ↾ 𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ↔ ((𝑐 ↾ 𝑅) ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ∧ Σ𝑡 ∈ 𝑅 ((𝑐 ↾ 𝑅)‘𝑡) = (𝐽 − (𝑐‘𝑍)))) |
181 | 174, 180 | sylibr 226 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))}) |
182 | | oveq2 6930 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = 𝑅 → ((0...𝑛) ↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 𝑅)) |
183 | | rabeq 3389 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((0...𝑛)
↑𝑚 𝑠) = ((0...𝑛) ↑𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
184 | 182, 183 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛}) |
185 | | sumeq1 14827 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑠 = 𝑅 → Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑐‘𝑡)) |
186 | 185 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑠 = 𝑅 → (Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛)) |
187 | 186 | rabbidv 3386 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
188 | 184, 187 | eqtrd 2814 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑠 = 𝑅 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑠) ∣ Σ𝑡 ∈ 𝑠 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
189 | 188 | mpteq2dv 4980 |
. . . . . . . . . . . . . . . 16
⊢ (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})) |
190 | | elpwg 4387 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑅 ∈ V → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇)) |
191 | 162, 190 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑅 ∈ 𝒫 𝑇 ↔ 𝑅 ⊆ 𝑇)) |
192 | 108, 191 | mpbird 249 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑅 ∈ 𝒫 𝑇) |
193 | 24 | mptex 6758 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛)
↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V |
194 | 193 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) ∈ V) |
195 | 8, 189, 192, 194 | fvmptd3 6564 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝐶‘𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
196 | 195 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
197 | | oveq2 6930 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 𝑚 → (0...𝑛) = (0...𝑚)) |
198 | 197 | oveq1d 6937 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑚 → ((0...𝑛) ↑𝑚 𝑅) = ((0...𝑚) ↑𝑚 𝑅)) |
199 | | rabeq 3389 |
. . . . . . . . . . . . . . . . . 18
⊢
(((0...𝑛)
↑𝑚 𝑅) = ((0...𝑚) ↑𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
200 | 198, 199 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
201 | | eqeq2 2789 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 𝑚 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚)) |
202 | 201 | rabbidv 3386 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}) |
203 | 200, 202 | eqtrd 2814 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑚 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}) |
204 | 203 | cbvmptv 4985 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ0
↦ {𝑐 ∈
((0...𝑛)
↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚}) |
205 | 204 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) = (𝑚 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})) |
206 | 196, 205 | eqtrd 2814 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐶‘𝑅) = (𝑚 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚})) |
207 | | fveq1 6445 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑐 = 𝑒 → (𝑐‘𝑡) = (𝑒‘𝑡)) |
208 | 207 | sumeq2sdv 14842 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑐 = 𝑒 → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 (𝑒‘𝑡)) |
209 | 208 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑐 = 𝑒 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚)) |
210 | 209 | cbvrabv 3396 |
. . . . . . . . . . . . . . . 16
⊢ {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} |
211 | 210 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → {𝑐 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑚} = {𝑒 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚}) |
212 | | oveq2 6930 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → (0...𝑚) = (0...(𝐽 − (𝑐‘𝑍)))) |
213 | 212 | oveq1d 6937 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → ((0...𝑚) ↑𝑚 𝑅) = ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅)) |
214 | | rabeq 3389 |
. . . . . . . . . . . . . . . 16
⊢
(((0...𝑚)
↑𝑚 𝑅) = ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) → {𝑒 ∈ ((0...𝑚) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚} = {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚}) |
215 | 213, 214 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (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}) |
216 | | eqeq2 2789 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝐽 − (𝑐‘𝑍)) → (Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = 𝑚 ↔ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍)))) |
217 | 216 | rabbidv 3386 |
. . . . . . . . . . . . . . 15
⊢ (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))}) |
218 | 211, 215,
217 | 3eqtrd 2818 |
. . . . . . . . . . . . . 14
⊢ (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))}) |
219 | 218 | adantl 475 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 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))}) |
220 | 56, 63 | jca 507 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐‘𝑍)))) |
221 | | elnn0z 11741 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 − (𝑐‘𝑍)) ∈ ℕ0 ↔ ((𝐽 − (𝑐‘𝑍)) ∈ ℤ ∧ 0 ≤ (𝐽 − (𝑐‘𝑍)))) |
222 | 220, 221 | sylibr 226 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈
ℕ0) |
223 | | ovex 6954 |
. . . . . . . . . . . . . . 15
⊢
((0...(𝐽 −
(𝑐‘𝑍))) ↑𝑚 𝑅) ∈ V |
224 | 223 | rabex 5049 |
. . . . . . . . . . . . . 14
⊢ {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ∈ V |
225 | 224 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} ∈ V) |
226 | 206, 219,
222, 225 | fvmptd 6548 |
. . . . . . . . . . . 12
⊢ ((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))}) |
227 | 226 | eqcomd 2784 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → {𝑒 ∈ ((0...(𝐽 − (𝑐‘𝑍))) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑒‘𝑡) = (𝐽 − (𝑐‘𝑍))} = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))) |
228 | 181, 227 | eleqtrd 2861 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))) |
229 | 70, 228 | jca 507 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽) ∧ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) |
230 | 1, 229 | jca 507 |
. . . . . . . 8
⊢ ((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)))))) |
231 | | ovexd 6956 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) ∈ V) |
232 | | vex 3401 |
. . . . . . . . . . 11
⊢ 𝑐 ∈ V |
233 | 232 | resex 5693 |
. . . . . . . . . 10
⊢ (𝑐 ↾ 𝑅) ∈ V |
234 | 233 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐 ↾ 𝑅) ∈ V) |
235 | | opeq12 4638 |
. . . . . . . . . . . 12
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → 〈𝑘, 𝑑〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
236 | 235 | eqeq2d 2788 |
. . . . . . . . . . 11
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈𝑘, 𝑑〉 ↔ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
237 | | eleq1 2847 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (𝐽 − (𝑐‘𝑍)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))) |
238 | 237 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (𝑘 ∈ (0...𝐽) ↔ (𝐽 − (𝑐‘𝑍)) ∈ (0...𝐽))) |
239 | | simpr 479 |
. . . . . . . . . . . . 13
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → 𝑑 = (𝑐 ↾ 𝑅)) |
240 | | fveq2 6446 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (𝐽 − (𝑐‘𝑍)) → ((𝐶‘𝑅)‘𝑘) = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))) |
241 | 240 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → ((𝐶‘𝑅)‘𝑘) = ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍)))) |
242 | 239, 241 | eleq12d 2853 |
. . . . . . . . . . . 12
⊢ ((𝑘 = (𝐽 − (𝑐‘𝑍)) ∧ 𝑑 = (𝑐 ↾ 𝑅)) → (𝑑 ∈ ((𝐶‘𝑅)‘𝑘) ↔ (𝑐 ↾ 𝑅) ∈ ((𝐶‘𝑅)‘(𝐽 − (𝑐‘𝑍))))) |
243 | 238, 242 | anbi12d 624 |
. . . . . . . . . . 11
⊢ ((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)))))) |
244 | 236, 243 | anbi12d 624 |
. . . . . . . . . 10
⊢ ((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))))))) |
245 | 244 | spc2egv 3497 |
. . . . . . . . 9
⊢ (((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))))) |
246 | 231, 234,
245 | syl2anc 579 |
. . . . . . . 8
⊢ ((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))))) |
247 | 230, 246 | mpd 15 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ∃𝑘∃𝑑(〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈𝑘, 𝑑〉 ∧ (𝑘 ∈ (0...𝐽) ∧ 𝑑 ∈ ((𝐶‘𝑅)‘𝑘)))) |
248 | | eliunxp 5505 |
. . . . . . 7
⊢
(〈(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)))) |
249 | 247, 248 | sylibr 226 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
250 | | dvnprodlem1.d |
. . . . . 6
⊢ 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
251 | 249, 250 | fmptd 6648 |
. . . . 5
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)⟶∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
252 | 93 | nfcri 2929 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑡 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) |
253 | 94, 252 | nfan 1946 |
. . . . . . . . . . 11
⊢
Ⅎ𝑡(𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
254 | 77, 253 | nfan 1946 |
. . . . . . . . . 10
⊢
Ⅎ𝑡(𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) |
255 | | nfv 1957 |
. . . . . . . . . 10
⊢
Ⅎ𝑡(𝐷‘𝑐) = (𝐷‘𝑒) |
256 | 254, 255 | nfan 1946 |
. . . . . . . . 9
⊢
Ⅎ𝑡((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) |
257 | 97 | eqcomd 2784 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
258 | 257 | adantlrr 711 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
259 | 258 | adantlr 705 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
260 | 250 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
261 | | opex 5164 |
. . . . . . . . . . . . . . . . . . . 20
⊢
〈(𝐽 −
(𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ V |
262 | 261 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 ∈ V) |
263 | 260, 262 | fvmpt2d 6554 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑐) = 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) |
264 | 263 | fveq2d 6450 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (2nd ‘(𝐷‘𝑐)) = (2nd ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
265 | 264 | fveq1d 6448 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2nd ‘(𝐷‘𝑐))‘𝑡) = ((2nd ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)‘𝑡)) |
266 | | ovex 6954 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐽 − (𝑐‘𝑍)) ∈ V |
267 | 266, 233 | op2nd 7454 |
. . . . . . . . . . . . . . . . . 18
⊢
(2nd ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) = (𝑐 ↾ 𝑅) |
268 | 267 | fveq1i 6447 |
. . . . . . . . . . . . . . . . 17
⊢
((2nd ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡) |
269 | 268 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((2nd
‘〈(𝐽 −
(𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)‘𝑡) = ((𝑐 ↾ 𝑅)‘𝑡)) |
270 | 265, 269 | eqtr2d 2815 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2nd ‘(𝐷‘𝑐))‘𝑡)) |
271 | 270 | adantrr 707 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2nd ‘(𝐷‘𝑐))‘𝑡)) |
272 | 271 | ad2antrr 716 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((𝑐 ↾ 𝑅)‘𝑡) = ((2nd ‘(𝐷‘𝑐))‘𝑡)) |
273 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑐) = (𝐷‘𝑒)) |
274 | | fveq1 6445 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = 𝑒 → (𝑐‘𝑍) = (𝑒‘𝑍)) |
275 | 274 | oveq2d 6938 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑐 = 𝑒 → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝑒‘𝑍))) |
276 | | reseq1 5636 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑐 = 𝑒 → (𝑐 ↾ 𝑅) = (𝑒 ↾ 𝑅)) |
277 | 275, 276 | opeq12d 4644 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑐 = 𝑒 → 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉 = 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) |
278 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
279 | | opex 5164 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
〈(𝐽 −
(𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉 ∈ V |
280 | 279 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉 ∈ V) |
281 | 250, 277,
278, 280 | fvmptd3 6564 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐷‘𝑒) = 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) |
282 | 281 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑒) = 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) |
283 | 273, 282 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐷‘𝑐) = 〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) |
284 | 283 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (2nd ‘(𝐷‘𝑐)) = (2nd ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
285 | 284 | adantlrl 710 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (2nd ‘(𝐷‘𝑐)) = (2nd ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
286 | 285 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ ((((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)〉)) |
287 | | ovex 6954 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐽 − (𝑒‘𝑍)) ∈ V |
288 | | vex 3401 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑒 ∈ V |
289 | 288 | resex 5693 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑒 ↾ 𝑅) ∈ V |
290 | 287, 289 | op2nd 7454 |
. . . . . . . . . . . . . . . . 17
⊢
(2nd ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) = (𝑒 ↾ 𝑅) |
291 | 290 | a1i 11 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2nd ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) = (𝑒 ↾ 𝑅)) |
292 | 286, 291 | eqtrd 2814 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (2nd ‘(𝐷‘𝑐)) = (𝑒 ↾ 𝑅)) |
293 | 292 | fveq1d 6448 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘(𝐷‘𝑐))‘𝑡) = ((𝑒 ↾ 𝑅)‘𝑡)) |
294 | | fvres 6465 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ 𝑅 → ((𝑒 ↾ 𝑅)‘𝑡) = (𝑒‘𝑡)) |
295 | 294 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((𝑒 ↾ 𝑅)‘𝑡) = (𝑒‘𝑡)) |
296 | 293, 295 | eqtrd 2814 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘(𝐷‘𝑐))‘𝑡) = (𝑒‘𝑡)) |
297 | 259, 272,
296 | 3eqtrd 2818 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
298 | 297 | adantlr 705 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
299 | | simpl 476 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍}))) |
300 | | elunnel1 3977 |
. . . . . . . . . . . . . 14
⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 ∈ {𝑍}) |
301 | | elsni 4415 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ {𝑍} → 𝑡 = 𝑍) |
302 | 300, 301 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍) |
303 | 302 | adantll 704 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍) |
304 | | simpr 479 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍) |
305 | 304 | fveq2d 6450 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑐‘𝑍)) |
306 | 3 | nn0cnd 11704 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝐽 ∈ ℂ) |
307 | 306 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ) |
308 | 307, 129 | nncand 10739 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝑐‘𝑍)) |
309 | 308 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍)))) |
310 | 309 | adantrr 707 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍)))) |
311 | 310 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐‘𝑍) = (𝐽 − (𝐽 − (𝑐‘𝑍)))) |
312 | 263 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1st ‘(𝐷‘𝑐)) = (1st ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
313 | 266, 233 | op1st 7453 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1st ‘〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) = (𝐽 − (𝑐‘𝑍)) |
314 | 313 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1st
‘〈(𝐽 −
(𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉) = (𝐽 − (𝑐‘𝑍))) |
315 | 312, 314 | eqtr2d 2815 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝑐‘𝑍)) = (1st ‘(𝐷‘𝑐))) |
316 | 315 | oveq2d 6938 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1st ‘(𝐷‘𝑐)))) |
317 | 316 | adantrr 707 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1st ‘(𝐷‘𝑐)))) |
318 | 317 | adantr 474 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (𝐽 − (𝑐‘𝑍))) = (𝐽 − (1st ‘(𝐷‘𝑐)))) |
319 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐷‘𝑐) = (𝐷‘𝑒) → (1st ‘(𝐷‘𝑐)) = (1st ‘(𝐷‘𝑒))) |
320 | 319 | adantl 475 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1st ‘(𝐷‘𝑐)) = (1st ‘(𝐷‘𝑒))) |
321 | 281 | fveq2d 6450 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (1st ‘(𝐷‘𝑒)) = (1st ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
322 | 321 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1st ‘(𝐷‘𝑒)) = (1st ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉)) |
323 | 287, 289 | op1st 7453 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(1st ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) = (𝐽 − (𝑒‘𝑍)) |
324 | 323 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1st ‘〈(𝐽 − (𝑒‘𝑍)), (𝑒 ↾ 𝑅)〉) = (𝐽 − (𝑒‘𝑍))) |
325 | 320, 322,
324 | 3eqtrd 2818 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (1st ‘(𝐷‘𝑐)) = (𝐽 − (𝑒‘𝑍))) |
326 | 325 | oveq2d 6938 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1st ‘(𝐷‘𝑐))) = (𝐽 − (𝐽 − (𝑒‘𝑍)))) |
327 | 306 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝐽 ∈ ℂ) |
328 | | zsscn 11736 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ℤ
⊆ ℂ |
329 | 6, 328 | sstri 3830 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(0...𝐽) ⊆
ℂ |
330 | 329 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (0...𝐽) ⊆ ℂ) |
331 | | eleq1w 2842 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = 𝑒 → (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↔ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) |
332 | 331 | anbi2d 622 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = 𝑒 → ((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ↔ (𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)))) |
333 | | feq1 6272 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = 𝑒 → (𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽) ↔ 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽))) |
334 | 332, 333 | imbi12d 336 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑐 = 𝑒 → (((𝜑 ∧ 𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑐:(𝑅 ∪ {𝑍})⟶(0...𝐽)) ↔ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽)))) |
335 | 334, 47 | chvarv 2361 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒:(𝑅 ∪ {𝑍})⟶(0...𝐽)) |
336 | 52 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
337 | 335, 336 | ffvelrnd 6624 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒‘𝑍) ∈ (0...𝐽)) |
338 | 330, 337 | sseldd 3822 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝑒‘𝑍) ∈ ℂ) |
339 | 327, 338 | nncand 10739 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → (𝐽 − (𝐽 − (𝑒‘𝑍))) = (𝑒‘𝑍)) |
340 | 339 | adantr 474 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (𝐽 − (𝑒‘𝑍))) = (𝑒‘𝑍)) |
341 | | eqidd 2779 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑒‘𝑍) = (𝑒‘𝑍)) |
342 | 326, 340,
341 | 3eqtrd 2818 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1st ‘(𝐷‘𝑐))) = (𝑒‘𝑍)) |
343 | 342 | adantlrl 710 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝐽 − (1st ‘(𝐷‘𝑐))) = (𝑒‘𝑍)) |
344 | 311, 318,
343 | 3eqtrd 2818 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐‘𝑍) = (𝑒‘𝑍)) |
345 | 344 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑍) = (𝑒‘𝑍)) |
346 | | fveq2 6446 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑍 → (𝑒‘𝑡) = (𝑒‘𝑍)) |
347 | 346 | eqcomd 2784 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 = 𝑍 → (𝑒‘𝑍) = (𝑒‘𝑡)) |
348 | 347 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑒‘𝑍) = (𝑒‘𝑡)) |
349 | 305, 345,
348 | 3eqtrd 2818 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
350 | 349 | adantlr 705 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
351 | 299, 303,
350 | syl2anc 579 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
352 | 298, 351 | pm2.61dan 803 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (𝑐‘𝑡) = (𝑒‘𝑡)) |
353 | 352 | ex 403 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑡 ∈ (𝑅 ∪ {𝑍}) → (𝑐‘𝑡) = (𝑒‘𝑡))) |
354 | 256, 353 | ralrimi 3139 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡)) |
355 | 72 | adantrr 707 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑐 Fn (𝑅 ∪ {𝑍})) |
356 | 355 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑐 Fn (𝑅 ∪ {𝑍})) |
357 | 335 | ffnd 6292 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) → 𝑒 Fn (𝑅 ∪ {𝑍})) |
358 | 357 | adantrl 706 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → 𝑒 Fn (𝑅 ∪ {𝑍})) |
359 | 358 | adantr 474 |
. . . . . . . . 9
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑒 Fn (𝑅 ∪ {𝑍})) |
360 | | eqfnfv 6574 |
. . . . . . . . 9
⊢ ((𝑐 Fn (𝑅 ∪ {𝑍}) ∧ 𝑒 Fn (𝑅 ∪ {𝑍})) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡))) |
361 | 356, 359,
360 | syl2anc 579 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → (𝑐 = 𝑒 ↔ ∀𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = (𝑒‘𝑡))) |
362 | 354, 361 | mpbird 249 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) ∧ (𝐷‘𝑐) = (𝐷‘𝑒)) → 𝑐 = 𝑒) |
363 | 362 | ex 403 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ∧ 𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽))) → ((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒)) |
364 | 363 | ralrimivva 3153 |
. . . . 5
⊢ (𝜑 → ∀𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)∀𝑒 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)((𝐷‘𝑐) = (𝐷‘𝑒) → 𝑐 = 𝑒)) |
365 | 251, 364 | jca 507 |
. . . 4
⊢ (𝜑 → (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))) |
366 | | dff13 6784 |
. . . 4
⊢ (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))) |
367 | 365, 366 | sylibr 226 |
. . 3
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
368 | | eliun 4757 |
. . . . . . . . . . 11
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) ↔ ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
369 | 368 | biimpi 208 |
. . . . . . . . . 10
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
370 | 369 | adantl 475 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
371 | | nfv 1957 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘𝜑 |
372 | | nfcv 2934 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘𝑝 |
373 | | nfiu1 4783 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑘∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
374 | 372, 373 | nfel 2946 |
. . . . . . . . . . 11
⊢
Ⅎ𝑘 𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
375 | 371, 374 | nfan 1946 |
. . . . . . . . . 10
⊢
Ⅎ𝑘(𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
376 | | nfv 1957 |
. . . . . . . . . 10
⊢
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} |
377 | | nfv 1957 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡 𝑘 ∈ (0...𝐽) |
378 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑡𝑝 |
379 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑡{𝑘} |
380 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑡𝑅 |
381 | 89, 380 | nffv 6456 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡(𝐶‘𝑅) |
382 | | nfcv 2934 |
. . . . . . . . . . . . . . . . . . . 20
⊢
Ⅎ𝑡𝑘 |
383 | 381, 382 | nffv 6456 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑡((𝐶‘𝑅)‘𝑘) |
384 | 379, 383 | nfxp 5388 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑡({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
385 | 378, 384 | nfel 2946 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑡 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) |
386 | 77, 377, 385 | nf3an 1948 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡(𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
387 | | 0zd 11740 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ∈
ℤ) |
388 | 4 | adantr 474 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ) |
389 | 388 | 3ad2antl1 1193 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝐽 ∈ ℤ) |
390 | | iftrue 4313 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 ∈ 𝑅 → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡)) |
391 | 390 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((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)) |
392 | | fzssz 12660 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(0...𝑘) ⊆
ℤ |
393 | 392 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → (0...𝑘) ⊆ ℤ) |
394 | | simp1 1127 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝜑) |
395 | | simp2 1128 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽)) |
396 | | xp2nd 7478 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (2nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘)) |
397 | 396 | 3ad2ant3 1126 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) ∈ ((𝐶‘𝑅)‘𝑘)) |
398 | 195 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐶‘𝑅) = (𝑛 ∈ ℕ0 ↦ {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛})) |
399 | | oveq2 6930 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑛 = 𝑘 → (0...𝑛) = (0...𝑘)) |
400 | 399 | oveq1d 6937 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑛 = 𝑘 → ((0...𝑛) ↑𝑚 𝑅) = ((0...𝑘) ↑𝑚 𝑅)) |
401 | | rabeq 3389 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((0...𝑛)
↑𝑚 𝑅) = ((0...𝑘) ↑𝑚 𝑅) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
402 | 400, 401 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛}) |
403 | | eqeq2 2789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑛 = 𝑘 → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛 ↔ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘)) |
404 | 403 | rabbidv 3386 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
405 | 402, 404 | eqtrd 2814 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑛 = 𝑘 → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
406 | 405 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑛 = 𝑘) → {𝑐 ∈ ((0...𝑛) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑛} = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
407 | | elfznn0 12751 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℕ0) |
408 | 407 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℕ0) |
409 | | ovex 6954 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
((0...𝑘)
↑𝑚 𝑅) ∈ V |
410 | 409 | rabex 5049 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V |
411 | 410 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ∈ V) |
412 | 398, 406,
408, 411 | fvmptd 6548 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
413 | 412 | 3adant3 1123 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((𝐶‘𝑅)‘𝑘) = {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
414 | 397, 413 | eleqtrd 2861 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) |
415 | | elrabi 3567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
((2nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} → (2nd ‘𝑝) ∈ ((0...𝑘) ↑𝑚
𝑅)) |
416 | 415 | 3ad2ant3 1126 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ (2nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘}) → (2nd ‘𝑝) ∈ ((0...𝑘) ↑𝑚
𝑅)) |
417 | 394, 395,
414, 416 | syl3anc 1439 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) ∈ ((0...𝑘) ↑𝑚
𝑅)) |
418 | | elmapi 8162 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((2nd ‘𝑝) ∈ ((0...𝑘) ↑𝑚 𝑅) → (2nd
‘𝑝):𝑅⟶(0...𝑘)) |
419 | 417, 418 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝):𝑅⟶(0...𝑘)) |
420 | 419 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (2nd ‘𝑝):𝑅⟶(0...𝑘)) |
421 | 420 | ffvelrnda 6623 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘)) |
422 | 393, 421 | sseldd 3822 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ ℤ) |
423 | 391, 422 | eqeltrd 2859 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
424 | | simpl 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍}))) |
425 | 302 | adantll 704 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 𝑡 = 𝑍) |
426 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → 𝑡 = 𝑍) |
427 | 134 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → ¬ 𝑍 ∈ 𝑅) |
428 | 426, 427 | eqneltrd 2878 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → ¬ 𝑡 ∈ 𝑅) |
429 | 428 | iffalsed 4318 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
430 | 429 | 3ad2antl1 1193 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
431 | 4 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ) |
432 | 431 | 3ad2antl1 1193 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 𝐽 ∈ ℤ) |
433 | | xp1st 7477 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ {𝑘}) |
434 | | elsni 4415 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
((1st ‘𝑝) ∈ {𝑘} → (1st ‘𝑝) = 𝑘) |
435 | 433, 434 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) = 𝑘) |
436 | 435 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) = 𝑘) |
437 | 6 | sseli 3817 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℤ) |
438 | 437 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ ℤ) |
439 | 436, 438 | eqeltrd 2859 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈
ℤ) |
440 | 439 | 3adant1 1121 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈
ℤ) |
441 | 440 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (1st ‘𝑝) ∈
ℤ) |
442 | 432, 441 | zsubcld 11839 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1st ‘𝑝)) ∈
ℤ) |
443 | 430, 442 | eqeltrd 2859 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
444 | 443 | adantlr 705 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
445 | 424, 425,
444 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
446 | 423, 445 | pm2.61dan 803 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ) |
447 | 387, 389,
446 | 3jca 1119 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → (0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℤ)) |
448 | 419 | ffvelrnda 6623 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘)) |
449 | | elfzle1 12661 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → 0 ≤ ((2nd ‘𝑝)‘𝑡)) |
450 | 448, 449 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 0 ≤ ((2nd
‘𝑝)‘𝑡)) |
451 | 390 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑡 ∈ 𝑅 → ((2nd ‘𝑝)‘𝑡) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
452 | 451 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
453 | 450, 452 | breqtrd 4912 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
454 | 453 | adantlr 705 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
455 | | simpll 757 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → (𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)))) |
456 | | elfzle2 12662 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ≤ 𝐽) |
457 | | elfzel2 12657 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℤ) |
458 | 457 | zred 11834 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝐽) → 𝐽 ∈ ℝ) |
459 | 113 | sseli 3817 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℝ) |
460 | 458, 459 | subge0d 10965 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 ∈ (0...𝐽) → (0 ≤ (𝐽 − 𝑘) ↔ 𝑘 ≤ 𝐽)) |
461 | 456, 460 | mpbird 249 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 ∈ (0...𝐽) → 0 ≤ (𝐽 − 𝑘)) |
462 | 461 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽 − 𝑘)) |
463 | 462 | 3ad2antl2 1194 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ (𝐽 − 𝑘)) |
464 | 394, 428 | sylan 575 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → ¬ 𝑡 ∈ 𝑅) |
465 | 464 | iffalsed 4318 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
466 | 436 | 3adant1 1121 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) = 𝑘) |
467 | 466 | oveq2d 6938 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) = (𝐽 − 𝑘)) |
468 | 467 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − (1st ‘𝑝)) = (𝐽 − 𝑘)) |
469 | 465, 468 | eqtr2d 2815 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
470 | 463, 469 | breqtrd 4912 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
471 | 455, 425,
470 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
472 | 454, 471 | pm2.61dan 803 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 0 ≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
473 | | simpl2 1201 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → 𝑘 ∈ (0...𝐽)) |
474 | 392 | sseli 3817 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd ‘𝑝)‘𝑡) ∈ ℤ) |
475 | 474 | zred 11834 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd ‘𝑝)‘𝑡) ∈ ℝ) |
476 | 475 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd ‘𝑝)‘𝑡) ∈ ℝ) |
477 | 459 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ) |
478 | 458 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ) |
479 | | elfzle2 12662 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) → ((2nd ‘𝑝)‘𝑡) ≤ 𝑘) |
480 | 479 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd ‘𝑝)‘𝑡) ≤ 𝑘) |
481 | 456 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ≤ 𝐽) |
482 | 476, 477,
478, 480, 481 | letrd 10533 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((2nd ‘𝑝)‘𝑡) ∈ (0...𝑘) ∧ 𝑘 ∈ (0...𝐽)) → ((2nd ‘𝑝)‘𝑡) ≤ 𝐽) |
483 | 448, 473,
482 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ≤ 𝐽) |
484 | 483 | adantlr 705 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ≤ 𝐽) |
485 | 391, 484 | eqbrtrd 4908 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽) |
486 | 469 | eqcomd 2784 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = (𝐽 − 𝑘)) |
487 | 408 | nn0ge0d 11705 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 0 ≤ 𝑘) |
488 | 458 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝐽 ∈ ℝ) |
489 | 459 | adantl 475 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → 𝑘 ∈ ℝ) |
490 | 488, 489 | subge02d 10967 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (0 ≤ 𝑘 ↔ (𝐽 − 𝑘) ≤ 𝐽)) |
491 | 487, 490 | mpbid 224 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽)) → (𝐽 − 𝑘) ≤ 𝐽) |
492 | 491 | adantr 474 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽)) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) ≤ 𝐽) |
493 | 492 | 3adantl3 1170 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → (𝐽 − 𝑘) ≤ 𝐽) |
494 | 486, 493 | eqbrtrd 4908 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 = 𝑍) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽) |
495 | 455, 425,
494 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ ¬ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽) |
496 | 485, 495 | pm2.61dan 803 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽) |
497 | 447, 472,
496 | jca32 511 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → ((0 ∈ ℤ ∧ 𝐽 ∈ ℤ ∧ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈ ℤ) ∧ (0
≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∧ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽))) |
498 | | elfz2 12650 |
. . . . . . . . . . . . . . . . 17
⊢ (if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈ (0...𝐽) ↔ ((0 ∈ ℤ
∧ 𝐽 ∈ ℤ
∧ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈ ℤ) ∧ (0
≤ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∧ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ≤ 𝐽))) |
499 | 497, 498 | sylibr 226 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈ (0...𝐽)) |
500 | | eqid 2778 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
501 | 386, 499,
500 | fmptdf 6651 |
. . . . . . . . . . . . . . 15
⊢ ((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)) |
502 | | ovexd 6956 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (0...𝐽) ∈ V) |
503 | 394, 20 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑅 ∪ {𝑍}) ∈ V) |
504 | 502, 503 | elmapd 8154 |
. . . . . . . . . . . . . . 15
⊢ ((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})) ↔ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))):(𝑅 ∪ {𝑍})⟶(0...𝐽))) |
505 | 501, 504 | mpbird 249 |
. . . . . . . . . . . . . 14
⊢ ((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}))) |
506 | | eqidd 2779 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((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))))) |
507 | | eleq1w 2842 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 = 𝑡 → (𝑟 ∈ 𝑅 ↔ 𝑡 ∈ 𝑅)) |
508 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑟 = 𝑡 → ((2nd ‘𝑝)‘𝑟) = ((2nd ‘𝑝)‘𝑡)) |
509 | 507, 508 | ifbieq1d 4330 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑟 = 𝑡 → if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
510 | 509 | adantl 475 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) ∧ 𝑟 = 𝑡) → if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))) = if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) |
511 | | simpr 479 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ (𝑅 ∪ {𝑍})) → 𝑡 ∈ (𝑅 ∪ {𝑍})) |
512 | 506, 510,
511, 446 | fvmptd 6548 |
. . . . . . . . . . . . . . . . . 18
⊢ (((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)))) |
513 | 512 | ex 403 |
. . . . . . . . . . . . . . . . 17
⊢ ((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))))) |
514 | 386, 513 | ralrimi 3139 |
. . . . . . . . . . . . . . . 16
⊢ ((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)))) |
515 | 514 | sumeq2d 14840 |
. . . . . . . . . . . . . . 15
⊢ ((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)))) |
516 | | nfcv 2934 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑡if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) |
517 | 394, 110 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑅 ∈ Fin) |
518 | 394, 48 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑍 ∈ 𝑇) |
519 | 394, 134 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ¬ 𝑍 ∈ 𝑅) |
520 | 390 | adantl 475 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡)) |
521 | 448, 474 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ ℤ) |
522 | 521 | zcnd 11835 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → ((2nd ‘𝑝)‘𝑡) ∈ ℂ) |
523 | 520, 522 | eqeltrd 2859 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑡 ∈ 𝑅) → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) ∈
ℂ) |
524 | | eleq1 2847 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑍 → (𝑡 ∈ 𝑅 ↔ 𝑍 ∈ 𝑅)) |
525 | | fveq2 6446 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑡 = 𝑍 → ((2nd ‘𝑝)‘𝑡) = ((2nd ‘𝑝)‘𝑍)) |
526 | 524, 525 | ifbieq1d 4330 |
. . . . . . . . . . . . . . . 16
⊢ (𝑡 = 𝑍 → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝)))) |
527 | 134 | adantr 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ¬ 𝑍 ∈ 𝑅) |
528 | 527 | iffalsed 4318 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
529 | 528 | 3adant2 1122 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
530 | 4 | 3ad2ant1 1124 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℤ) |
531 | 530, 440 | zsubcld 11839 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) ∈
ℤ) |
532 | 531 | zcnd 11835 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝐽 − (1st ‘𝑝)) ∈
ℂ) |
533 | 529, 532 | eqeltrd 2859 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) ∈
ℂ) |
534 | 386, 516,
517, 518, 519, 523, 526, 533 | fsumsplitsn 14881 |
. . . . . . . . . . . . . . 15
⊢ ((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))))) |
535 | 390 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡))) |
536 | 386, 535 | ralrimi 3139 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∀𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = ((2nd
‘𝑝)‘𝑡)) |
537 | 536 | sumeq2d 14840 |
. . . . . . . . . . . . . . . . . 18
⊢ ((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)) |
538 | | eqidd 2779 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑐 = (2nd ‘𝑝) → 𝑅 = 𝑅) |
539 | | simpl 476 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑐 = (2nd ‘𝑝) ∧ 𝑡 ∈ 𝑅) → 𝑐 = (2nd ‘𝑝)) |
540 | 539 | fveq1d 6448 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑐 = (2nd ‘𝑝) ∧ 𝑡 ∈ 𝑅) → (𝑐‘𝑡) = ((2nd ‘𝑝)‘𝑡)) |
541 | 538, 540 | sumeq12rdv 14845 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑐 = (2nd ‘𝑝) → Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡)) |
542 | 541 | eqeq1d 2780 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑐 = (2nd ‘𝑝) → (Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘 ↔ Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡) = 𝑘)) |
543 | 542 | elrab 3572 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((2nd ‘𝑝) ∈ {𝑐 ∈ ((0...𝑘) ↑𝑚 𝑅) ∣ Σ𝑡 ∈ 𝑅 (𝑐‘𝑡) = 𝑘} ↔ ((2nd ‘𝑝) ∈ ((0...𝑘) ↑𝑚
𝑅) ∧ Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡) = 𝑘)) |
544 | 414, 543 | sylib 210 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ((2nd ‘𝑝) ∈ ((0...𝑘) ↑𝑚
𝑅) ∧ Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡) = 𝑘)) |
545 | 544 | simprd 491 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 ((2nd ‘𝑝)‘𝑡) = 𝑘) |
546 | 537, 545 | eqtrd 2814 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → Σ𝑡 ∈ 𝑅 if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = 𝑘) |
547 | 519 | iffalsed 4318 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
548 | 547, 467 | eqtrd 2814 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → if(𝑍 ∈ 𝑅, ((2nd ‘𝑝)‘𝑍), (𝐽 − (1st ‘𝑝))) = (𝐽 − 𝑘)) |
549 | 546, 548 | oveq12d 6940 |
. . . . . . . . . . . . . . . 16
⊢ ((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))) |
550 | 329 | sseli 3817 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝐽) → 𝑘 ∈ ℂ) |
551 | 550 | 3ad2ant2 1125 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ ℂ) |
552 | 394, 306 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐽 ∈ ℂ) |
553 | 551, 552 | pncan3d 10737 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 + (𝐽 − 𝑘)) = 𝐽) |
554 | 549, 553 | eqtrd 2814 |
. . . . . . . . . . . . . . 15
⊢ ((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) |
555 | 515, 534,
554 | 3eqtrd 2818 |
. . . . . . . . . . . . . 14
⊢ ((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) |
556 | 505, 555 | jca 507 |
. . . . . . . . . . . . 13
⊢ ((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})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡) = 𝐽)) |
557 | | eleq1w 2842 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑟 → (𝑡 ∈ 𝑅 ↔ 𝑟 ∈ 𝑅)) |
558 | | fveq2 6446 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑡 = 𝑟 → ((2nd ‘𝑝)‘𝑡) = ((2nd ‘𝑝)‘𝑟)) |
559 | 557, 558 | ifbieq1d 4330 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑡 = 𝑟 → if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))) = if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝)))) |
560 | 559 | cbvmptv 4985 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝)))) |
561 | 560 | eqeq2i 2790 |
. . . . . . . . . . . . . . . . 17
⊢ (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))))) |
562 | 561 | biimpi 208 |
. . . . . . . . . . . . . . . 16
⊢ (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))))) |
563 | | fveq1 6445 |
. . . . . . . . . . . . . . . . 17
⊢ (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)) |
564 | 563 | sumeq2sdv 14842 |
. . . . . . . . . . . . . . . 16
⊢ (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)) |
565 | 562, 564 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (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)) |
566 | 565 | eqeq1d 2780 |
. . . . . . . . . . . . . 14
⊢ (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)) |
567 | 566 | elrab 3572 |
. . . . . . . . . . . . 13
⊢ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} ↔ ((𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) ∈ ((0...𝐽) ↑𝑚
(𝑅 ∪ {𝑍})) ∧ Σ𝑡 ∈ (𝑅 ∪ {𝑍})((𝑟 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))‘𝑡) = 𝐽)) |
568 | 556, 567 | sylibr 226 |
. . . . . . . . . . . 12
⊢ ((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}) |
569 | 568 | 3exp 1109 |
. . . . . . . . . . 11
⊢ (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}))) |
570 | 569 | adantr 474 |
. . . . . . . . . 10
⊢ ((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}))) |
571 | 375, 376,
570 | rexlimd 3208 |
. . . . . . . . 9
⊢ ((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})) |
572 | 370, 571 | mpd 15 |
. . . . . . . 8
⊢ ((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}) |
573 | 39 | eqcomd 2784 |
. . . . . . . . 9
⊢ (𝜑 → {𝑐 ∈ ((0...𝐽) ↑𝑚 (𝑅 ∪ {𝑍})) ∣ Σ𝑡 ∈ (𝑅 ∪ {𝑍})(𝑐‘𝑡) = 𝐽} = ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)) |
574 | 573 | adantr 474 |
. . . . . . . 8
⊢ ((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)) |
575 | 572, 574 | eleqtrd 2861 |
. . . . . . 7
⊢ ((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)) |
576 | 250 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝐷 = (𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽) ↦ 〈(𝐽 − (𝑐‘𝑍)), (𝑐 ↾ 𝑅)〉)) |
577 | | simpr 479 |
. . . . . . . . . . . . . . 15
⊢ ((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))))) |
578 | 560 | a1i 11 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 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 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))))) |
579 | 577, 578 | eqtrd 2814 |
. . . . . . . . . . . . . 14
⊢ ((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))))) |
580 | | simpr 479 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 = 𝑍) → 𝑟 = 𝑍) |
581 | 134 | adantr 474 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑟 = 𝑍) → ¬ 𝑍 ∈ 𝑅) |
582 | 580, 581 | eqneltrd 2878 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑟 = 𝑍) → ¬ 𝑟 ∈ 𝑅) |
583 | 582 | iffalsed 4318 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑟 = 𝑍) → if(𝑟 ∈ 𝑅, ((2nd ‘𝑝)‘𝑟), (𝐽 − (1st ‘𝑝))) = (𝐽 − (1st ‘𝑝))) |
584 | 583 | adantlr 705 |
. . . . . . . . . . . . . 14
⊢ (((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))) |
585 | 52 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → 𝑍 ∈ (𝑅 ∪ {𝑍})) |
586 | | ovexd 6956 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝐽 − (1st ‘𝑝)) ∈ V) |
587 | 579, 584,
585, 586 | fvmptd 6548 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝑐‘𝑍) = (𝐽 − (1st ‘𝑝))) |
588 | 587 | oveq2d 6938 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → (𝐽 − (𝑐‘𝑍)) = (𝐽 − (𝐽 − (1st ‘𝑝)))) |
589 | 588 | adantlr 705 |
. . . . . . . . . . 11
⊢ (((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)))) |
590 | 306 | ad2antrr 716 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) ∧ 𝑐 = (𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝))))) → 𝐽 ∈ ℂ) |
591 | | nfv 1957 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(1st ‘𝑝) ∈ (0...𝐽) |
592 | | simpl 476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑘 ∈ (0...𝐽)) |
593 | | simpr 479 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ∈ (0...𝐽) ∧ (1st ‘𝑝) = 𝑘) → (1st ‘𝑝) = 𝑘) |
594 | | simpl 476 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑘 ∈ (0...𝐽) ∧ (1st ‘𝑝) = 𝑘) → 𝑘 ∈ (0...𝐽)) |
595 | 593, 594 | eqeltrd 2859 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑘 ∈ (0...𝐽) ∧ (1st ‘𝑝) = 𝑘) → (1st ‘𝑝) ∈ (0...𝐽)) |
596 | 592, 436,
595 | syl2anc 579 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (1st ‘𝑝) ∈ (0...𝐽)) |
597 | 596 | ex 403 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽))) |
598 | 597 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽)))) |
599 | 374, 591,
598 | rexlimd 3208 |
. . . . . . . . . . . . . . . 16
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽))) |
600 | 369, 599 | mpd 15 |
. . . . . . . . . . . . . . 15
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈ (0...𝐽)) |
601 | 6 | sseli 3817 |
. . . . . . . . . . . . . . 15
⊢
((1st ‘𝑝) ∈ (0...𝐽) → (1st ‘𝑝) ∈
ℤ) |
602 | 600, 601 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈
ℤ) |
603 | 602 | zcnd 11835 |
. . . . . . . . . . . . 13
⊢ (𝑝 ∈ ∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (1st ‘𝑝) ∈
ℂ) |
604 | 603 | ad2antlr 717 |
. . . . . . . . . . . 12
⊢ (((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) |
605 | 590, 604 | nncand 10739 |
. . . . . . . . . . 11
⊢ (((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)) |
606 | 589, 605 | eqtrd 2814 |
. . . . . . . . . 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‘𝑍)) = (1st ‘𝑝)) |
607 | | reseq1 5636 |
. . . . . . . . . . . 12
⊢ (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)) |
608 | 607 | adantl 475 |
. . . . . . . . . . 11
⊢ (((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)) |
609 | 73 | a1i 11 |
. . . . . . . . . . . 12
⊢ (((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})) |
610 | 609 | resmptd 5702 |
. . . . . . . . . . 11
⊢ (((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))))) |
611 | | nfv 1957 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘(𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝) |
612 | 390 | mpteq2ia 4975 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (𝑡 ∈ 𝑅 ↦ ((2nd ‘𝑝)‘𝑡)) |
613 | 612 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (𝑡 ∈ 𝑅 ↦ ((2nd ‘𝑝)‘𝑡))) |
614 | 419 | feqmptd 6509 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (2nd ‘𝑝) = (𝑡 ∈ 𝑅 ↦ ((2nd ‘𝑝)‘𝑡))) |
615 | 613, 614 | eqtr4d 2817 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝐽) ∧ 𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝)) |
616 | 615 | 3exp 1109 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝)))) |
617 | 616 | adantr 474 |
. . . . . . . . . . . . . 14
⊢ ((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)))) |
618 | 375, 611,
617 | rexlimd 3208 |
. . . . . . . . . . . . 13
⊢ ((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))) |
619 | 370, 618 | mpd 15 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑡 ∈ 𝑅 ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))) = (2nd
‘𝑝)) |
620 | 619 | adantr 474 |
. . . . . . . . . . 11
⊢ (((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)) |
621 | 608, 610,
620 | 3eqtrd 2818 |
. . . . . . . . . 10
⊢ (((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)) |
622 | 606, 621 | opeq12d 4644 |
. . . . . . . . 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)), (𝑐 ↾ 𝑅)〉 = 〈(1st ‘𝑝), (2nd ‘𝑝)〉) |
623 | | opex 5164 |
. . . . . . . . . 10
⊢
〈(1st ‘𝑝), (2nd ‘𝑝)〉 ∈ V |
624 | 623 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 ∈
V) |
625 | 576, 622,
575, 624 | fvmptd 6548 |
. . . . . . . 8
⊢ ((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)〉) |
626 | | nfv 1957 |
. . . . . . . . . 10
⊢
Ⅎ𝑘〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝 |
627 | | 1st2nd2 7484 |
. . . . . . . . . . . . 13
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 𝑝 = 〈(1st ‘𝑝), (2nd ‘𝑝)〉) |
628 | 627 | eqcomd 2784 |
. . . . . . . . . . . 12
⊢ (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝) |
629 | 628 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝)) |
630 | 629 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (𝑘 ∈ (0...𝐽) → (𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝))) |
631 | 375, 626,
630 | rexlimd 3208 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → (∃𝑘 ∈ (0...𝐽)𝑝 ∈ ({𝑘} × ((𝐶‘𝑅)‘𝑘)) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝)) |
632 | 370, 631 | mpd 15 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 〈(1st ‘𝑝), (2nd ‘𝑝)〉 = 𝑝) |
633 | 625, 632 | eqtr2d 2815 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → 𝑝 = (𝐷‘(𝑡 ∈ (𝑅 ∪ {𝑍}) ↦ if(𝑡 ∈ 𝑅, ((2nd ‘𝑝)‘𝑡), (𝐽 − (1st ‘𝑝)))))) |
634 | | fveq2 6446 |
. . . . . . . 8
⊢ (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)))))) |
635 | 634 | rspceeqv 3529 |
. . . . . . 7
⊢ (((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)) |
636 | 575, 633,
635 | syl2anc 579 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) → ∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)) |
637 | 636 | ralrimiva 3148 |
. . . . 5
⊢ (𝜑 → ∀𝑝 ∈ ∪
𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))∃𝑐 ∈ ((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)𝑝 = (𝐷‘𝑐)) |
638 | 251, 637 | jca 507 |
. . . 4
⊢ (𝜑 → (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))) |
639 | | dffo3 6638 |
. . . 4
⊢ (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))) |
640 | 638, 639 | sylibr 226 |
. . 3
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |
641 | 367, 640 | jca 507 |
. 2
⊢ (𝜑 → (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)))) |
642 | | df-f1o 6142 |
. 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)))) |
643 | 641, 642 | sylibr 226 |
1
⊢ (𝜑 → 𝐷:((𝐶‘(𝑅 ∪ {𝑍}))‘𝐽)–1-1-onto→∪ 𝑘 ∈ (0...𝐽)({𝑘} × ((𝐶‘𝑅)‘𝑘))) |