Step | Hyp | Ref
| Expression |
1 | | poimirlem2.3 |
. . . . 5
⊢ (𝜑 → 𝑈:(1...𝑁)–1-1-onto→(1...𝑁)) |
2 | | f1of 6137 |
. . . . 5
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)⟶(1...𝑁)) |
3 | 1, 2 | syl 17 |
. . . 4
⊢ (𝜑 → 𝑈:(1...𝑁)⟶(1...𝑁)) |
4 | | poimir.0 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℕ) |
5 | 4 | nncnd 11036 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℂ) |
6 | | npcan1 10455 |
. . . . . . . 8
⊢ (𝑁 ∈ ℂ → ((𝑁 − 1) + 1) = 𝑁) |
7 | 5, 6 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁) |
8 | 4 | nnzd 11481 |
. . . . . . . 8
⊢ (𝜑 → 𝑁 ∈ ℤ) |
9 | | peano2zm 11420 |
. . . . . . . 8
⊢ (𝑁 ∈ ℤ → (𝑁 − 1) ∈
ℤ) |
10 | | uzid 11702 |
. . . . . . . 8
⊢ ((𝑁 − 1) ∈ ℤ
→ (𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1))) |
11 | | peano2uz 11741 |
. . . . . . . 8
⊢ ((𝑁 − 1) ∈
(ℤ≥‘(𝑁 − 1)) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
12 | 8, 9, 10, 11 | 4syl 19 |
. . . . . . 7
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑁 − 1))) |
13 | 7, 12 | eqeltrrd 2702 |
. . . . . 6
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑁 − 1))) |
14 | | fzss2 12381 |
. . . . . 6
⊢ (𝑁 ∈
(ℤ≥‘(𝑁 − 1)) → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
15 | 13, 14 | syl 17 |
. . . . 5
⊢ (𝜑 → (1...(𝑁 − 1)) ⊆ (1...𝑁)) |
16 | | poimirlem1.4 |
. . . . 5
⊢ (𝜑 → 𝑀 ∈ (1...(𝑁 − 1))) |
17 | 15, 16 | sseldd 3604 |
. . . 4
⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
18 | 3, 17 | ffvelrnd 6360 |
. . 3
⊢ (𝜑 → (𝑈‘𝑀) ∈ (1...𝑁)) |
19 | | fzp1elp1 12394 |
. . . . . 6
⊢ (𝑀 ∈ (1...(𝑁 − 1)) → (𝑀 + 1) ∈ (1...((𝑁 − 1) + 1))) |
20 | 16, 19 | syl 17 |
. . . . 5
⊢ (𝜑 → (𝑀 + 1) ∈ (1...((𝑁 − 1) + 1))) |
21 | 7 | oveq2d 6666 |
. . . . 5
⊢ (𝜑 → (1...((𝑁 − 1) + 1)) = (1...𝑁)) |
22 | 20, 21 | eleqtrd 2703 |
. . . 4
⊢ (𝜑 → (𝑀 + 1) ∈ (1...𝑁)) |
23 | 3, 22 | ffvelrnd 6360 |
. . 3
⊢ (𝜑 → (𝑈‘(𝑀 + 1)) ∈ (1...𝑁)) |
24 | | imassrn 5477 |
. . . . . . . . . 10
⊢ (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ ran 𝑈 |
25 | | frn 6053 |
. . . . . . . . . . 11
⊢ (𝑈:(1...𝑁)⟶(1...𝑁) → ran 𝑈 ⊆ (1...𝑁)) |
26 | 1, 2, 25 | 3syl 18 |
. . . . . . . . . 10
⊢ (𝜑 → ran 𝑈 ⊆ (1...𝑁)) |
27 | 24, 26 | syl5ss 3614 |
. . . . . . . . 9
⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (1...𝑁)) |
28 | 27 | sselda 3603 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (1...𝑁)) |
29 | | poimirlem2.2 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇:(1...𝑁)⟶ℤ) |
30 | 29 | ffvelrnda 6359 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) ∈ ℤ) |
31 | 30 | zred 11482 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) ∈ ℝ) |
32 | 31 | ltp1d 10954 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) < ((𝑇‘𝑛) + 1)) |
33 | 31, 32 | ltned 10173 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) ≠ ((𝑇‘𝑛) + 1)) |
34 | 28, 33 | syldan 487 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (𝑇‘𝑛) ≠ ((𝑇‘𝑛) + 1)) |
35 | | poimirlem2.1 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹 = (𝑦 ∈ (0...(𝑁 − 1)) ↦ ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))))) |
36 | | breq1 4656 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑀 − 1) → (𝑦 < 𝑀 ↔ (𝑀 − 1) < 𝑀)) |
37 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑀 − 1) → 𝑦 = (𝑀 − 1)) |
38 | 36, 37 | ifbieq1d 4109 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑀 − 1) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if((𝑀 − 1) < 𝑀, (𝑀 − 1), (𝑦 + 1))) |
39 | | elfzelz 12342 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ ℤ) |
40 | 16, 39 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑀 ∈ ℤ) |
41 | 40 | zred 11482 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝑀 ∈ ℝ) |
42 | 41 | ltm1d 10956 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀 − 1) < 𝑀) |
43 | 42 | iftrued 4094 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → if((𝑀 − 1) < 𝑀, (𝑀 − 1), (𝑦 + 1)) = (𝑀 − 1)) |
44 | 38, 43 | sylan9eqr 2678 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑀 − 1)) |
45 | 44 | csbeq1d 3540 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑀 − 1) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) |
46 | 8, 9 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝑁 − 1) ∈ ℤ) |
47 | | elfzm1b 12418 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑀 ∈ ℤ ∧ (𝑁 − 1) ∈ ℤ)
→ (𝑀 ∈
(1...(𝑁 − 1)) ↔
(𝑀 − 1) ∈
(0...((𝑁 − 1) −
1)))) |
48 | 40, 46, 47 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑀 ∈ (1...(𝑁 − 1)) ↔ (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1)))) |
49 | 16, 48 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑀 − 1) ∈ (0...((𝑁 − 1) − 1))) |
50 | | oveq2 6658 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = (𝑀 − 1) → (1...𝑗) = (1...(𝑀 − 1))) |
51 | 50 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑀 − 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑀 − 1)))) |
52 | 51 | xpeq1d 5138 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑀 − 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 − 1))) × {1})) |
53 | 52 | adantl 482 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 − 1))) × {1})) |
54 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = (𝑀 − 1) → (𝑗 + 1) = ((𝑀 − 1) + 1)) |
55 | 40 | zcnd 11483 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑀 ∈ ℂ) |
56 | | npcan1 10455 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ ℂ → ((𝑀 − 1) + 1) = 𝑀) |
57 | 55, 56 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑀 − 1) + 1) = 𝑀) |
58 | 54, 57 | sylan9eqr 2678 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑗 + 1) = 𝑀) |
59 | 58 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑗 + 1)...𝑁) = (𝑀...𝑁)) |
60 | 59 | imaeq2d 5466 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (𝑀...𝑁))) |
61 | 60 | xpeq1d 5138 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (𝑀...𝑁)) × {0})) |
62 | 53, 61 | uneq12d 3768 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))) |
63 | 62 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 = (𝑀 − 1)) → (𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))) |
64 | 49, 63 | csbied 3560 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ⦋(𝑀 − 1) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))) |
65 | 64 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋(𝑀 − 1) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))) |
66 | 45, 65 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = (𝑀 − 1)) → ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))) |
67 | 46 | zcnd 11483 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑁 − 1) ∈ ℂ) |
68 | | npcan1 10455 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 − 1) ∈ ℂ
→ (((𝑁 − 1)
− 1) + 1) = (𝑁
− 1)) |
69 | 67, 68 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((𝑁 − 1) − 1) + 1) = (𝑁 − 1)) |
70 | | peano2zm 11420 |
. . . . . . . . . . . . . . 15
⊢ ((𝑁 − 1) ∈ ℤ
→ ((𝑁 − 1)
− 1) ∈ ℤ) |
71 | | uzid 11702 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 − 1) − 1) ∈
ℤ → ((𝑁 −
1) − 1) ∈ (ℤ≥‘((𝑁 − 1) − 1))) |
72 | | peano2uz 11741 |
. . . . . . . . . . . . . . 15
⊢ (((𝑁 − 1) − 1) ∈
(ℤ≥‘((𝑁 − 1) − 1)) → (((𝑁 − 1) − 1) + 1)
∈ (ℤ≥‘((𝑁 − 1) − 1))) |
73 | 46, 70, 71, 72 | 4syl 19 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (((𝑁 − 1) − 1) + 1) ∈
(ℤ≥‘((𝑁 − 1) − 1))) |
74 | 69, 73 | eqeltrrd 2702 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 − 1) ∈
(ℤ≥‘((𝑁 − 1) − 1))) |
75 | | fzss2 12381 |
. . . . . . . . . . . . 13
⊢ ((𝑁 − 1) ∈
(ℤ≥‘((𝑁 − 1) − 1)) → (0...((𝑁 − 1) − 1)) ⊆
(0...(𝑁 −
1))) |
76 | 74, 75 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝜑 → (0...((𝑁 − 1) − 1)) ⊆ (0...(𝑁 − 1))) |
77 | 76, 49 | sseldd 3604 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑀 − 1) ∈ (0...(𝑁 − 1))) |
78 | | ovexd 6680 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))) ∈ V) |
79 | 35, 66, 77, 78 | fvmptd 6288 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹‘(𝑀 − 1)) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))) |
80 | 79 | fveq1d 6193 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛)) |
81 | 80 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛)) |
82 | | ffn 6045 |
. . . . . . . . . . . 12
⊢ (𝑇:(1...𝑁)⟶ℤ → 𝑇 Fn (1...𝑁)) |
83 | 29, 82 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 Fn (1...𝑁)) |
84 | 83 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑇 Fn (1...𝑁)) |
85 | | 1ex 10035 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
V |
86 | | fnconstg 6093 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
V → ((𝑈 “
(1...(𝑀 − 1)))
× {1}) Fn (𝑈 “
(1...(𝑀 −
1)))) |
87 | 85, 86 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) |
88 | | c0ex 10034 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
89 | | fnconstg 6093 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
V → ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))) |
90 | 88, 89 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)) |
91 | 87, 90 | pm3.2i 471 |
. . . . . . . . . . . . 13
⊢ (((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))) |
92 | | dff1o3 6143 |
. . . . . . . . . . . . . . . 16
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) ↔ (𝑈:(1...𝑁)–onto→(1...𝑁) ∧ Fun ◡𝑈)) |
93 | 92 | simprbi 480 |
. . . . . . . . . . . . . . 15
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → Fun ◡𝑈) |
94 | | imain 5974 |
. . . . . . . . . . . . . . 15
⊢ (Fun
◡𝑈 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁)))) |
95 | 1, 93, 94 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁)))) |
96 | | fzdisj 12368 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 − 1) < 𝑀 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅) |
97 | 42, 96 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1...(𝑀 − 1)) ∩ (𝑀...𝑁)) = ∅) |
98 | 97 | imaeq2d 5466 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = (𝑈 “ ∅)) |
99 | | ima0 5481 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 “ ∅) =
∅ |
100 | 98, 99 | syl6eq 2672 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 “ ((1...(𝑀 − 1)) ∩ (𝑀...𝑁))) = ∅) |
101 | 95, 100 | eqtr3d 2658 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅) |
102 | | fnun 5997 |
. . . . . . . . . . . . 13
⊢
(((((𝑈 “
(1...(𝑀 − 1)))
× {1}) Fn (𝑈 “
(1...(𝑀 − 1))) ∧
((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁))) ∧ ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅) → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁)))) |
103 | 91, 101, 102 | sylancr 695 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁)))) |
104 | | elfzuz 12338 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈
(ℤ≥‘1)) |
105 | 16, 104 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑀 ∈
(ℤ≥‘1)) |
106 | 57, 105 | eqeltrd 2701 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝑀 − 1) + 1) ∈
(ℤ≥‘1)) |
107 | | peano2zm 11420 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑀 ∈ ℤ → (𝑀 − 1) ∈
ℤ) |
108 | | uzid 11702 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 − 1) ∈ ℤ
→ (𝑀 − 1) ∈
(ℤ≥‘(𝑀 − 1))) |
109 | | peano2uz 11741 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑀 − 1) ∈
(ℤ≥‘(𝑀 − 1)) → ((𝑀 − 1) + 1) ∈
(ℤ≥‘(𝑀 − 1))) |
110 | 40, 107, 108, 109 | 4syl 19 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ((𝑀 − 1) + 1) ∈
(ℤ≥‘(𝑀 − 1))) |
111 | 57, 110 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘(𝑀 − 1))) |
112 | | peano2uz 11741 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑀 ∈
(ℤ≥‘(𝑀 − 1)) → (𝑀 + 1) ∈
(ℤ≥‘(𝑀 − 1))) |
113 | | uzss 11708 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑀 + 1) ∈
(ℤ≥‘(𝑀 − 1)) →
(ℤ≥‘(𝑀 + 1)) ⊆
(ℤ≥‘(𝑀 − 1))) |
114 | 111, 112,
113 | 3syl 18 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 →
(ℤ≥‘(𝑀 + 1)) ⊆
(ℤ≥‘(𝑀 − 1))) |
115 | | elfzuz3 12339 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑀 ∈ (1...(𝑁 − 1)) → (𝑁 − 1) ∈
(ℤ≥‘𝑀)) |
116 | | eluzp1p1 11713 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑁 − 1) ∈
(ℤ≥‘𝑀) → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
117 | 16, 115, 116 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → ((𝑁 − 1) + 1) ∈
(ℤ≥‘(𝑀 + 1))) |
118 | 7, 117 | eqeltrrd 2702 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) |
119 | 114, 118 | sseldd 3604 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) |
120 | | fzsplit2 12366 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝑀 − 1) + 1) ∈
(ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘(𝑀 − 1))) → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁))) |
121 | 106, 119,
120 | syl2anc 693 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁))) |
122 | 57 | oveq1d 6665 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (((𝑀 − 1) + 1)...𝑁) = (𝑀...𝑁)) |
123 | 122 | uneq2d 3767 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((1...(𝑀 − 1)) ∪ (((𝑀 − 1) + 1)...𝑁)) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) |
124 | 121, 123 | eqtrd 2656 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) |
125 | 124 | imaeq2d 5466 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁)))) |
126 | | imaundi 5545 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 “ ((1...(𝑀 − 1)) ∪ (𝑀...𝑁))) = ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) |
127 | 125, 126 | syl6eq 2672 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 “ (1...𝑁)) = ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁)))) |
128 | | f1ofo 6144 |
. . . . . . . . . . . . . . 15
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–onto→(1...𝑁)) |
129 | | foima 6120 |
. . . . . . . . . . . . . . 15
⊢ (𝑈:(1...𝑁)–onto→(1...𝑁) → (𝑈 “ (1...𝑁)) = (1...𝑁)) |
130 | 1, 128, 129 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (1...𝑁)) |
131 | 127, 130 | eqtr3d 2658 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) = (1...𝑁)) |
132 | 131 | fneq2d 5982 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 − 1))) ∪ (𝑈 “ (𝑀...𝑁))) ↔ (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁))) |
133 | 103, 132 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)) |
134 | 133 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})) Fn (1...𝑁)) |
135 | | ovexd 6680 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (1...𝑁) ∈ V) |
136 | | inidm 3822 |
. . . . . . . . . 10
⊢
((1...𝑁) ∩
(1...𝑁)) = (1...𝑁) |
137 | | eqidd 2623 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) = (𝑇‘𝑛)) |
138 | 101 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅) |
139 | | fzss2 12381 |
. . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘(𝑀 + 1)) → (𝑀...(𝑀 + 1)) ⊆ (𝑀...𝑁)) |
140 | | imass2 5501 |
. . . . . . . . . . . . . . 15
⊢ ((𝑀...(𝑀 + 1)) ⊆ (𝑀...𝑁) → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (𝑀...𝑁))) |
141 | 118, 139,
140 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (𝑀...𝑁))) |
142 | 141 | sselda 3603 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (𝑈 “ (𝑀...𝑁))) |
143 | | fvun2 6270 |
. . . . . . . . . . . . . 14
⊢ ((((𝑈 “ (1...(𝑀 − 1))) × {1}) Fn (𝑈 “ (1...(𝑀 − 1))) ∧ ((𝑈 “ (𝑀...𝑁)) × {0}) Fn (𝑈 “ (𝑀...𝑁)) ∧ (((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑀...𝑁)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛)) |
144 | 87, 90, 143 | mp3an12 1414 |
. . . . . . . . . . . . 13
⊢ ((((𝑈 “ (1...(𝑀 − 1))) ∩ (𝑈 “ (𝑀...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (𝑀...𝑁))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛)) |
145 | 138, 142,
144 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛)) |
146 | 88 | fvconst2 6469 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (𝑈 “ (𝑀...𝑁)) → (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛) = 0) |
147 | 142, 146 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (𝑀...𝑁)) × {0})‘𝑛) = 0) |
148 | 145, 147 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = 0) |
149 | 148 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0}))‘𝑛) = 0) |
150 | 84, 134, 135, 135, 136, 137, 149 | ofval 6906 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛) = ((𝑇‘𝑛) + 0)) |
151 | 28, 150 | mpdan 702 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 − 1))) × {1}) ∪ ((𝑈 “ (𝑀...𝑁)) × {0})))‘𝑛) = ((𝑇‘𝑛) + 0)) |
152 | 30 | zcnd 11483 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → (𝑇‘𝑛) ∈ ℂ) |
153 | 152 | addid1d 10236 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇‘𝑛) + 0) = (𝑇‘𝑛)) |
154 | 28, 153 | syldan 487 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇‘𝑛) + 0) = (𝑇‘𝑛)) |
155 | 81, 151, 154 | 3eqtrd 2660 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) = (𝑇‘𝑛)) |
156 | | breq1 4656 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑀 → (𝑦 < 𝑀 ↔ 𝑀 < 𝑀)) |
157 | | oveq1 6657 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑀 → (𝑦 + 1) = (𝑀 + 1)) |
158 | 156, 157 | ifbieq2d 4111 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑀 → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = if(𝑀 < 𝑀, 𝑦, (𝑀 + 1))) |
159 | 41 | ltnrd 10171 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ¬ 𝑀 < 𝑀) |
160 | 159 | iffalsed 4097 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → if(𝑀 < 𝑀, 𝑦, (𝑀 + 1)) = (𝑀 + 1)) |
161 | 158, 160 | sylan9eqr 2678 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) = (𝑀 + 1)) |
162 | 161 | csbeq1d 3540 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = ⦋(𝑀 + 1) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})))) |
163 | | ovex 6678 |
. . . . . . . . . . . . 13
⊢ (𝑀 + 1) ∈ V |
164 | | oveq2 6658 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑀 + 1) → (1...𝑗) = (1...(𝑀 + 1))) |
165 | 164 | imaeq2d 5466 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑀 + 1) → (𝑈 “ (1...𝑗)) = (𝑈 “ (1...(𝑀 + 1)))) |
166 | 165 | xpeq1d 5138 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑀 + 1) → ((𝑈 “ (1...𝑗)) × {1}) = ((𝑈 “ (1...(𝑀 + 1))) × {1})) |
167 | | oveq1 6657 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = (𝑀 + 1) → (𝑗 + 1) = ((𝑀 + 1) + 1)) |
168 | 167 | oveq1d 6665 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = (𝑀 + 1) → ((𝑗 + 1)...𝑁) = (((𝑀 + 1) + 1)...𝑁)) |
169 | 168 | imaeq2d 5466 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = (𝑀 + 1) → (𝑈 “ ((𝑗 + 1)...𝑁)) = (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) |
170 | 169 | xpeq1d 5138 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = (𝑀 + 1) → ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}) = ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) |
171 | 166, 170 | uneq12d 3768 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = (𝑀 + 1) → (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0})) = (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))) |
172 | 171 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ (𝑗 = (𝑀 + 1) → (𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))) |
173 | 163, 172 | csbie 3559 |
. . . . . . . . . . . 12
⊢
⦋(𝑀 +
1) / 𝑗⦌(𝑇 ∘𝑓 +
(((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))) |
174 | 162, 173 | syl6eq 2672 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑦 = 𝑀) → ⦋if(𝑦 < 𝑀, 𝑦, (𝑦 + 1)) / 𝑗⦌(𝑇 ∘𝑓 + (((𝑈 “ (1...𝑗)) × {1}) ∪ ((𝑈 “ ((𝑗 + 1)...𝑁)) × {0}))) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))) |
175 | | 1eluzge0 11732 |
. . . . . . . . . . . . 13
⊢ 1 ∈
(ℤ≥‘0) |
176 | | fzss1 12380 |
. . . . . . . . . . . . 13
⊢ (1 ∈
(ℤ≥‘0) → (1...(𝑁 − 1)) ⊆ (0...(𝑁 − 1))) |
177 | 175, 176 | ax-mp 5 |
. . . . . . . . . . . 12
⊢
(1...(𝑁 − 1))
⊆ (0...(𝑁 −
1)) |
178 | 177, 16 | sseldi 3601 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ (0...(𝑁 − 1))) |
179 | | ovexd 6680 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))) ∈ V) |
180 | 35, 174, 178, 179 | fvmptd 6288 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐹‘𝑀) = (𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))) |
181 | 180 | fveq1d 6193 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐹‘𝑀)‘𝑛) = ((𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛)) |
182 | 181 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘𝑀)‘𝑛) = ((𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛)) |
183 | | fnconstg 6093 |
. . . . . . . . . . . . . . 15
⊢ (1 ∈
V → ((𝑈 “
(1...(𝑀 + 1))) × {1})
Fn (𝑈 “ (1...(𝑀 + 1)))) |
184 | 85, 183 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) |
185 | | fnconstg 6093 |
. . . . . . . . . . . . . . 15
⊢ (0 ∈
V → ((𝑈 “
(((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) |
186 | 88, 185 | ax-mp 5 |
. . . . . . . . . . . . . 14
⊢ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)) |
187 | 184, 186 | pm3.2i 471 |
. . . . . . . . . . . . 13
⊢ (((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) |
188 | | imain 5974 |
. . . . . . . . . . . . . . . 16
⊢ (Fun
◡𝑈 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))) |
189 | 1, 93, 188 | 3syl 18 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))) |
190 | | peano2re 10209 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑀 ∈ ℝ → (𝑀 + 1) ∈
ℝ) |
191 | 41, 190 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝑀 + 1) ∈ ℝ) |
192 | 191 | ltp1d 10954 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑀 + 1) < ((𝑀 + 1) + 1)) |
193 | | fzdisj 12368 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 + 1) < ((𝑀 + 1) + 1) → ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁)) = ∅) |
194 | 192, 193 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁)) = ∅) |
195 | 194 | imaeq2d 5466 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑈 “ ((1...(𝑀 + 1)) ∩ (((𝑀 + 1) + 1)...𝑁))) = (𝑈 “ ∅)) |
196 | 189, 195 | eqtr3d 2658 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = (𝑈 “ ∅)) |
197 | 196, 99 | syl6eq 2672 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅) |
198 | | fnun 5997 |
. . . . . . . . . . . . 13
⊢
(((((𝑈 “
(1...(𝑀 + 1))) × {1})
Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) ∧ ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅) → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))) |
199 | 187, 197,
198 | sylancr 695 |
. . . . . . . . . . . 12
⊢ (𝜑 → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))) |
200 | | fzsplit 12367 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑀 + 1) ∈ (1...𝑁) → (1...𝑁) = ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) |
201 | 22, 200 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (1...𝑁) = ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) |
202 | 201 | imaeq2d 5466 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑈 “ (1...𝑁)) = (𝑈 “ ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁)))) |
203 | | imaundi 5545 |
. . . . . . . . . . . . . . 15
⊢ (𝑈 “ ((1...(𝑀 + 1)) ∪ (((𝑀 + 1) + 1)...𝑁))) = ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) |
204 | 202, 203 | syl6eq 2672 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 “ (1...𝑁)) = ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁)))) |
205 | 204, 130 | eqtr3d 2658 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = (1...𝑁)) |
206 | 205 | fneq2d 5982 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn ((𝑈 “ (1...(𝑀 + 1))) ∪ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) ↔ (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁))) |
207 | 199, 206 | mpbid 222 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
208 | 207 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})) Fn (1...𝑁)) |
209 | 197 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅) |
210 | | fzss1 12380 |
. . . . . . . . . . . . . . 15
⊢ (𝑀 ∈
(ℤ≥‘1) → (𝑀...(𝑀 + 1)) ⊆ (1...(𝑀 + 1))) |
211 | | imass2 5501 |
. . . . . . . . . . . . . . 15
⊢ ((𝑀...(𝑀 + 1)) ⊆ (1...(𝑀 + 1)) → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (1...(𝑀 + 1)))) |
212 | 105, 210,
211 | 3syl 18 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) ⊆ (𝑈 “ (1...(𝑀 + 1)))) |
213 | 212 | sselda 3603 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1)))) |
214 | | fvun1 6269 |
. . . . . . . . . . . . . 14
⊢ ((((𝑈 “ (1...(𝑀 + 1))) × {1}) Fn (𝑈 “ (1...(𝑀 + 1))) ∧ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}) Fn (𝑈 “ (((𝑀 + 1) + 1)...𝑁)) ∧ (((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛)) |
215 | 184, 186,
214 | mp3an12 1414 |
. . . . . . . . . . . . 13
⊢ ((((𝑈 “ (1...(𝑀 + 1))) ∩ (𝑈 “ (((𝑀 + 1) + 1)...𝑁))) = ∅ ∧ 𝑛 ∈ (𝑈 “ (1...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛)) |
216 | 209, 213,
215 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛)) |
217 | 85 | fvconst2 6469 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ (𝑈 “ (1...(𝑀 + 1))) → (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛) = 1) |
218 | 213, 217 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → (((𝑈 “ (1...(𝑀 + 1))) × {1})‘𝑛) = 1) |
219 | 216, 218 | eqtrd 2656 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1) |
220 | 219 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0}))‘𝑛) = 1) |
221 | 84, 208, 135, 135, 136, 137, 220 | ofval 6906 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) ∧ 𝑛 ∈ (1...𝑁)) → ((𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇‘𝑛) + 1)) |
222 | 28, 221 | mpdan 702 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝑇 ∘𝑓 + (((𝑈 “ (1...(𝑀 + 1))) × {1}) ∪ ((𝑈 “ (((𝑀 + 1) + 1)...𝑁)) × {0})))‘𝑛) = ((𝑇‘𝑛) + 1)) |
223 | 182, 222 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘𝑀)‘𝑛) = ((𝑇‘𝑛) + 1)) |
224 | 34, 155, 223 | 3netr4d 2871 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))) → ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) |
225 | 224 | ralrimiva 2966 |
. . . . 5
⊢ (𝜑 → ∀𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) |
226 | | fzpr 12396 |
. . . . . . . . 9
⊢ (𝑀 ∈ ℤ → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) |
227 | 16, 39, 226 | 3syl 18 |
. . . . . . . 8
⊢ (𝜑 → (𝑀...(𝑀 + 1)) = {𝑀, (𝑀 + 1)}) |
228 | 227 | imaeq2d 5466 |
. . . . . . 7
⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) = (𝑈 “ {𝑀, (𝑀 + 1)})) |
229 | | f1ofn 6138 |
. . . . . . . . 9
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈 Fn (1...𝑁)) |
230 | 1, 229 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑈 Fn (1...𝑁)) |
231 | | fnimapr 6262 |
. . . . . . . 8
⊢ ((𝑈 Fn (1...𝑁) ∧ 𝑀 ∈ (1...𝑁) ∧ (𝑀 + 1) ∈ (1...𝑁)) → (𝑈 “ {𝑀, (𝑀 + 1)}) = {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))}) |
232 | 230, 17, 22, 231 | syl3anc 1326 |
. . . . . . 7
⊢ (𝜑 → (𝑈 “ {𝑀, (𝑀 + 1)}) = {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))}) |
233 | 228, 232 | eqtrd 2656 |
. . . . . 6
⊢ (𝜑 → (𝑈 “ (𝑀...(𝑀 + 1))) = {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))}) |
234 | 233 | raleqdv 3144 |
. . . . 5
⊢ (𝜑 → (∀𝑛 ∈ (𝑈 “ (𝑀...(𝑀 + 1)))((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ∀𝑛 ∈ {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛))) |
235 | 225, 234 | mpbid 222 |
. . . 4
⊢ (𝜑 → ∀𝑛 ∈ {(𝑈‘𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) |
236 | | fvex 6201 |
. . . . 5
⊢ (𝑈‘𝑀) ∈ V |
237 | | fvex 6201 |
. . . . 5
⊢ (𝑈‘(𝑀 + 1)) ∈ V |
238 | | fveq2 6191 |
. . . . . 6
⊢ (𝑛 = (𝑈‘𝑀) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀))) |
239 | | fveq2 6191 |
. . . . . 6
⊢ (𝑛 = (𝑈‘𝑀) → ((𝐹‘𝑀)‘𝑛) = ((𝐹‘𝑀)‘(𝑈‘𝑀))) |
240 | 238, 239 | neeq12d 2855 |
. . . . 5
⊢ (𝑛 = (𝑈‘𝑀) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)))) |
241 | | fveq2 6191 |
. . . . . 6
⊢ (𝑛 = (𝑈‘(𝑀 + 1)) → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1)))) |
242 | | fveq2 6191 |
. . . . . 6
⊢ (𝑛 = (𝑈‘(𝑀 + 1)) → ((𝐹‘𝑀)‘𝑛) = ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))) |
243 | 241, 242 | neeq12d 2855 |
. . . . 5
⊢ (𝑛 = (𝑈‘(𝑀 + 1)) → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1))))) |
244 | 236, 237,
240, 243 | ralpr 4238 |
. . . 4
⊢
(∀𝑛 ∈
{(𝑈‘𝑀), (𝑈‘(𝑀 + 1))} ((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1))))) |
245 | 235, 244 | sylib 208 |
. . 3
⊢ (𝜑 → (((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1))))) |
246 | 41 | ltp1d 10954 |
. . . . 5
⊢ (𝜑 → 𝑀 < (𝑀 + 1)) |
247 | 41, 246 | ltned 10173 |
. . . 4
⊢ (𝜑 → 𝑀 ≠ (𝑀 + 1)) |
248 | | f1of1 6136 |
. . . . . . 7
⊢ (𝑈:(1...𝑁)–1-1-onto→(1...𝑁) → 𝑈:(1...𝑁)–1-1→(1...𝑁)) |
249 | 1, 248 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑈:(1...𝑁)–1-1→(1...𝑁)) |
250 | | f1veqaeq 6514 |
. . . . . 6
⊢ ((𝑈:(1...𝑁)–1-1→(1...𝑁) ∧ (𝑀 ∈ (1...𝑁) ∧ (𝑀 + 1) ∈ (1...𝑁))) → ((𝑈‘𝑀) = (𝑈‘(𝑀 + 1)) → 𝑀 = (𝑀 + 1))) |
251 | 249, 17, 22, 250 | syl12anc 1324 |
. . . . 5
⊢ (𝜑 → ((𝑈‘𝑀) = (𝑈‘(𝑀 + 1)) → 𝑀 = (𝑀 + 1))) |
252 | 251 | necon3d 2815 |
. . . 4
⊢ (𝜑 → (𝑀 ≠ (𝑀 + 1) → (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1)))) |
253 | 247, 252 | mpd 15 |
. . 3
⊢ (𝜑 → (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1))) |
254 | 240 | anbi1d 741 |
. . . . 5
⊢ (𝑛 = (𝑈‘𝑀) → ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)))) |
255 | | neeq1 2856 |
. . . . 5
⊢ (𝑛 = (𝑈‘𝑀) → (𝑛 ≠ 𝑚 ↔ (𝑈‘𝑀) ≠ 𝑚)) |
256 | 254, 255 | anbi12d 747 |
. . . 4
⊢ (𝑛 = (𝑈‘𝑀) → (((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ (𝑈‘𝑀) ≠ 𝑚))) |
257 | | fveq2 6191 |
. . . . . . 7
⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝐹‘(𝑀 − 1))‘𝑚) = ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1)))) |
258 | | fveq2 6191 |
. . . . . . 7
⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝐹‘𝑀)‘𝑚) = ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))) |
259 | 257, 258 | neeq12d 2855 |
. . . . . 6
⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → (((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚) ↔ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1))))) |
260 | 259 | anbi2d 740 |
. . . . 5
⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → ((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ↔ (((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))))) |
261 | | neeq2 2857 |
. . . . 5
⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → ((𝑈‘𝑀) ≠ 𝑚 ↔ (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1)))) |
262 | 260, 261 | anbi12d 747 |
. . . 4
⊢ (𝑚 = (𝑈‘(𝑀 + 1)) → (((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ (𝑈‘𝑀) ≠ 𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))) ∧ (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1))))) |
263 | 256, 262 | rspc2ev 3324 |
. . 3
⊢ (((𝑈‘𝑀) ∈ (1...𝑁) ∧ (𝑈‘(𝑀 + 1)) ∈ (1...𝑁) ∧ ((((𝐹‘(𝑀 − 1))‘(𝑈‘𝑀)) ≠ ((𝐹‘𝑀)‘(𝑈‘𝑀)) ∧ ((𝐹‘(𝑀 − 1))‘(𝑈‘(𝑀 + 1))) ≠ ((𝐹‘𝑀)‘(𝑈‘(𝑀 + 1)))) ∧ (𝑈‘𝑀) ≠ (𝑈‘(𝑀 + 1)))) → ∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚)) |
264 | 18, 23, 245, 253, 263 | syl112anc 1330 |
. 2
⊢ (𝜑 → ∃𝑛 ∈ (1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚)) |
265 | | dfrex2 2996 |
. . 3
⊢
(∃𝑛 ∈
(1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ¬ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚)) |
266 | | fveq2 6191 |
. . . . . 6
⊢ (𝑛 = 𝑚 → ((𝐹‘(𝑀 − 1))‘𝑛) = ((𝐹‘(𝑀 − 1))‘𝑚)) |
267 | | fveq2 6191 |
. . . . . 6
⊢ (𝑛 = 𝑚 → ((𝐹‘𝑀)‘𝑛) = ((𝐹‘𝑀)‘𝑚)) |
268 | 266, 267 | neeq12d 2855 |
. . . . 5
⊢ (𝑛 = 𝑚 → (((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚))) |
269 | 268 | rmo4 3399 |
. . . 4
⊢
(∃*𝑛 ∈
(1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁)∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚)) |
270 | | dfral2 2994 |
. . . . . 6
⊢
(∀𝑚 ∈
(1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ¬ ∃𝑚 ∈ (1...𝑁) ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚)) |
271 | | df-ne 2795 |
. . . . . . . . 9
⊢ (𝑛 ≠ 𝑚 ↔ ¬ 𝑛 = 𝑚) |
272 | 271 | anbi2i 730 |
. . . . . . . 8
⊢
(((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ ¬ 𝑛 = 𝑚)) |
273 | | annim 441 |
. . . . . . . 8
⊢
(((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ ¬ 𝑛 = 𝑚) ↔ ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚)) |
274 | 272, 273 | bitri 264 |
. . . . . . 7
⊢
(((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚)) |
275 | 274 | rexbii 3041 |
. . . . . 6
⊢
(∃𝑚 ∈
(1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ∃𝑚 ∈ (1...𝑁) ¬ ((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚)) |
276 | 270, 275 | xchbinxr 325 |
. . . . 5
⊢
(∀𝑚 ∈
(1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚)) |
277 | 276 | ralbii 2980 |
. . . 4
⊢
(∀𝑛 ∈
(1...𝑁)∀𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) → 𝑛 = 𝑚) ↔ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚)) |
278 | 269, 277 | bitri 264 |
. . 3
⊢
(∃*𝑛 ∈
(1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ↔ ∀𝑛 ∈ (1...𝑁) ¬ ∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚)) |
279 | 265, 278 | xchbinxr 325 |
. 2
⊢
(∃𝑛 ∈
(1...𝑁)∃𝑚 ∈ (1...𝑁)((((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛) ∧ ((𝐹‘(𝑀 − 1))‘𝑚) ≠ ((𝐹‘𝑀)‘𝑚)) ∧ 𝑛 ≠ 𝑚) ↔ ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) |
280 | 264, 279 | sylib 208 |
1
⊢ (𝜑 → ¬ ∃*𝑛 ∈ (1...𝑁)((𝐹‘(𝑀 − 1))‘𝑛) ≠ ((𝐹‘𝑀)‘𝑛)) |