Proof of Theorem seqf1oglem1
Step | Hyp | Ref
| Expression |
1 | | seqf1olem.7 |
. 2
⊢ 𝐽 = (𝑘 ∈ (𝑀...𝑁) ↦ (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) |
2 | | seqf1olem.5 |
. . . . 5
⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
3 | | f1of 5492 |
. . . . 5
⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
4 | 2, 3 | syl 14 |
. . . 4
⊢ (𝜑 → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
5 | 4 | adantr 276 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
6 | | fzelp1 10130 |
. . . . 5
⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
7 | 6 | adantl 277 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
8 | | fzp1elp1 10131 |
. . . . 5
⊢ (𝑘 ∈ (𝑀...𝑁) → (𝑘 + 1) ∈ (𝑀...(𝑁 + 1))) |
9 | 8 | adantl 277 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 + 1) ∈ (𝑀...(𝑁 + 1))) |
10 | | elfzelz 10081 |
. . . . 5
⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℤ) |
11 | | seqf1olem.8 |
. . . . . . 7
⊢ 𝐾 = (◡𝐹‘(𝑁 + 1)) |
12 | | f1ocnv 5505 |
. . . . . . . . . . 11
⊢ (𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
13 | 2, 12 | syl 14 |
. . . . . . . . . 10
⊢ (𝜑 → ◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
14 | | f1of1 5491 |
. . . . . . . . . 10
⊢ (◡𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) → ◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1))) |
15 | 13, 14 | syl 14 |
. . . . . . . . 9
⊢ (𝜑 → ◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1))) |
16 | | f1f 5451 |
. . . . . . . . 9
⊢ (◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)) → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
17 | 15, 16 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
18 | | seqf1o.4 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
19 | | peano2uz 9638 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
20 | 18, 19 | syl 14 |
. . . . . . . . 9
⊢ (𝜑 → (𝑁 + 1) ∈
(ℤ≥‘𝑀)) |
21 | | eluzfz2 10088 |
. . . . . . . . 9
⊢ ((𝑁 + 1) ∈
(ℤ≥‘𝑀) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
22 | 20, 21 | syl 14 |
. . . . . . . 8
⊢ (𝜑 → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
23 | 17, 22 | ffvelcdmd 5686 |
. . . . . . 7
⊢ (𝜑 → (◡𝐹‘(𝑁 + 1)) ∈ (𝑀...(𝑁 + 1))) |
24 | 11, 23 | eqeltrid 2280 |
. . . . . 6
⊢ (𝜑 → 𝐾 ∈ (𝑀...(𝑁 + 1))) |
25 | 24 | elfzelzd 10082 |
. . . . 5
⊢ (𝜑 → 𝐾 ∈ ℤ) |
26 | | zdclt 9384 |
. . . . 5
⊢ ((𝑘 ∈ ℤ ∧ 𝐾 ∈ ℤ) →
DECID 𝑘 <
𝐾) |
27 | 10, 25, 26 | syl2anr 290 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → DECID 𝑘 < 𝐾) |
28 | 7, 9, 27 | ifcldcd 3593 |
. . 3
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) ∈ (𝑀...(𝑁 + 1))) |
29 | 5, 28 | ffvelcdmd 5686 |
. 2
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ∈ (𝑀...(𝑁 + 1))) |
30 | 17 | adantr 276 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
31 | | fzelp1 10130 |
. . . . . 6
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
32 | 31 | adantl 277 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
33 | 30, 32 | ffvelcdmd 5686 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1))) |
34 | 33 | elfzelzd 10082 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (◡𝐹‘𝑥) ∈ ℤ) |
35 | | peano2zm 9345 |
. . . 4
⊢ ((◡𝐹‘𝑥) ∈ ℤ → ((◡𝐹‘𝑥) − 1) ∈ ℤ) |
36 | 34, 35 | syl 14 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((◡𝐹‘𝑥) − 1) ∈ ℤ) |
37 | 25 | adantr 276 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝐾 ∈ ℤ) |
38 | | zdclt 9384 |
. . . 4
⊢ (((◡𝐹‘𝑥) ∈ ℤ ∧ 𝐾 ∈ ℤ) → DECID
(◡𝐹‘𝑥) < 𝐾) |
39 | 34, 37, 38 | syl2anc 411 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → DECID (◡𝐹‘𝑥) < 𝐾) |
40 | 34, 36, 39 | ifcldcd 3593 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) ∈
ℤ) |
41 | | iftrue 3562 |
. . . . . . . . 9
⊢ (𝑘 < 𝐾 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = 𝑘) |
42 | 41 | fveq2d 5550 |
. . . . . . . 8
⊢ (𝑘 < 𝐾 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘𝑘)) |
43 | 42 | eqeq2d 2205 |
. . . . . . 7
⊢ (𝑘 < 𝐾 → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘𝑘))) |
44 | 43 | adantl 277 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘𝑘))) |
45 | | simprr 531 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑥 = (𝐹‘𝑘)) |
46 | 5, 7 | ffvelcdmd 5686 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1))) |
47 | 46 | elfzelzd 10082 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℤ) |
48 | | eluzel2 9587 |
. . . . . . . . . . . . . 14
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
49 | 18, 48 | syl 14 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑀 ∈ ℤ) |
50 | 49 | adantr 276 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) |
51 | | uzssz 9602 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
52 | 51, 18 | sselid 3177 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑁 ∈ ℤ) |
53 | 52 | adantr 276 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℤ) |
54 | | fzdcel 10096 |
. . . . . . . . . . . 12
⊢ (((𝐹‘𝑘) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID
(𝐹‘𝑘) ∈ (𝑀...𝑁)) |
55 | 47, 50, 53, 54 | syl3anc 1249 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → DECID (𝐹‘𝑘) ∈ (𝑀...𝑁)) |
56 | 55 | adantr 276 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → DECID (𝐹‘𝑘) ∈ (𝑀...𝑁)) |
57 | 10 | zred 9429 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ ℝ) |
58 | 57 | ad2antlr 489 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ ℝ) |
59 | | simprl 529 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 < 𝐾) |
60 | 58, 59 | gtned 8122 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐾 ≠ 𝑘) |
61 | 4 | ad2antrr 488 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
62 | | fzssp1 10123 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑀...𝑁) ⊆ (𝑀...(𝑁 + 1)) |
63 | | simplr 528 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ (𝑀...𝑁)) |
64 | 62, 63 | sselid 3177 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
65 | 61, 64 | ffvelcdmd 5686 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1))) |
66 | | elfzp1 10128 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1)))) |
67 | 18, 66 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1)))) |
68 | 67 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1)))) |
69 | 65, 68 | mpbid 147 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) ∈ (𝑀...𝑁) ∨ (𝐹‘𝑘) = (𝑁 + 1))) |
70 | 69 | ord 725 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (¬ (𝐹‘𝑘) ∈ (𝑀...𝑁) → (𝐹‘𝑘) = (𝑁 + 1))) |
71 | 2 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
72 | | f1ocnvfv 5814 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑘) = (𝑁 + 1) → (◡𝐹‘(𝑁 + 1)) = 𝑘)) |
73 | 71, 64, 72 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = (𝑁 + 1) → (◡𝐹‘(𝑁 + 1)) = 𝑘)) |
74 | 11 | eqeq1i 2201 |
. . . . . . . . . . . . . 14
⊢ (𝐾 = 𝑘 ↔ (◡𝐹‘(𝑁 + 1)) = 𝑘) |
75 | 73, 74 | imbitrrdi 162 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = (𝑁 + 1) → 𝐾 = 𝑘)) |
76 | 70, 75 | syld 45 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (¬ (𝐹‘𝑘) ∈ (𝑀...𝑁) → 𝐾 = 𝑘)) |
77 | 76 | a1d 22 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (DECID (𝐹‘𝑘) ∈ (𝑀...𝑁) → (¬ (𝐹‘𝑘) ∈ (𝑀...𝑁) → 𝐾 = 𝑘))) |
78 | 77 | necon1addc 2440 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (DECID (𝐹‘𝑘) ∈ (𝑀...𝑁) → (𝐾 ≠ 𝑘 → (𝐹‘𝑘) ∈ (𝑀...𝑁)))) |
79 | 56, 60, 78 | mp2d 47 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) ∈ (𝑀...𝑁)) |
80 | 45, 79 | eqeltrd 2270 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑥 ∈ (𝑀...𝑁)) |
81 | 45 | eqcomd 2199 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝐹‘𝑘) = 𝑥) |
82 | | f1ocnvfv 5814 |
. . . . . . . . . . . . 13
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑘 ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘𝑘) = 𝑥 → (◡𝐹‘𝑥) = 𝑘)) |
83 | 71, 64, 82 | syl2anc 411 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → ((𝐹‘𝑘) = 𝑥 → (◡𝐹‘𝑥) = 𝑘)) |
84 | 81, 83 | mpd 13 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (◡𝐹‘𝑥) = 𝑘) |
85 | 84, 59 | eqbrtrd 4051 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (◡𝐹‘𝑥) < 𝐾) |
86 | | iftrue 3562 |
. . . . . . . . . 10
⊢ ((◡𝐹‘𝑥) < 𝐾 → if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) = (◡𝐹‘𝑥)) |
87 | 85, 86 | syl 14 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) = (◡𝐹‘𝑥)) |
88 | 87, 84 | eqtr2d 2227 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))) |
89 | 80, 88 | jca 306 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘𝑘))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)))) |
90 | 89 | expr 375 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ 𝑘 < 𝐾) → (𝑥 = (𝐹‘𝑘) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))))) |
91 | 44, 90 | sylbid 150 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))))) |
92 | | iffalse 3565 |
. . . . . . . . 9
⊢ (¬
𝑘 < 𝐾 → if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)) = (𝑘 + 1)) |
93 | 92 | fveq2d 5550 |
. . . . . . . 8
⊢ (¬
𝑘 < 𝐾 → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘(𝑘 + 1))) |
94 | 93 | eqeq2d 2205 |
. . . . . . 7
⊢ (¬
𝑘 < 𝐾 → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘(𝑘 + 1)))) |
95 | 94 | adantl 277 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ ¬ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) ↔ 𝑥 = (𝐹‘(𝑘 + 1)))) |
96 | | simprr 531 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑥 = (𝐹‘(𝑘 + 1))) |
97 | 5, 9 | ffvelcdmd 5686 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1))) |
98 | 97 | elfzelzd 10082 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘(𝑘 + 1)) ∈ ℤ) |
99 | | fzdcel 10096 |
. . . . . . . . . . . 12
⊢ (((𝐹‘(𝑘 + 1)) ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID
(𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁)) |
100 | 98, 50, 53, 99 | syl3anc 1249 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → DECID (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁)) |
101 | 100 | adantr 276 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → DECID (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁)) |
102 | 25 | zred 9429 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ∈ ℝ) |
103 | 102 | ad2antrr 488 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ∈ ℝ) |
104 | 57 | ad2antlr 489 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 ∈ ℝ) |
105 | | peano2re 8145 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈
ℝ) |
106 | 104, 105 | syl 14 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑘 + 1) ∈ ℝ) |
107 | | simprl 529 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ¬ 𝑘 < 𝐾) |
108 | 103, 104,
107 | nltled 8130 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ≤ 𝑘) |
109 | 104 | ltp1d 8939 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 < (𝑘 + 1)) |
110 | 103, 104,
106, 108, 109 | lelttrd 8134 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 < (𝑘 + 1)) |
111 | 103, 110 | ltned 8123 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐾 ≠ (𝑘 + 1)) |
112 | 4 | ad2antrr 488 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
113 | 8 | ad2antlr 489 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑘 + 1) ∈ (𝑀...(𝑁 + 1))) |
114 | 112, 113 | ffvelcdmd 5686 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1))) |
115 | | elfzp1 10128 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1)))) |
116 | 18, 115 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1)))) |
117 | 116 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...(𝑁 + 1)) ↔ ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1)))) |
118 | 114, 117 | mpbid 147 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) ∨ (𝐹‘(𝑘 + 1)) = (𝑁 + 1))) |
119 | 118 | ord 725 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (¬ (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → (𝐹‘(𝑘 + 1)) = (𝑁 + 1))) |
120 | 2 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
121 | | f1ocnvfv 5814 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ (𝑘 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘(𝑘 + 1)) = (𝑁 + 1) → (◡𝐹‘(𝑁 + 1)) = (𝑘 + 1))) |
122 | 120, 113,
121 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = (𝑁 + 1) → (◡𝐹‘(𝑁 + 1)) = (𝑘 + 1))) |
123 | 11 | eqeq1i 2201 |
. . . . . . . . . . . . . 14
⊢ (𝐾 = (𝑘 + 1) ↔ (◡𝐹‘(𝑁 + 1)) = (𝑘 + 1)) |
124 | 122, 123 | imbitrrdi 162 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = (𝑁 + 1) → 𝐾 = (𝑘 + 1))) |
125 | 119, 124 | syld 45 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (¬ (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → 𝐾 = (𝑘 + 1))) |
126 | 125 | a1d 22 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (DECID (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → (¬ (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → 𝐾 = (𝑘 + 1)))) |
127 | 126 | necon1addc 2440 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (DECID (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁) → (𝐾 ≠ (𝑘 + 1) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁)))) |
128 | 101, 111,
127 | mp2d 47 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) ∈ (𝑀...𝑁)) |
129 | 96, 128 | eqeltrd 2270 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑥 ∈ (𝑀...𝑁)) |
130 | 96 | eqcomd 2199 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝐹‘(𝑘 + 1)) = 𝑥) |
131 | | f1ocnvfv 5814 |
. . . . . . . . . . . . . . 15
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ (𝑘 + 1) ∈ (𝑀...(𝑁 + 1))) → ((𝐹‘(𝑘 + 1)) = 𝑥 → (◡𝐹‘𝑥) = (𝑘 + 1))) |
132 | 120, 113,
131 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝐹‘(𝑘 + 1)) = 𝑥 → (◡𝐹‘𝑥) = (𝑘 + 1))) |
133 | 130, 132 | mpd 13 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (◡𝐹‘𝑥) = (𝑘 + 1)) |
134 | 133 | breq1d 4039 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((◡𝐹‘𝑥) < 𝐾 ↔ (𝑘 + 1) < 𝐾)) |
135 | | lttr 8083 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℝ ∧ (𝑘 + 1) ∈ ℝ ∧ 𝐾 ∈ ℝ) → ((𝑘 < (𝑘 + 1) ∧ (𝑘 + 1) < 𝐾) → 𝑘 < 𝐾)) |
136 | 104, 106,
103, 135 | syl3anc 1249 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 < (𝑘 + 1) ∧ (𝑘 + 1) < 𝐾) → 𝑘 < 𝐾)) |
137 | 109, 136 | mpand 429 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 + 1) < 𝐾 → 𝑘 < 𝐾)) |
138 | 134, 137 | sylbid 150 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((◡𝐹‘𝑥) < 𝐾 → 𝑘 < 𝐾)) |
139 | 107, 138 | mtod 664 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ¬ (◡𝐹‘𝑥) < 𝐾) |
140 | | iffalse 3565 |
. . . . . . . . . 10
⊢ (¬
(◡𝐹‘𝑥) < 𝐾 → if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) = ((◡𝐹‘𝑥) − 1)) |
141 | 139, 140 | syl 14 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) = ((◡𝐹‘𝑥) − 1)) |
142 | 133 | oveq1d 5925 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((◡𝐹‘𝑥) − 1) = ((𝑘 + 1) − 1)) |
143 | 104 | recnd 8038 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 ∈ ℂ) |
144 | | ax-1cn 7955 |
. . . . . . . . . 10
⊢ 1 ∈
ℂ |
145 | | pncan 8215 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 + 1)
− 1) = 𝑘) |
146 | 143, 144,
145 | sylancl 413 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → ((𝑘 + 1) − 1) = 𝑘) |
147 | 141, 142,
146 | 3eqtrrd 2231 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))) |
148 | 129, 147 | jca 306 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ (¬ 𝑘 < 𝐾 ∧ 𝑥 = (𝐹‘(𝑘 + 1)))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)))) |
149 | 148 | expr 375 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ ¬ 𝑘 < 𝐾) → (𝑥 = (𝐹‘(𝑘 + 1)) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))))) |
150 | 95, 149 | sylbid 150 |
. . . . 5
⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) ∧ ¬ 𝑘 < 𝐾) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))))) |
151 | | exmiddc 837 |
. . . . . 6
⊢
(DECID 𝑘 < 𝐾 → (𝑘 < 𝐾 ∨ ¬ 𝑘 < 𝐾)) |
152 | 27, 151 | syl 14 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑘 < 𝐾 ∨ ¬ 𝑘 < 𝐾)) |
153 | 91, 150, 152 | mpjaodan 799 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))))) |
154 | 153 | expimpd 363 |
. . 3
⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) → (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))))) |
155 | 86 | eqeq2d 2205 |
. . . . . . 7
⊢ ((◡𝐹‘𝑥) < 𝐾 → (𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) ↔ 𝑘 = (◡𝐹‘𝑥))) |
156 | 155 | adantl 277 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) ↔ 𝑘 = (◡𝐹‘𝑥))) |
157 | 49 | ad2antrr 488 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑀 ∈ ℤ) |
158 | | eluzelz 9591 |
. . . . . . . . . . 11
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ ℤ) |
159 | 18, 158 | syl 14 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℤ) |
160 | 159 | ad2antrr 488 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑁 ∈ ℤ) |
161 | | simprr 531 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 = (◡𝐹‘𝑥)) |
162 | 17 | ad2antrr 488 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
163 | | simplr 528 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑥 ∈ (𝑀...𝑁)) |
164 | 62, 163 | sselid 3177 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
165 | 162, 164 | ffvelcdmd 5686 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1))) |
166 | 161, 165 | eqeltrd 2270 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 ∈ (𝑀...(𝑁 + 1))) |
167 | 166 | elfzelzd 10082 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 ∈ ℤ) |
168 | | elfzle1 10083 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (𝑀...(𝑁 + 1)) → 𝑀 ≤ 𝑘) |
169 | 166, 168 | syl 14 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑀 ≤ 𝑘) |
170 | 167 | zred 9429 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 ∈ ℝ) |
171 | 102 | ad2antrr 488 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝐾 ∈ ℝ) |
172 | 159 | peano2zd 9432 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑁 + 1) ∈ ℤ) |
173 | 172 | zred 9429 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑁 + 1) ∈ ℝ) |
174 | 173 | ad2antrr 488 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (𝑁 + 1) ∈ ℝ) |
175 | | simprl 529 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (◡𝐹‘𝑥) < 𝐾) |
176 | 161, 175 | eqbrtrd 4051 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 < 𝐾) |
177 | | elfzle2 10084 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝐾 ≤ (𝑁 + 1)) |
178 | 24, 177 | syl 14 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐾 ≤ (𝑁 + 1)) |
179 | 178 | ad2antrr 488 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝐾 ≤ (𝑁 + 1)) |
180 | 170, 171,
174, 176, 179 | ltletrd 8432 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 < (𝑁 + 1)) |
181 | | zleltp1 9362 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1))) |
182 | 167, 160,
181 | syl2anc 411 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (𝑘 ≤ 𝑁 ↔ 𝑘 < (𝑁 + 1))) |
183 | 180, 182 | mpbird 167 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 ≤ 𝑁) |
184 | 157, 160,
167, 169, 183 | elfzd 10072 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑘 ∈ (𝑀...𝑁)) |
185 | 176, 42 | syl 14 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘𝑘)) |
186 | 161 | fveq2d 5550 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (𝐹‘𝑘) = (𝐹‘(◡𝐹‘𝑥))) |
187 | 2 | ad2antrr 488 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
188 | | f1ocnvfv2 5813 |
. . . . . . . . . 10
⊢ ((𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1))) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) |
189 | 187, 164,
188 | syl2anc 411 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) |
190 | 185, 186,
189 | 3eqtrrd 2231 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) |
191 | 184, 190 | jca 306 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ((◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = (◡𝐹‘𝑥))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))) |
192 | 191 | expr 375 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (◡𝐹‘𝑥) < 𝐾) → (𝑘 = (◡𝐹‘𝑥) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))) |
193 | 156, 192 | sylbid 150 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))) |
194 | 140 | eqeq2d 2205 |
. . . . . . 7
⊢ (¬
(◡𝐹‘𝑥) < 𝐾 → (𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) ↔ 𝑘 = ((◡𝐹‘𝑥) − 1))) |
195 | 194 | adantl 277 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ¬ (◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) ↔ 𝑘 = ((◡𝐹‘𝑥) − 1))) |
196 | 49 | ad2antrr 488 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑀 ∈ ℤ) |
197 | 159 | ad2antrr 488 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑁 ∈ ℤ) |
198 | | simprr 531 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 = ((◡𝐹‘𝑥) − 1)) |
199 | 17 | ad2antrr 488 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ◡𝐹:(𝑀...(𝑁 + 1))⟶(𝑀...(𝑁 + 1))) |
200 | | simplr 528 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑥 ∈ (𝑀...𝑁)) |
201 | 62, 200 | sselid 3177 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑥 ∈ (𝑀...(𝑁 + 1))) |
202 | 199, 201 | ffvelcdmd 5686 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1))) |
203 | 202 | elfzelzd 10082 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ∈ ℤ) |
204 | 203, 35 | syl 14 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ((◡𝐹‘𝑥) − 1) ∈ ℤ) |
205 | 198, 204 | eqeltrd 2270 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 ∈ ℤ) |
206 | 49 | zred 9429 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ∈ ℝ) |
207 | 206 | ad2antrr 488 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑀 ∈ ℝ) |
208 | 102 | ad2antrr 488 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 ∈ ℝ) |
209 | 205 | zred 9429 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 ∈ ℝ) |
210 | | elfzle1 10083 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ (𝑀...(𝑁 + 1)) → 𝑀 ≤ 𝐾) |
211 | 24, 210 | syl 14 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑀 ≤ 𝐾) |
212 | 211 | ad2antrr 488 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑀 ≤ 𝐾) |
213 | | elfzelz 10081 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ∈ ℤ) |
214 | 213 | adantl 277 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℤ) |
215 | 214 | zred 9429 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ∈ ℝ) |
216 | 159 | zred 9429 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑁 ∈ ℝ) |
217 | 216 | adantr 276 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 ∈ ℝ) |
218 | 173 | adantr 276 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ∈ ℝ) |
219 | | elfzle2 10084 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ (𝑀...𝑁) → 𝑥 ≤ 𝑁) |
220 | 219 | adantl 277 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 ≤ 𝑁) |
221 | 217 | ltp1d 8939 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑁 < (𝑁 + 1)) |
222 | 215, 217,
218, 220, 221 | lelttrd 8134 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → 𝑥 < (𝑁 + 1)) |
223 | 215, 222 | gtned 8122 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑁 + 1) ≠ 𝑥) |
224 | 223 | adantr 276 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝑁 + 1) ≠ 𝑥) |
225 | 15 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1))) |
226 | 22 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝑁 + 1) ∈ (𝑀...(𝑁 + 1))) |
227 | | f1fveq 5807 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((◡𝐹:(𝑀...(𝑁 + 1))–1-1→(𝑀...(𝑁 + 1)) ∧ ((𝑁 + 1) ∈ (𝑀...(𝑁 + 1)) ∧ 𝑥 ∈ (𝑀...(𝑁 + 1)))) → ((◡𝐹‘(𝑁 + 1)) = (◡𝐹‘𝑥) ↔ (𝑁 + 1) = 𝑥)) |
228 | 225, 226,
201, 227 | syl12anc 1247 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ((◡𝐹‘(𝑁 + 1)) = (◡𝐹‘𝑥) ↔ (𝑁 + 1) = 𝑥)) |
229 | 228 | necon3bid 2405 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ((◡𝐹‘(𝑁 + 1)) ≠ (◡𝐹‘𝑥) ↔ (𝑁 + 1) ≠ 𝑥)) |
230 | 224, 229 | mpbird 167 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘(𝑁 + 1)) ≠ (◡𝐹‘𝑥)) |
231 | 11 | neeq1i 2379 |
. . . . . . . . . . . . . . . 16
⊢ (𝐾 ≠ (◡𝐹‘𝑥) ↔ (◡𝐹‘(𝑁 + 1)) ≠ (◡𝐹‘𝑥)) |
232 | 230, 231 | sylibr 134 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 ≠ (◡𝐹‘𝑥)) |
233 | 232 | necomd 2450 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ≠ 𝐾) |
234 | 25 | ad2antrr 488 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 ∈ ℤ) |
235 | | zapne 9381 |
. . . . . . . . . . . . . . 15
⊢ (((◡𝐹‘𝑥) ∈ ℤ ∧ 𝐾 ∈ ℤ) → ((◡𝐹‘𝑥) # 𝐾 ↔ (◡𝐹‘𝑥) ≠ 𝐾)) |
236 | 203, 234,
235 | syl2anc 411 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ((◡𝐹‘𝑥) # 𝐾 ↔ (◡𝐹‘𝑥) ≠ 𝐾)) |
237 | 233, 236 | mpbird 167 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) # 𝐾) |
238 | 203 | zred 9429 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ∈ ℝ) |
239 | | simprl 529 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ¬ (◡𝐹‘𝑥) < 𝐾) |
240 | 208, 238,
239 | nltled 8130 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 ≤ (◡𝐹‘𝑥)) |
241 | 208, 238,
240 | leltapd 8648 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝐾 < (◡𝐹‘𝑥) ↔ (◡𝐹‘𝑥) # 𝐾)) |
242 | 237, 241 | mpbird 167 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 < (◡𝐹‘𝑥)) |
243 | | zltlem1 9364 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ ℤ ∧ (◡𝐹‘𝑥) ∈ ℤ) → (𝐾 < (◡𝐹‘𝑥) ↔ 𝐾 ≤ ((◡𝐹‘𝑥) − 1))) |
244 | 234, 203,
243 | syl2anc 411 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝐾 < (◡𝐹‘𝑥) ↔ 𝐾 ≤ ((◡𝐹‘𝑥) − 1))) |
245 | 242, 244 | mpbid 147 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 ≤ ((◡𝐹‘𝑥) − 1)) |
246 | 245, 198 | breqtrrd 4057 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐾 ≤ 𝑘) |
247 | 207, 208,
209, 212, 246 | letrd 8133 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑀 ≤ 𝑘) |
248 | | elfzle2 10084 |
. . . . . . . . . . . 12
⊢ ((◡𝐹‘𝑥) ∈ (𝑀...(𝑁 + 1)) → (◡𝐹‘𝑥) ≤ (𝑁 + 1)) |
249 | 202, 248 | syl 14 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ≤ (𝑁 + 1)) |
250 | 216 | ad2antrr 488 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑁 ∈ ℝ) |
251 | | 1re 8008 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℝ |
252 | | lesubadd 8443 |
. . . . . . . . . . . . 13
⊢ (((◡𝐹‘𝑥) ∈ ℝ ∧ 1 ∈ ℝ ∧
𝑁 ∈ ℝ) →
(((◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (◡𝐹‘𝑥) ≤ (𝑁 + 1))) |
253 | 251, 252 | mp3an2 1336 |
. . . . . . . . . . . 12
⊢ (((◡𝐹‘𝑥) ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (◡𝐹‘𝑥) ≤ (𝑁 + 1))) |
254 | 238, 250,
253 | syl2anc 411 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (((◡𝐹‘𝑥) − 1) ≤ 𝑁 ↔ (◡𝐹‘𝑥) ≤ (𝑁 + 1))) |
255 | 249, 254 | mpbird 167 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ((◡𝐹‘𝑥) − 1) ≤ 𝑁) |
256 | 198, 255 | eqbrtrd 4051 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 ≤ 𝑁) |
257 | 196, 197,
205, 247, 256 | elfzd 10072 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑘 ∈ (𝑀...𝑁)) |
258 | 208, 209,
246 | lensymd 8131 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → ¬ 𝑘 < 𝐾) |
259 | 258, 93 | syl 14 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))) = (𝐹‘(𝑘 + 1))) |
260 | 198 | oveq1d 5925 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝑘 + 1) = (((◡𝐹‘𝑥) − 1) + 1)) |
261 | 203 | zcnd 9430 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (◡𝐹‘𝑥) ∈ ℂ) |
262 | | npcan 8218 |
. . . . . . . . . . . 12
⊢ (((◡𝐹‘𝑥) ∈ ℂ ∧ 1 ∈ ℂ)
→ (((◡𝐹‘𝑥) − 1) + 1) = (◡𝐹‘𝑥)) |
263 | 261, 144,
262 | sylancl 413 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (((◡𝐹‘𝑥) − 1) + 1) = (◡𝐹‘𝑥)) |
264 | 260, 263 | eqtrd 2226 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝑘 + 1) = (◡𝐹‘𝑥)) |
265 | 264 | fveq2d 5550 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝐹‘(𝑘 + 1)) = (𝐹‘(◡𝐹‘𝑥))) |
266 | 2 | ad2antrr 488 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝐹:(𝑀...(𝑁 + 1))–1-1-onto→(𝑀...(𝑁 + 1))) |
267 | 266, 201,
188 | syl2anc 411 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝐹‘(◡𝐹‘𝑥)) = 𝑥) |
268 | 259, 265,
267 | 3eqtrrd 2231 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) |
269 | 257, 268 | jca 306 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ (¬ (◡𝐹‘𝑥) < 𝐾 ∧ 𝑘 = ((◡𝐹‘𝑥) − 1))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1))))) |
270 | 269 | expr 375 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ¬ (◡𝐹‘𝑥) < 𝐾) → (𝑘 = ((◡𝐹‘𝑥) − 1) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))) |
271 | 195, 270 | sylbid 150 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) ∧ ¬ (◡𝐹‘𝑥) < 𝐾) → (𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))) |
272 | | exmiddc 837 |
. . . . . 6
⊢
(DECID (◡𝐹‘𝑥) < 𝐾 → ((◡𝐹‘𝑥) < 𝐾 ∨ ¬ (◡𝐹‘𝑥) < 𝐾)) |
273 | 39, 272 | syl 14 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → ((◡𝐹‘𝑥) < 𝐾 ∨ ¬ (◡𝐹‘𝑥) < 𝐾)) |
274 | 193, 271,
273 | mpjaodan 799 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀...𝑁)) → (𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1)) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))) |
275 | 274 | expimpd 363 |
. . 3
⊢ (𝜑 → ((𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))) → (𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))))) |
276 | 154, 275 | impbid 129 |
. 2
⊢ (𝜑 → ((𝑘 ∈ (𝑀...𝑁) ∧ 𝑥 = (𝐹‘if(𝑘 < 𝐾, 𝑘, (𝑘 + 1)))) ↔ (𝑥 ∈ (𝑀...𝑁) ∧ 𝑘 = if((◡𝐹‘𝑥) < 𝐾, (◡𝐹‘𝑥), ((◡𝐹‘𝑥) − 1))))) |
277 | 1, 29, 40, 276 | f1od 6113 |
1
⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)) |