Step |
Hyp |
Ref |
Expression |
1 |
|
seqf1o.1 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ) ) → ( 𝑥 + 𝑦 ) ∈ 𝑆 ) |
2 |
|
seqf1o.2 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶 ) ) → ( 𝑥 + 𝑦 ) = ( 𝑦 + 𝑥 ) ) |
3 |
|
seqf1o.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆 ) ) → ( ( 𝑥 + 𝑦 ) + 𝑧 ) = ( 𝑥 + ( 𝑦 + 𝑧 ) ) ) |
4 |
|
seqf1o.4 |
⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
5 |
|
seqf1o.5 |
⊢ ( 𝜑 → 𝐶 ⊆ 𝑆 ) |
6 |
|
seqf1olem.5 |
⊢ ( 𝜑 → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
7 |
|
seqf1olem.6 |
⊢ ( 𝜑 → 𝐺 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ 𝐶 ) |
8 |
|
seqf1olem.7 |
⊢ 𝐽 = ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↦ ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) |
9 |
|
seqf1olem.8 |
⊢ 𝐾 = ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) |
10 |
|
fvexd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ∈ V ) |
11 |
|
fvex |
⊢ ( ◡ 𝐹 ‘ 𝑥 ) ∈ V |
12 |
|
ovex |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ∈ V |
13 |
11 12
|
ifex |
⊢ if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ∈ V |
14 |
13
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ∈ V ) |
15 |
|
iftrue |
⊢ ( 𝑘 < 𝐾 → if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) = 𝑘 ) |
16 |
15
|
fveq2d |
⊢ ( 𝑘 < 𝐾 → ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) = ( 𝐹 ‘ 𝑘 ) ) |
17 |
16
|
eqeq2d |
⊢ ( 𝑘 < 𝐾 → ( 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ↔ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) |
18 |
17
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ 𝑘 < 𝐾 ) → ( 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ↔ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) |
19 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝑥 = ( 𝐹 ‘ 𝑘 ) ) |
20 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) → 𝑘 ∈ ℤ ) |
21 |
20
|
zred |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) → 𝑘 ∈ ℝ ) |
22 |
21
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝑘 ∈ ℝ ) |
23 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝑘 < 𝐾 ) |
24 |
22 23
|
gtned |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝐾 ≠ 𝑘 ) |
25 |
|
f1of |
⊢ ( 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
26 |
6 25
|
syl |
⊢ ( 𝜑 → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
27 |
26
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
28 |
|
fzssp1 |
⊢ ( 𝑀 ... 𝑁 ) ⊆ ( 𝑀 ... ( 𝑁 + 1 ) ) |
29 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
30 |
28 29
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
31 |
27 30
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
32 |
|
elfzp1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ 𝑘 ) = ( 𝑁 + 1 ) ) ) ) |
33 |
4 32
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ 𝑘 ) = ( 𝑁 + 1 ) ) ) ) |
34 |
33
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ 𝑘 ) = ( 𝑁 + 1 ) ) ) ) |
35 |
31 34
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ 𝑘 ) = ( 𝑁 + 1 ) ) ) |
36 |
35
|
ord |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ¬ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝑁 + 1 ) ) ) |
37 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
38 |
|
f1ocnvfv |
⊢ ( ( 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) = ( 𝑁 + 1 ) → ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = 𝑘 ) ) |
39 |
37 30 38
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) = ( 𝑁 + 1 ) → ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = 𝑘 ) ) |
40 |
9
|
eqeq1i |
⊢ ( 𝐾 = 𝑘 ↔ ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = 𝑘 ) |
41 |
39 40
|
syl6ibr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) = ( 𝑁 + 1 ) → 𝐾 = 𝑘 ) ) |
42 |
36 41
|
syld |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ¬ ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... 𝑁 ) → 𝐾 = 𝑘 ) ) |
43 |
42
|
necon1ad |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐾 ≠ 𝑘 → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... 𝑁 ) ) ) |
44 |
24 43
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 𝑀 ... 𝑁 ) ) |
45 |
19 44
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
46 |
19
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝐹 ‘ 𝑘 ) = 𝑥 ) |
47 |
|
f1ocnvfv |
⊢ ( ( 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ∧ 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) = 𝑥 → ( ◡ 𝐹 ‘ 𝑥 ) = 𝑘 ) ) |
48 |
37 30 47
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ( 𝐹 ‘ 𝑘 ) = 𝑥 → ( ◡ 𝐹 ‘ 𝑥 ) = 𝑘 ) ) |
49 |
46 48
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) = 𝑘 ) |
50 |
49 23
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) |
51 |
|
iftrue |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) = ( ◡ 𝐹 ‘ 𝑥 ) ) |
52 |
50 51
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) = ( ◡ 𝐹 ‘ 𝑥 ) ) |
53 |
52 49
|
eqtr2d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
54 |
45 53
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ 𝑘 ) ) ) → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) ) |
55 |
54
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ 𝑘 < 𝐾 ) → ( 𝑥 = ( 𝐹 ‘ 𝑘 ) → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) ) ) |
56 |
18 55
|
sylbid |
⊢ ( ( ( 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 ) ) ) ) ) |
57 |
|
iffalse |
⊢ ( ¬ 𝑘 < 𝐾 → if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) = ( 𝑘 + 1 ) ) |
58 |
57
|
fveq2d |
⊢ ( ¬ 𝑘 < 𝐾 → ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
59 |
58
|
eqeq2d |
⊢ ( ¬ 𝑘 < 𝐾 → ( 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ↔ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
60 |
59
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ¬ 𝑘 < 𝐾 ) → ( 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ↔ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) |
61 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
62 |
|
f1ocnv |
⊢ ( 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
63 |
6 62
|
syl |
⊢ ( 𝜑 → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
64 |
|
f1of1 |
⊢ ( ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
65 |
63 64
|
syl |
⊢ ( 𝜑 → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
66 |
|
f1f |
⊢ ( ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1→ ( 𝑀 ... ( 𝑁 + 1 ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
67 |
65 66
|
syl |
⊢ ( 𝜑 → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
68 |
|
peano2uz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
69 |
4 68
|
syl |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
70 |
|
eluzfz2 |
⊢ ( ( 𝑁 + 1 ) ∈ ( ℤ≥ ‘ 𝑀 ) → ( 𝑁 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
71 |
69 70
|
syl |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
72 |
67 71
|
ffvelrnd |
⊢ ( 𝜑 → ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
73 |
9 72
|
eqeltrid |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
74 |
|
elfzelz |
⊢ ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝐾 ∈ ℤ ) |
75 |
73 74
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
76 |
75
|
zred |
⊢ ( 𝜑 → 𝐾 ∈ ℝ ) |
77 |
76
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐾 ∈ ℝ ) |
78 |
21
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑘 ∈ ℝ ) |
79 |
|
peano2re |
⊢ ( 𝑘 ∈ ℝ → ( 𝑘 + 1 ) ∈ ℝ ) |
80 |
78 79
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝑘 + 1 ) ∈ ℝ ) |
81 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ¬ 𝑘 < 𝐾 ) |
82 |
77 78 81
|
nltled |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐾 ≤ 𝑘 ) |
83 |
78
|
ltp1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑘 < ( 𝑘 + 1 ) ) |
84 |
77 78 80 82 83
|
lelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐾 < ( 𝑘 + 1 ) ) |
85 |
77 84
|
ltned |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐾 ≠ ( 𝑘 + 1 ) ) |
86 |
26
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
87 |
|
fzp1elp1 |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) → ( 𝑘 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
88 |
87
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝑘 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
89 |
86 88
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
90 |
|
elfzp1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) ) ) ) |
91 |
4 90
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) ) ) ) |
92 |
91
|
ad2antrr |
⊢ ( ( ( 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 ) ) ) ) |
93 |
89 92
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) ) ) |
94 |
93
|
ord |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ¬ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) ) ) |
95 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
96 |
|
f1ocnvfv |
⊢ ( ( 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 ) ) ) |
97 |
95 88 96
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) → ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = ( 𝑘 + 1 ) ) ) |
98 |
9
|
eqeq1i |
⊢ ( 𝐾 = ( 𝑘 + 1 ) ↔ ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = ( 𝑘 + 1 ) ) |
99 |
97 98
|
syl6ibr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) → 𝐾 = ( 𝑘 + 1 ) ) ) |
100 |
94 99
|
syld |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ¬ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) → 𝐾 = ( 𝑘 + 1 ) ) ) |
101 |
100
|
necon1ad |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝐾 ≠ ( 𝑘 + 1 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ) ) |
102 |
85 101
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ) |
103 |
61 102
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
104 |
61
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = 𝑥 ) |
105 |
|
f1ocnvfv |
⊢ ( ( 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ∧ ( 𝑘 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) = 𝑥 → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝑘 + 1 ) ) ) |
106 |
95 88 105
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) = 𝑥 → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝑘 + 1 ) ) ) |
107 |
104 106
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝑘 + 1 ) ) |
108 |
107
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ↔ ( 𝑘 + 1 ) < 𝐾 ) ) |
109 |
|
lttr |
⊢ ( ( 𝑘 ∈ ℝ ∧ ( 𝑘 + 1 ) ∈ ℝ ∧ 𝐾 ∈ ℝ ) → ( ( 𝑘 < ( 𝑘 + 1 ) ∧ ( 𝑘 + 1 ) < 𝐾 ) → 𝑘 < 𝐾 ) ) |
110 |
78 80 77 109
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝑘 < ( 𝑘 + 1 ) ∧ ( 𝑘 + 1 ) < 𝐾 ) → 𝑘 < 𝐾 ) ) |
111 |
83 110
|
mpand |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝑘 + 1 ) < 𝐾 → 𝑘 < 𝐾 ) ) |
112 |
108 111
|
sylbid |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → 𝑘 < 𝐾 ) ) |
113 |
81 112
|
mtod |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) |
114 |
|
iffalse |
⊢ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) |
115 |
113 114
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) |
116 |
107
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) = ( ( 𝑘 + 1 ) − 1 ) ) |
117 |
78
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑘 ∈ ℂ ) |
118 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
119 |
|
pncan |
⊢ ( ( 𝑘 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑘 + 1 ) − 1 ) = 𝑘 ) |
120 |
117 118 119
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝑘 + 1 ) − 1 ) = 𝑘 ) |
121 |
115 116 120
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
122 |
103 121
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) ) |
123 |
122
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ¬ 𝑘 < 𝐾 ) → ( 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) ) ) |
124 |
60 123
|
sylbid |
⊢ ( ( ( 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 ) ) ) ) ) |
125 |
56 124
|
pm2.61dan |
⊢ ( ( 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 ) ) ) ) ) |
126 |
125
|
expimpd |
⊢ ( 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 ) ) ) ) ) |
127 |
51
|
eqeq2d |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → ( 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ↔ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) |
128 |
127
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) → ( 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ↔ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) |
129 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) |
130 |
67
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
131 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
132 |
28 131
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑥 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
133 |
130 132
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
134 |
129 133
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
135 |
|
elfzle1 |
⊢ ( 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝑀 ≤ 𝑘 ) |
136 |
134 135
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑀 ≤ 𝑘 ) |
137 |
|
elfzelz |
⊢ ( 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝑘 ∈ ℤ ) |
138 |
134 137
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ∈ ℤ ) |
139 |
138
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ∈ ℝ ) |
140 |
76
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝐾 ∈ ℝ ) |
141 |
|
eluzelz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ ) |
142 |
4 141
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
143 |
142
|
peano2zd |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℤ ) |
144 |
143
|
zred |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℝ ) |
145 |
144
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝑁 + 1 ) ∈ ℝ ) |
146 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) |
147 |
129 146
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 < 𝐾 ) |
148 |
|
elfzle2 |
⊢ ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝐾 ≤ ( 𝑁 + 1 ) ) |
149 |
73 148
|
syl |
⊢ ( 𝜑 → 𝐾 ≤ ( 𝑁 + 1 ) ) |
150 |
149
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝐾 ≤ ( 𝑁 + 1 ) ) |
151 |
139 140 145 147 150
|
ltletrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 < ( 𝑁 + 1 ) ) |
152 |
142
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑁 ∈ ℤ ) |
153 |
|
zleltp1 |
⊢ ( ( 𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑘 ≤ 𝑁 ↔ 𝑘 < ( 𝑁 + 1 ) ) ) |
154 |
138 152 153
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝑘 ≤ 𝑁 ↔ 𝑘 < ( 𝑁 + 1 ) ) ) |
155 |
151 154
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ≤ 𝑁 ) |
156 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
157 |
4 156
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
158 |
157
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑀 ∈ ℤ ) |
159 |
|
elfz |
⊢ ( ( 𝑘 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁 ) ) ) |
160 |
138 158 152 159
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁 ) ) ) |
161 |
136 155 160
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
162 |
147 16
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) = ( 𝐹 ‘ 𝑘 ) ) |
163 |
129
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) ) |
164 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
165 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ∧ 𝑥 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
166 |
164 132 165
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
167 |
162 163 166
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) |
168 |
161 167
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) ) |
169 |
168
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) → ( 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) ) ) |
170 |
128 169
|
sylbid |
⊢ ( ( ( 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 ) ) ) ) ) ) |
171 |
114
|
eqeq2d |
⊢ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → ( 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ↔ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
172 |
171
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) → ( 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ↔ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
173 |
157
|
zred |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
174 |
173
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑀 ∈ ℝ ) |
175 |
76
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ∈ ℝ ) |
176 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) |
177 |
67
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
178 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
179 |
28 178
|
sseldi |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑥 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
180 |
177 179
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
181 |
|
elfzelz |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℤ ) |
182 |
180 181
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℤ ) |
183 |
|
peano2zm |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℤ → ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ∈ ℤ ) |
184 |
182 183
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ∈ ℤ ) |
185 |
176 184
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 ∈ ℤ ) |
186 |
185
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 ∈ ℝ ) |
187 |
|
elfzle1 |
⊢ ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝑀 ≤ 𝐾 ) |
188 |
73 187
|
syl |
⊢ ( 𝜑 → 𝑀 ≤ 𝐾 ) |
189 |
188
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑀 ≤ 𝐾 ) |
190 |
182
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
191 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) |
192 |
175 190 191
|
nltled |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ≤ ( ◡ 𝐹 ‘ 𝑥 ) ) |
193 |
|
elfzelz |
⊢ ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → 𝑥 ∈ ℤ ) |
194 |
193
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑥 ∈ ℤ ) |
195 |
194
|
zred |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑥 ∈ ℝ ) |
196 |
142
|
zred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
197 |
196
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑁 ∈ ℝ ) |
198 |
144
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝑁 + 1 ) ∈ ℝ ) |
199 |
|
elfzle2 |
⊢ ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → 𝑥 ≤ 𝑁 ) |
200 |
199
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑥 ≤ 𝑁 ) |
201 |
197
|
ltp1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑁 < ( 𝑁 + 1 ) ) |
202 |
195 197 198 200 201
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑥 < ( 𝑁 + 1 ) ) |
203 |
195 202
|
gtned |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝑁 + 1 ) ≠ 𝑥 ) |
204 |
203
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑁 + 1 ) ≠ 𝑥 ) |
205 |
65
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
206 |
71
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑁 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
207 |
|
f1fveq |
⊢ ( ( ◡ 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 ) ) |
208 |
205 206 179 207
|
syl12anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = ( ◡ 𝐹 ‘ 𝑥 ) ↔ ( 𝑁 + 1 ) = 𝑥 ) ) |
209 |
208
|
necon3bid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) ≠ ( ◡ 𝐹 ‘ 𝑥 ) ↔ ( 𝑁 + 1 ) ≠ 𝑥 ) ) |
210 |
204 209
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) ≠ ( ◡ 𝐹 ‘ 𝑥 ) ) |
211 |
9
|
neeq1i |
⊢ ( 𝐾 ≠ ( ◡ 𝐹 ‘ 𝑥 ) ↔ ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) ≠ ( ◡ 𝐹 ‘ 𝑥 ) ) |
212 |
210 211
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ≠ ( ◡ 𝐹 ‘ 𝑥 ) ) |
213 |
212
|
necomd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ≠ 𝐾 ) |
214 |
175 190 192 213
|
leneltd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 < ( ◡ 𝐹 ‘ 𝑥 ) ) |
215 |
75
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ∈ ℤ ) |
216 |
|
zltlem1 |
⊢ ( ( 𝐾 ∈ ℤ ∧ ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℤ ) → ( 𝐾 < ( ◡ 𝐹 ‘ 𝑥 ) ↔ 𝐾 ≤ ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
217 |
215 182 216
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝐾 < ( ◡ 𝐹 ‘ 𝑥 ) ↔ 𝐾 ≤ ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
218 |
214 217
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ≤ ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) |
219 |
218 176
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ≤ 𝑘 ) |
220 |
174 175 186 189 219
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑀 ≤ 𝑘 ) |
221 |
|
elfzle2 |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) |
222 |
180 221
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) |
223 |
196
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑁 ∈ ℝ ) |
224 |
|
1re |
⊢ 1 ∈ ℝ |
225 |
|
lesubadd |
⊢ ( ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ≤ 𝑁 ↔ ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) ) |
226 |
224 225
|
mp3an2 |
⊢ ( ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ≤ 𝑁 ↔ ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) ) |
227 |
190 223 226
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ≤ 𝑁 ↔ ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) ) |
228 |
222 227
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ≤ 𝑁 ) |
229 |
176 228
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 ≤ 𝑁 ) |
230 |
157
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑀 ∈ ℤ ) |
231 |
142
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑁 ∈ ℤ ) |
232 |
185 230 231 159
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ↔ ( 𝑀 ≤ 𝑘 ∧ 𝑘 ≤ 𝑁 ) ) ) |
233 |
220 229 232
|
mpbir2and |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
234 |
175 186 219
|
lensymd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ¬ 𝑘 < 𝐾 ) |
235 |
234 58
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
236 |
176
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑘 + 1 ) = ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) + 1 ) ) |
237 |
182
|
zcnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
238 |
|
npcan |
⊢ ( ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) + 1 ) = ( ◡ 𝐹 ‘ 𝑥 ) ) |
239 |
237 118 238
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) + 1 ) = ( ◡ 𝐹 ‘ 𝑥 ) ) |
240 |
236 239
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑘 + 1 ) = ( ◡ 𝐹 ‘ 𝑥 ) ) |
241 |
240
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) ) |
242 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
243 |
242 179 165
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
244 |
235 241 243
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) |
245 |
233 244
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) ) |
246 |
245
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) → ( 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) ) ) |
247 |
172 246
|
sylbid |
⊢ ( ( ( 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 ) ) ) ) ) ) |
248 |
170 247
|
pm2.61dan |
⊢ ( ( 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 ) ) ) ) ) ) |
249 |
248
|
expimpd |
⊢ ( 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 ) ) ) ) ) ) |
250 |
126 249
|
impbid |
⊢ ( 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 ) ) ) ) ) |
251 |
8 10 14 250
|
f1od |
⊢ ( 𝜑 → 𝐽 : ( 𝑀 ... 𝑁 ) –1-1-onto→ ( 𝑀 ... 𝑁 ) ) |