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
|
sselid |
⊢ ( ( ( 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 |
73
|
elfzelzd |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
75 |
74
|
zred |
⊢ ( 𝜑 → 𝐾 ∈ ℝ ) |
76 |
75
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐾 ∈ ℝ ) |
77 |
21
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑘 ∈ ℝ ) |
78 |
|
peano2re |
⊢ ( 𝑘 ∈ ℝ → ( 𝑘 + 1 ) ∈ ℝ ) |
79 |
77 78
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝑘 + 1 ) ∈ ℝ ) |
80 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ¬ 𝑘 < 𝐾 ) |
81 |
76 77 80
|
nltled |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐾 ≤ 𝑘 ) |
82 |
77
|
ltp1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑘 < ( 𝑘 + 1 ) ) |
83 |
76 77 79 81 82
|
lelttrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐾 < ( 𝑘 + 1 ) ) |
84 |
76 83
|
ltned |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐾 ≠ ( 𝑘 + 1 ) ) |
85 |
26
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
86 |
|
fzp1elp1 |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) → ( 𝑘 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
87 |
86
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝑘 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
88 |
85 87
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
89 |
|
elfzp1 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) ) ) ) |
90 |
4 89
|
syl |
⊢ ( 𝜑 → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ↔ ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) ) ) ) |
91 |
90
|
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 ) ) ) ) |
92 |
88 91
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ∨ ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) ) ) |
93 |
92
|
ord |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ¬ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) ) ) |
94 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
95 |
|
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 ) ) ) |
96 |
94 87 95
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) → ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = ( 𝑘 + 1 ) ) ) |
97 |
9
|
eqeq1i |
⊢ ( 𝐾 = ( 𝑘 + 1 ) ↔ ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = ( 𝑘 + 1 ) ) |
98 |
96 97
|
syl6ibr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝑁 + 1 ) → 𝐾 = ( 𝑘 + 1 ) ) ) |
99 |
93 98
|
syld |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ¬ ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) → 𝐾 = ( 𝑘 + 1 ) ) ) |
100 |
99
|
necon1ad |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝐾 ≠ ( 𝑘 + 1 ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ) ) |
101 |
84 100
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) ∈ ( 𝑀 ... 𝑁 ) ) |
102 |
61 101
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
103 |
61
|
eqcomd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = 𝑥 ) |
104 |
|
f1ocnvfv |
⊢ ( ( 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ∧ ( 𝑘 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) = 𝑥 → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝑘 + 1 ) ) ) |
105 |
94 87 104
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝐹 ‘ ( 𝑘 + 1 ) ) = 𝑥 → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝑘 + 1 ) ) ) |
106 |
103 105
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) = ( 𝑘 + 1 ) ) |
107 |
106
|
breq1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ↔ ( 𝑘 + 1 ) < 𝐾 ) ) |
108 |
|
lttr |
⊢ ( ( 𝑘 ∈ ℝ ∧ ( 𝑘 + 1 ) ∈ ℝ ∧ 𝐾 ∈ ℝ ) → ( ( 𝑘 < ( 𝑘 + 1 ) ∧ ( 𝑘 + 1 ) < 𝐾 ) → 𝑘 < 𝐾 ) ) |
109 |
77 79 76 108
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝑘 < ( 𝑘 + 1 ) ∧ ( 𝑘 + 1 ) < 𝐾 ) → 𝑘 < 𝐾 ) ) |
110 |
82 109
|
mpand |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝑘 + 1 ) < 𝐾 → 𝑘 < 𝐾 ) ) |
111 |
107 110
|
sylbid |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → 𝑘 < 𝐾 ) ) |
112 |
80 111
|
mtod |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) |
113 |
|
iffalse |
⊢ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) |
114 |
112 113
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) |
115 |
106
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) = ( ( 𝑘 + 1 ) − 1 ) ) |
116 |
77
|
recnd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑘 ∈ ℂ ) |
117 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
118 |
|
pncan |
⊢ ( ( 𝑘 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝑘 + 1 ) − 1 ) = 𝑘 ) |
119 |
116 117 118
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( ( 𝑘 + 1 ) − 1 ) = 𝑘 ) |
120 |
114 115 119
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
121 |
102 120
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ 𝑘 < 𝐾 ∧ 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) ) → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) ) |
122 |
121
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ¬ 𝑘 < 𝐾 ) → ( 𝑥 = ( 𝐹 ‘ ( 𝑘 + 1 ) ) → ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) ) ) |
123 |
60 122
|
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 ) ) ) ) ) |
124 |
56 123
|
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 ) ) ) ) ) |
125 |
124
|
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 ) ) ) ) ) |
126 |
51
|
eqeq2d |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → ( 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ↔ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) |
127 |
126
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) → ( 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ↔ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) |
128 |
|
eluzel2 |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑀 ∈ ℤ ) |
129 |
4 128
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
130 |
129
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑀 ∈ ℤ ) |
131 |
|
eluzelz |
⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → 𝑁 ∈ ℤ ) |
132 |
4 131
|
syl |
⊢ ( 𝜑 → 𝑁 ∈ ℤ ) |
133 |
132
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑁 ∈ ℤ ) |
134 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) |
135 |
67
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
136 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
137 |
28 136
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑥 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
138 |
135 137
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
139 |
134 138
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
140 |
139
|
elfzelzd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ∈ ℤ ) |
141 |
|
elfzle1 |
⊢ ( 𝑘 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝑀 ≤ 𝑘 ) |
142 |
139 141
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑀 ≤ 𝑘 ) |
143 |
140
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ∈ ℝ ) |
144 |
75
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝐾 ∈ ℝ ) |
145 |
132
|
peano2zd |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℤ ) |
146 |
145
|
zred |
⊢ ( 𝜑 → ( 𝑁 + 1 ) ∈ ℝ ) |
147 |
146
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝑁 + 1 ) ∈ ℝ ) |
148 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) |
149 |
134 148
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 < 𝐾 ) |
150 |
|
elfzle2 |
⊢ ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝐾 ≤ ( 𝑁 + 1 ) ) |
151 |
73 150
|
syl |
⊢ ( 𝜑 → 𝐾 ≤ ( 𝑁 + 1 ) ) |
152 |
151
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝐾 ≤ ( 𝑁 + 1 ) ) |
153 |
143 144 147 149 152
|
ltletrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 < ( 𝑁 + 1 ) ) |
154 |
|
zleltp1 |
⊢ ( ( 𝑘 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑘 ≤ 𝑁 ↔ 𝑘 < ( 𝑁 + 1 ) ) ) |
155 |
140 133 154
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝑘 ≤ 𝑁 ↔ 𝑘 < ( 𝑁 + 1 ) ) ) |
156 |
153 155
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ≤ 𝑁 ) |
157 |
130 133 140 142 156
|
elfzd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
158 |
149 16
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) = ( 𝐹 ‘ 𝑘 ) ) |
159 |
134
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) ) |
160 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
161 |
|
f1ocnvfv2 |
⊢ ( ( 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ∧ 𝑥 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
162 |
160 137 161
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
163 |
158 159 162
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) |
164 |
157 163
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) ) ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) ) |
165 |
164
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) → ( 𝑘 = ( ◡ 𝐹 ‘ 𝑥 ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) ) ) |
166 |
127 165
|
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 ) ) ) ) ) ) |
167 |
113
|
eqeq2d |
⊢ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 → ( 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ↔ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
168 |
167
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) → ( 𝑘 = if ( ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 , ( ◡ 𝐹 ‘ 𝑥 ) , ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ↔ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
169 |
129
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑀 ∈ ℤ ) |
170 |
132
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑁 ∈ ℤ ) |
171 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) |
172 |
67
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) ⟶ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
173 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) |
174 |
28 173
|
sselid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑥 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
175 |
172 174
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
176 |
175
|
elfzelzd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℤ ) |
177 |
|
peano2zm |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℤ → ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ∈ ℤ ) |
178 |
176 177
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ∈ ℤ ) |
179 |
171 178
|
eqeltrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 ∈ ℤ ) |
180 |
129
|
zred |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
181 |
180
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑀 ∈ ℝ ) |
182 |
75
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ∈ ℝ ) |
183 |
179
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 ∈ ℝ ) |
184 |
|
elfzle1 |
⊢ ( 𝐾 ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → 𝑀 ≤ 𝐾 ) |
185 |
73 184
|
syl |
⊢ ( 𝜑 → 𝑀 ≤ 𝐾 ) |
186 |
185
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑀 ≤ 𝐾 ) |
187 |
176
|
zred |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℝ ) |
188 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) |
189 |
182 187 188
|
nltled |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ≤ ( ◡ 𝐹 ‘ 𝑥 ) ) |
190 |
|
elfzelz |
⊢ ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → 𝑥 ∈ ℤ ) |
191 |
190
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑥 ∈ ℤ ) |
192 |
191
|
zred |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑥 ∈ ℝ ) |
193 |
132
|
zred |
⊢ ( 𝜑 → 𝑁 ∈ ℝ ) |
194 |
193
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑁 ∈ ℝ ) |
195 |
146
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝑁 + 1 ) ∈ ℝ ) |
196 |
|
elfzle2 |
⊢ ( 𝑥 ∈ ( 𝑀 ... 𝑁 ) → 𝑥 ≤ 𝑁 ) |
197 |
196
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑥 ≤ 𝑁 ) |
198 |
194
|
ltp1d |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑁 < ( 𝑁 + 1 ) ) |
199 |
192 194 195 197 198
|
lelttrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → 𝑥 < ( 𝑁 + 1 ) ) |
200 |
192 199
|
gtned |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) → ( 𝑁 + 1 ) ≠ 𝑥 ) |
201 |
200
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑁 + 1 ) ≠ 𝑥 ) |
202 |
65
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ◡ 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
203 |
71
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑁 + 1 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
204 |
|
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 ) ) |
205 |
202 203 174 204
|
syl12anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) = ( ◡ 𝐹 ‘ 𝑥 ) ↔ ( 𝑁 + 1 ) = 𝑥 ) ) |
206 |
205
|
necon3bid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) ≠ ( ◡ 𝐹 ‘ 𝑥 ) ↔ ( 𝑁 + 1 ) ≠ 𝑥 ) ) |
207 |
201 206
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) ≠ ( ◡ 𝐹 ‘ 𝑥 ) ) |
208 |
9
|
neeq1i |
⊢ ( 𝐾 ≠ ( ◡ 𝐹 ‘ 𝑥 ) ↔ ( ◡ 𝐹 ‘ ( 𝑁 + 1 ) ) ≠ ( ◡ 𝐹 ‘ 𝑥 ) ) |
209 |
207 208
|
sylibr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ≠ ( ◡ 𝐹 ‘ 𝑥 ) ) |
210 |
209
|
necomd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ≠ 𝐾 ) |
211 |
182 187 189 210
|
leneltd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 < ( ◡ 𝐹 ‘ 𝑥 ) ) |
212 |
74
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ∈ ℤ ) |
213 |
|
zltlem1 |
⊢ ( ( 𝐾 ∈ ℤ ∧ ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℤ ) → ( 𝐾 < ( ◡ 𝐹 ‘ 𝑥 ) ↔ 𝐾 ≤ ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
214 |
212 176 213
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝐾 < ( ◡ 𝐹 ‘ 𝑥 ) ↔ 𝐾 ≤ ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) |
215 |
211 214
|
mpbid |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ≤ ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) |
216 |
215 171
|
breqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐾 ≤ 𝑘 ) |
217 |
181 182 183 186 216
|
letrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑀 ≤ 𝑘 ) |
218 |
|
elfzle2 |
⊢ ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ( 𝑀 ... ( 𝑁 + 1 ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) |
219 |
175 218
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) |
220 |
193
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑁 ∈ ℝ ) |
221 |
|
1re |
⊢ 1 ∈ ℝ |
222 |
|
lesubadd |
⊢ ( ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ≤ 𝑁 ↔ ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) ) |
223 |
221 222
|
mp3an2 |
⊢ ( ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ≤ 𝑁 ↔ ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) ) |
224 |
187 220 223
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ≤ 𝑁 ↔ ( ◡ 𝐹 ‘ 𝑥 ) ≤ ( 𝑁 + 1 ) ) ) |
225 |
219 224
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ≤ 𝑁 ) |
226 |
171 225
|
eqbrtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 ≤ 𝑁 ) |
227 |
169 170 179 217 226
|
elfzd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑘 ∈ ( 𝑀 ... 𝑁 ) ) |
228 |
182 183 216
|
lensymd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ¬ 𝑘 < 𝐾 ) |
229 |
228 58
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) = ( 𝐹 ‘ ( 𝑘 + 1 ) ) ) |
230 |
171
|
oveq1d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑘 + 1 ) = ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) + 1 ) ) |
231 |
176
|
zcnd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℂ ) |
232 |
|
npcan |
⊢ ( ( ( ◡ 𝐹 ‘ 𝑥 ) ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) + 1 ) = ( ◡ 𝐹 ‘ 𝑥 ) ) |
233 |
231 117 232
|
sylancl |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) + 1 ) = ( ◡ 𝐹 ‘ 𝑥 ) ) |
234 |
230 233
|
eqtrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑘 + 1 ) = ( ◡ 𝐹 ‘ 𝑥 ) ) |
235 |
234
|
fveq2d |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝐹 ‘ ( 𝑘 + 1 ) ) = ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) ) |
236 |
6
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝐹 : ( 𝑀 ... ( 𝑁 + 1 ) ) –1-1-onto→ ( 𝑀 ... ( 𝑁 + 1 ) ) ) |
237 |
236 174 161
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝐹 ‘ ( ◡ 𝐹 ‘ 𝑥 ) ) = 𝑥 ) |
238 |
229 235 237
|
3eqtrrd |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) |
239 |
227 238
|
jca |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ( ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ∧ 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) ) ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) ) |
240 |
239
|
expr |
⊢ ( ( ( 𝜑 ∧ 𝑥 ∈ ( 𝑀 ... 𝑁 ) ) ∧ ¬ ( ◡ 𝐹 ‘ 𝑥 ) < 𝐾 ) → ( 𝑘 = ( ( ◡ 𝐹 ‘ 𝑥 ) − 1 ) → ( 𝑘 ∈ ( 𝑀 ... 𝑁 ) ∧ 𝑥 = ( 𝐹 ‘ if ( 𝑘 < 𝐾 , 𝑘 , ( 𝑘 + 1 ) ) ) ) ) ) |
241 |
168 240
|
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 ) ) ) ) ) ) |
242 |
166 241
|
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 ) ) ) ) ) ) |
243 |
242
|
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 ) ) ) ) ) ) |
244 |
125 243
|
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 ) ) ) ) ) |
245 |
8 10 14 244
|
f1od |
⊢ ( 𝜑 → 𝐽 : ( 𝑀 ... 𝑁 ) –1-1-onto→ ( 𝑀 ... 𝑁 ) ) |