Step |
Hyp |
Ref |
Expression |
1 |
|
cncfiooicc.x |
⊢ Ⅎ 𝑥 𝜑 |
2 |
|
cncfiooicc.g |
⊢ 𝐺 = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ if ( 𝑥 = 𝐴 , 𝑅 , if ( 𝑥 = 𝐵 , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) |
3 |
|
cncfiooicc.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
4 |
|
cncfiooicc.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
5 |
|
cncfiooicc.f |
⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) ) |
6 |
|
cncfiooicc.l |
⊢ ( 𝜑 → 𝐿 ∈ ( 𝐹 limℂ 𝐵 ) ) |
7 |
|
cncfiooicc.r |
⊢ ( 𝜑 → 𝑅 ∈ ( 𝐹 limℂ 𝐴 ) ) |
8 |
|
nfv |
⊢ Ⅎ 𝑥 ( 𝜑 ∧ 𝐴 < 𝐵 ) |
9 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐴 ∈ ℝ ) |
10 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐵 ∈ ℝ ) |
11 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐴 < 𝐵 ) |
12 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐹 ∈ ( ( 𝐴 (,) 𝐵 ) –cn→ ℂ ) ) |
13 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐿 ∈ ( 𝐹 limℂ 𝐵 ) ) |
14 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝑅 ∈ ( 𝐹 limℂ 𝐴 ) ) |
15 |
8 2 9 10 11 12 13 14
|
cncfiooicclem1 |
⊢ ( ( 𝜑 ∧ 𝐴 < 𝐵 ) → 𝐺 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
16 |
|
limccl |
⊢ ( 𝐹 limℂ 𝐴 ) ⊆ ℂ |
17 |
16 7
|
sseldi |
⊢ ( 𝜑 → 𝑅 ∈ ℂ ) |
18 |
17
|
snssd |
⊢ ( 𝜑 → { 𝑅 } ⊆ ℂ ) |
19 |
|
ssid |
⊢ ℂ ⊆ ℂ |
20 |
19
|
a1i |
⊢ ( 𝜑 → ℂ ⊆ ℂ ) |
21 |
|
cncfss |
⊢ ( ( { 𝑅 } ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( { 𝐴 } –cn→ { 𝑅 } ) ⊆ ( { 𝐴 } –cn→ ℂ ) ) |
22 |
18 20 21
|
syl2anc |
⊢ ( 𝜑 → ( { 𝐴 } –cn→ { 𝑅 } ) ⊆ ( { 𝐴 } –cn→ ℂ ) ) |
23 |
22
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( { 𝐴 } –cn→ { 𝑅 } ) ⊆ ( { 𝐴 } –cn→ ℂ ) ) |
24 |
3
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
25 |
|
iccid |
⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 [,] 𝐴 ) = { 𝐴 } ) |
26 |
24 25
|
syl |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐴 ) = { 𝐴 } ) |
27 |
|
oveq2 |
⊢ ( 𝐴 = 𝐵 → ( 𝐴 [,] 𝐴 ) = ( 𝐴 [,] 𝐵 ) ) |
28 |
26 27
|
sylan9req |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → { 𝐴 } = ( 𝐴 [,] 𝐵 ) ) |
29 |
28
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝐴 [,] 𝐵 ) = { 𝐴 } ) |
30 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 𝐵 ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) |
31 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 𝐵 ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 [,] 𝐵 ) = { 𝐴 } ) |
32 |
30 31
|
eleqtrd |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 𝐵 ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥 ∈ { 𝐴 } ) |
33 |
|
elsni |
⊢ ( 𝑥 ∈ { 𝐴 } → 𝑥 = 𝐴 ) |
34 |
32 33
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 𝐵 ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑥 = 𝐴 ) |
35 |
34
|
iftrued |
⊢ ( ( ( 𝜑 ∧ 𝐴 = 𝐵 ) ∧ 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑥 = 𝐴 , 𝑅 , if ( 𝑥 = 𝐵 , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) = 𝑅 ) |
36 |
29 35
|
mpteq12dva |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ if ( 𝑥 = 𝐴 , 𝑅 , if ( 𝑥 = 𝐵 , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ { 𝐴 } ↦ 𝑅 ) ) |
37 |
2 36
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐺 = ( 𝑥 ∈ { 𝐴 } ↦ 𝑅 ) ) |
38 |
3
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
39 |
38
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐴 ∈ ℂ ) |
40 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝑅 ∈ ℂ ) |
41 |
|
cncfdmsn |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑅 ∈ ℂ ) → ( 𝑥 ∈ { 𝐴 } ↦ 𝑅 ) ∈ ( { 𝐴 } –cn→ { 𝑅 } ) ) |
42 |
39 40 41
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( 𝑥 ∈ { 𝐴 } ↦ 𝑅 ) ∈ ( { 𝐴 } –cn→ { 𝑅 } ) ) |
43 |
37 42
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐺 ∈ ( { 𝐴 } –cn→ { 𝑅 } ) ) |
44 |
23 43
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐺 ∈ ( { 𝐴 } –cn→ ℂ ) ) |
45 |
28
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → ( { 𝐴 } –cn→ ℂ ) = ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
46 |
44 45
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝐴 = 𝐵 ) → 𝐺 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
47 |
46
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ 𝐴 = 𝐵 ) → 𝐺 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
48 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → 𝜑 ) |
49 |
|
eqcom |
⊢ ( 𝐵 = 𝐴 ↔ 𝐴 = 𝐵 ) |
50 |
49
|
biimpi |
⊢ ( 𝐵 = 𝐴 → 𝐴 = 𝐵 ) |
51 |
50
|
con3i |
⊢ ( ¬ 𝐴 = 𝐵 → ¬ 𝐵 = 𝐴 ) |
52 |
51
|
adantl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐵 = 𝐴 ) |
53 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → ¬ 𝐴 < 𝐵 ) |
54 |
|
pm4.56 |
⊢ ( ( ¬ 𝐵 = 𝐴 ∧ ¬ 𝐴 < 𝐵 ) ↔ ¬ ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ) |
55 |
54
|
biimpi |
⊢ ( ( ¬ 𝐵 = 𝐴 ∧ ¬ 𝐴 < 𝐵 ) → ¬ ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ) |
56 |
52 53 55
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → ¬ ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ) |
57 |
48 4
|
syl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 ∈ ℝ ) |
58 |
48 3
|
syl |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → 𝐴 ∈ ℝ ) |
59 |
57 58
|
lttrid |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → ( 𝐵 < 𝐴 ↔ ¬ ( 𝐵 = 𝐴 ∨ 𝐴 < 𝐵 ) ) ) |
60 |
56 59
|
mpbird |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → 𝐵 < 𝐴 ) |
61 |
|
0ss |
⊢ ∅ ⊆ ℂ |
62 |
|
eqid |
⊢ ( TopOpen ‘ ℂfld ) = ( TopOpen ‘ ℂfld ) |
63 |
62
|
cnfldtop |
⊢ ( TopOpen ‘ ℂfld ) ∈ Top |
64 |
|
rest0 |
⊢ ( ( TopOpen ‘ ℂfld ) ∈ Top → ( ( TopOpen ‘ ℂfld ) ↾t ∅ ) = { ∅ } ) |
65 |
63 64
|
ax-mp |
⊢ ( ( TopOpen ‘ ℂfld ) ↾t ∅ ) = { ∅ } |
66 |
65
|
eqcomi |
⊢ { ∅ } = ( ( TopOpen ‘ ℂfld ) ↾t ∅ ) |
67 |
62 66 66
|
cncfcn |
⊢ ( ( ∅ ⊆ ℂ ∧ ∅ ⊆ ℂ ) → ( ∅ –cn→ ∅ ) = ( { ∅ } Cn { ∅ } ) ) |
68 |
61 61 67
|
mp2an |
⊢ ( ∅ –cn→ ∅ ) = ( { ∅ } Cn { ∅ } ) |
69 |
|
cncfss |
⊢ ( ( ∅ ⊆ ℂ ∧ ℂ ⊆ ℂ ) → ( ∅ –cn→ ∅ ) ⊆ ( ∅ –cn→ ℂ ) ) |
70 |
61 19 69
|
mp2an |
⊢ ( ∅ –cn→ ∅ ) ⊆ ( ∅ –cn→ ℂ ) |
71 |
68 70
|
eqsstrri |
⊢ ( { ∅ } Cn { ∅ } ) ⊆ ( ∅ –cn→ ℂ ) |
72 |
2
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐺 = ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ if ( 𝑥 = 𝐴 , 𝑅 , if ( 𝑥 = 𝐵 , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
73 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐵 < 𝐴 ) |
74 |
24
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐴 ∈ ℝ* ) |
75 |
4
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
76 |
75
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐵 ∈ ℝ* ) |
77 |
|
icc0 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) ) |
78 |
74 76 77
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( ( 𝐴 [,] 𝐵 ) = ∅ ↔ 𝐵 < 𝐴 ) ) |
79 |
73 78
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( 𝐴 [,] 𝐵 ) = ∅ ) |
80 |
79
|
mpteq1d |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↦ if ( 𝑥 = 𝐴 , 𝑅 , if ( 𝑥 = 𝐵 , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) = ( 𝑥 ∈ ∅ ↦ if ( 𝑥 = 𝐴 , 𝑅 , if ( 𝑥 = 𝐵 , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) ) |
81 |
|
mpt0 |
⊢ ( 𝑥 ∈ ∅ ↦ if ( 𝑥 = 𝐴 , 𝑅 , if ( 𝑥 = 𝐵 , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) = ∅ |
82 |
81
|
a1i |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( 𝑥 ∈ ∅ ↦ if ( 𝑥 = 𝐴 , 𝑅 , if ( 𝑥 = 𝐵 , 𝐿 , ( 𝐹 ‘ 𝑥 ) ) ) ) = ∅ ) |
83 |
72 80 82
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐺 = ∅ ) |
84 |
|
0cnf |
⊢ ∅ ∈ ( { ∅ } Cn { ∅ } ) |
85 |
83 84
|
eqeltrdi |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐺 ∈ ( { ∅ } Cn { ∅ } ) ) |
86 |
71 85
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐺 ∈ ( ∅ –cn→ ℂ ) ) |
87 |
79
|
eqcomd |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ∅ = ( 𝐴 [,] 𝐵 ) ) |
88 |
87
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → ( ∅ –cn→ ℂ ) = ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
89 |
86 88
|
eleqtrd |
⊢ ( ( 𝜑 ∧ 𝐵 < 𝐴 ) → 𝐺 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
90 |
48 60 89
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) ∧ ¬ 𝐴 = 𝐵 ) → 𝐺 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
91 |
47 90
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ ¬ 𝐴 < 𝐵 ) → 𝐺 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |
92 |
15 91
|
pm2.61dan |
⊢ ( 𝜑 → 𝐺 ∈ ( ( 𝐴 [,] 𝐵 ) –cn→ ℂ ) ) |