Metamath Proof Explorer


Theorem pcopt

Description: Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010) (Revised by Mario Carneiro, 20-Dec-2013)

Ref Expression
Hypothesis pcopt.1 𝑃 = ( ( 0 [,] 1 ) × { 𝑌 } )
Assertion pcopt ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑃 ( *𝑝𝐽 ) 𝐹 ) ( ≃ph𝐽 ) 𝐹 )

Proof

Step Hyp Ref Expression
1 pcopt.1 𝑃 = ( ( 0 [,] 1 ) × { 𝑌 } )
2 1 fveq1i ( 𝑃 ‘ ( 2 · 𝑥 ) ) = ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( 2 · 𝑥 ) )
3 simpr ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝐹 ‘ 0 ) = 𝑌 )
4 iiuni ( 0 [,] 1 ) = II
5 eqid 𝐽 = 𝐽
6 4 5 cnf ( 𝐹 ∈ ( II Cn 𝐽 ) → 𝐹 : ( 0 [,] 1 ) ⟶ 𝐽 )
7 6 adantr ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 𝐹 : ( 0 [,] 1 ) ⟶ 𝐽 )
8 0elunit 0 ∈ ( 0 [,] 1 )
9 ffvelrn ( ( 𝐹 : ( 0 [,] 1 ) ⟶ 𝐽 ∧ 0 ∈ ( 0 [,] 1 ) ) → ( 𝐹 ‘ 0 ) ∈ 𝐽 )
10 7 8 9 sylancl ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝐹 ‘ 0 ) ∈ 𝐽 )
11 3 10 eqeltrrd ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 𝑌 𝐽 )
12 elii1 ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↔ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) )
13 iihalf1 ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) → ( 2 · 𝑥 ) ∈ ( 0 [,] 1 ) )
14 12 13 sylbir ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → ( 2 · 𝑥 ) ∈ ( 0 [,] 1 ) )
15 fvconst2g ( ( 𝑌 𝐽 ∧ ( 2 · 𝑥 ) ∈ ( 0 [,] 1 ) ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( 2 · 𝑥 ) ) = 𝑌 )
16 11 14 15 syl2an ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( 2 · 𝑥 ) ) = 𝑌 )
17 2 16 eqtrid ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝑃 ‘ ( 2 · 𝑥 ) ) = 𝑌 )
18 simplr ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝐹 ‘ 0 ) = 𝑌 )
19 17 18 eqtr4d ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝑃 ‘ ( 2 · 𝑥 ) ) = ( 𝐹 ‘ 0 ) )
20 19 ifeq1d ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝑃 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) )
21 20 expr ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝑃 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) )
22 iffalse ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝑃 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) )
23 iffalse ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) )
24 22 23 eqtr4d ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝑃 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) )
25 21 24 pm2.61d1 ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝑃 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) )
26 25 mpteq2dva ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝑃 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) )
27 cntop2 ( 𝐹 ∈ ( II Cn 𝐽 ) → 𝐽 ∈ Top )
28 27 adantr ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 𝐽 ∈ Top )
29 toptopon2 ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ 𝐽 ) )
30 28 29 sylib ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 𝐽 ∈ ( TopOn ‘ 𝐽 ) )
31 1 pcoptcl ( ( 𝐽 ∈ ( TopOn ‘ 𝐽 ) ∧ 𝑌 𝐽 ) → ( 𝑃 ∈ ( II Cn 𝐽 ) ∧ ( 𝑃 ‘ 0 ) = 𝑌 ∧ ( 𝑃 ‘ 1 ) = 𝑌 ) )
32 30 11 31 syl2anc ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑃 ∈ ( II Cn 𝐽 ) ∧ ( 𝑃 ‘ 0 ) = 𝑌 ∧ ( 𝑃 ‘ 1 ) = 𝑌 ) )
33 32 simp1d ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 𝑃 ∈ ( II Cn 𝐽 ) )
34 simpl ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 𝐹 ∈ ( II Cn 𝐽 ) )
35 33 34 pcoval ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑃 ( *𝑝𝐽 ) 𝐹 ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝑃 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) )
36 iffalse ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) = ( ( 2 · 𝑥 ) − 1 ) )
37 36 adantl ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) = ( ( 2 · 𝑥 ) − 1 ) )
38 elii2 ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) )
39 iihalf2 ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 2 · 𝑥 ) − 1 ) ∈ ( 0 [,] 1 ) )
40 38 39 syl ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 2 · 𝑥 ) − 1 ) ∈ ( 0 [,] 1 ) )
41 37 40 eqeltrd ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( 0 [,] 1 ) )
42 41 ex ( 𝑥 ∈ ( 0 [,] 1 ) → ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( 0 [,] 1 ) ) )
43 iftrue ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) = 0 )
44 43 8 eqeltrdi ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( 0 [,] 1 ) )
45 42 44 pm2.61d2 ( 𝑥 ∈ ( 0 [,] 1 ) → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( 0 [,] 1 ) )
46 45 adantl ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( 0 [,] 1 ) )
47 eqid ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) )
48 47 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) )
49 7 feqmptd ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 𝐹 = ( 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝐹𝑦 ) ) )
50 fveq2 ( 𝑦 = if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) → ( 𝐹𝑦 ) = ( 𝐹 ‘ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) )
51 fvif ( 𝐹 ‘ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) )
52 50 51 eqtrdi ( 𝑦 = if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) → ( 𝐹𝑦 ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) )
53 46 48 49 52 fmptco ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ 0 ) , ( 𝐹 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) )
54 26 35 53 3eqtr4d ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑃 ( *𝑝𝐽 ) 𝐹 ) = ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) ) )
55 iitopon II ∈ ( TopOn ‘ ( 0 [,] 1 ) )
56 55 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) )
57 56 cnmptid ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 𝑥 ) ∈ ( II Cn II ) )
58 8 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 0 ∈ ( 0 [,] 1 ) )
59 56 56 58 cnmptc ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 0 ) ∈ ( II Cn II ) )
60 eqid ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) )
61 eqid ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
62 eqid ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
63 dfii2 II = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) )
64 0re 0 ∈ ℝ
65 64 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 0 ∈ ℝ )
66 1re 1 ∈ ℝ
67 66 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → 1 ∈ ℝ )
68 halfre ( 1 / 2 ) ∈ ℝ
69 halfge0 0 ≤ ( 1 / 2 )
70 halflt1 ( 1 / 2 ) < 1
71 68 66 70 ltleii ( 1 / 2 ) ≤ 1
72 elicc01 ( ( 1 / 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 1 ) )
73 68 69 71 72 mpbir3an ( 1 / 2 ) ∈ ( 0 [,] 1 )
74 73 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 1 / 2 ) ∈ ( 0 [,] 1 ) )
75 simprl ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → 𝑦 = ( 1 / 2 ) )
76 75 oveq2d ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = ( 2 · ( 1 / 2 ) ) )
77 2cn 2 ∈ ℂ
78 2ne0 2 ≠ 0
79 77 78 recidi ( 2 · ( 1 / 2 ) ) = 1
80 76 79 eqtrdi ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = 1 )
81 80 oveq1d ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( ( 2 · 𝑦 ) − 1 ) = ( 1 − 1 ) )
82 1m1e0 ( 1 − 1 ) = 0
83 81 82 eqtr2di ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → 0 = ( ( 2 · 𝑦 ) − 1 ) )
84 retopon ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )
85 iccssre ( ( 0 ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ )
86 64 68 85 mp2an ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ
87 resttopon ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) )
88 84 86 87 mp2an ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) )
89 88 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) )
90 89 56 56 58 cnmpt2c ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 0 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn II ) )
91 iccssre ( ( ( 1 / 2 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ )
92 68 66 91 mp2an ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ
93 resttopon ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) )
94 84 92 93 mp2an ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) )
95 94 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) )
96 95 56 cnmpt1st ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ) )
97 62 iihalf2cn ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II )
98 97 a1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) )
99 oveq2 ( 𝑥 = 𝑦 → ( 2 · 𝑥 ) = ( 2 · 𝑦 ) )
100 99 oveq1d ( 𝑥 = 𝑦 → ( ( 2 · 𝑥 ) − 1 ) = ( ( 2 · 𝑦 ) − 1 ) )
101 95 56 96 95 98 100 cnmpt21 ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( ( 2 · 𝑦 ) − 1 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn II ) )
102 60 61 62 63 65 67 74 56 83 90 101 cnmpopc ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑦 ∈ ( 0 [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑦 ) − 1 ) ) ) ∈ ( ( II ×t II ) Cn II ) )
103 breq1 ( 𝑦 = 𝑥 → ( 𝑦 ≤ ( 1 / 2 ) ↔ 𝑥 ≤ ( 1 / 2 ) ) )
104 oveq2 ( 𝑦 = 𝑥 → ( 2 · 𝑦 ) = ( 2 · 𝑥 ) )
105 104 oveq1d ( 𝑦 = 𝑥 → ( ( 2 · 𝑦 ) − 1 ) = ( ( 2 · 𝑥 ) − 1 ) )
106 103 105 ifbieq2d ( 𝑦 = 𝑥 → if ( 𝑦 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑦 ) − 1 ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) )
107 106 adantr ( ( 𝑦 = 𝑥𝑧 = 0 ) → if ( 𝑦 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑦 ) − 1 ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) )
108 56 57 59 56 56 102 107 cnmpt12 ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) ∈ ( II Cn II ) )
109 id ( 𝑥 = 0 → 𝑥 = 0 )
110 109 69 eqbrtrdi ( 𝑥 = 0 → 𝑥 ≤ ( 1 / 2 ) )
111 110 43 syl ( 𝑥 = 0 → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) = 0 )
112 c0ex 0 ∈ V
113 111 47 112 fvmpt ( 0 ∈ ( 0 [,] 1 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) ‘ 0 ) = 0 )
114 8 113 mp1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) ‘ 0 ) = 0 )
115 1elunit 1 ∈ ( 0 [,] 1 )
116 68 66 ltnlei ( ( 1 / 2 ) < 1 ↔ ¬ 1 ≤ ( 1 / 2 ) )
117 70 116 mpbi ¬ 1 ≤ ( 1 / 2 )
118 breq1 ( 𝑥 = 1 → ( 𝑥 ≤ ( 1 / 2 ) ↔ 1 ≤ ( 1 / 2 ) ) )
119 117 118 mtbiri ( 𝑥 = 1 → ¬ 𝑥 ≤ ( 1 / 2 ) )
120 119 36 syl ( 𝑥 = 1 → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) = ( ( 2 · 𝑥 ) − 1 ) )
121 oveq2 ( 𝑥 = 1 → ( 2 · 𝑥 ) = ( 2 · 1 ) )
122 2t1e2 ( 2 · 1 ) = 2
123 121 122 eqtrdi ( 𝑥 = 1 → ( 2 · 𝑥 ) = 2 )
124 123 oveq1d ( 𝑥 = 1 → ( ( 2 · 𝑥 ) − 1 ) = ( 2 − 1 ) )
125 2m1e1 ( 2 − 1 ) = 1
126 124 125 eqtrdi ( 𝑥 = 1 → ( ( 2 · 𝑥 ) − 1 ) = 1 )
127 120 126 eqtrd ( 𝑥 = 1 → if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) = 1 )
128 1ex 1 ∈ V
129 127 47 128 fvmpt ( 1 ∈ ( 0 [,] 1 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) ‘ 1 ) = 1 )
130 115 129 mp1i ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) ‘ 1 ) = 1 )
131 34 108 114 130 reparpht ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , 0 , ( ( 2 · 𝑥 ) − 1 ) ) ) ) ( ≃ph𝐽 ) 𝐹 )
132 54 131 eqbrtrd ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 0 ) = 𝑌 ) → ( 𝑃 ( *𝑝𝐽 ) 𝐹 ) ( ≃ph𝐽 ) 𝐹 )