Metamath Proof Explorer

Theorem pcopt2

Description: Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015)

Ref Expression
Hypothesis pcopt.1 ⊢ 𝑃 = ( ( 0 [,] 1 ) × { 𝑌 } )
Assertion pcopt2 ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝑃 ) ( ≃ph ‘ 𝐽 ) 𝐹 )

Proof

Step Hyp Ref Expression
1 pcopt.1 ⊢ 𝑃 = ( ( 0 [,] 1 ) × { 𝑌 } )
2 1 fveq1i ⊢ ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( ( 2 · 𝑥 ) − 1 ) )
3 simpr ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ‘ 1 ) = 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 ‘ 1 ) = 𝑌 ) → 𝐹 : ( 0 [,] 1 ) ⟶ ∪ 𝐽 )
8 1elunit ⊢ 1 ∈ ( 0 [,] 1 )
9 ffvelcdm ⊢ ( ( 𝐹 : ( 0 [,] 1 ) ⟶ ∪ 𝐽 ∧ 1 ∈ ( 0 [,] 1 ) ) → ( 𝐹 ‘ 1 ) ∈ ∪ 𝐽 )
10 7 8 9 sylancl ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ‘ 1 ) ∈ ∪ 𝐽 )
11 3 10 eqeltrrd ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝑌 ∈ ∪ 𝐽 )
12 elii2 ⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) )
13 iihalf2 ⊢ ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 2 · 𝑥 ) − 1 ) ∈ ( 0 [,] 1 ) )
14 12 13 syl ⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → ( ( 2 · 𝑥 ) − 1 ) ∈ ( 0 [,] 1 ) )
15 fvconst2g ⊢ ( ( 𝑌 ∈ ∪ 𝐽 ∧ ( ( 2 · 𝑥 ) − 1 ) ∈ ( 0 [,] 1 ) ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( ( 2 · 𝑥 ) − 1 ) ) = 𝑌 )
16 11 14 15 syl2an ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( ( ( 0 [,] 1 ) × { 𝑌 } ) ‘ ( ( 2 · 𝑥 ) − 1 ) ) = 𝑌 )
17 2 16 eqtrid ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = 𝑌 )
18 simplr ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝐹 ‘ 1 ) = 𝑌 )
19 17 18 eqtr4d ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) ) → ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = ( 𝐹 ‘ 1 ) )
20 19 anassrs ⊢ ( ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑥 ≤ ( 1 / 2 ) ) → ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) = ( 𝐹 ‘ 1 ) )
21 20 ifeq2da ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) )
22 21 mpteq2dva ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) ) )
23 simpl ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐹 ∈ ( II Cn 𝐽 ) )
24 cntop2 ⊢ ( 𝐹 ∈ ( II Cn 𝐽 ) → 𝐽 ∈ Top )
25 24 adantr ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐽 ∈ Top )
26 toptopon2 ⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) )
27 25 26 sylib ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) )
28 1 pcoptcl ⊢ ( ( 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) ∧ 𝑌 ∈ ∪ 𝐽 ) → ( 𝑃 ∈ ( II Cn 𝐽 ) ∧ ( 𝑃 ‘ 0 ) = 𝑌 ∧ ( 𝑃 ‘ 1 ) = 𝑌 ) )
29 27 11 28 syl2anc ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑃 ∈ ( II Cn 𝐽 ) ∧ ( 𝑃 ‘ 0 ) = 𝑌 ∧ ( 𝑃 ‘ 1 ) = 𝑌 ) )
30 29 simp1d ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝑃 ∈ ( II Cn 𝐽 ) )
31 23 30 pcoval ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝑃 ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝑃 ‘ ( ( 2 · 𝑥 ) − 1 ) ) ) ) )
32 iftrue ⊢ ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = ( 2 · 𝑥 ) )
33 32 adantl ⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = ( 2 · 𝑥 ) )
34 elii1 ⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↔ ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) )
35 iihalf1 ⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) → ( 2 · 𝑥 ) ∈ ( 0 [,] 1 ) )
36 34 35 sylbir ⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → ( 2 · 𝑥 ) ∈ ( 0 [,] 1 ) )
37 33 36 eqeltrd ⊢ ( ( 𝑥 ∈ ( 0 [,] 1 ) ∧ 𝑥 ≤ ( 1 / 2 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) )
38 37 ex ⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → ( 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) ) )
39 iffalse ⊢ ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = 1 )
40 39 8 eqeltrdi ⊢ ( ¬ 𝑥 ≤ ( 1 / 2 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) )
41 38 40 pm2.61d1 ⊢ ( 𝑥 ∈ ( 0 [,] 1 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) )
42 41 adantl ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ 𝑥 ∈ ( 0 [,] 1 ) ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ∈ ( 0 [,] 1 ) )
43 eqidd ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) )
44 7 feqmptd ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 𝐹 = ( 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 𝐹 ‘ 𝑦 ) ) )
45 fveq2 ⊢ ( 𝑦 = if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) )
46 fvif ⊢ ( 𝐹 ‘ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) )
47 45 46 eqtrdi ⊢ ( 𝑦 = if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) → ( 𝐹 ‘ 𝑦 ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) )
48 42 43 44 47 fmptco ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑥 ) ) , ( 𝐹 ‘ 1 ) ) ) )
49 22 31 48 3eqtr4d ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝑃 ) = ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ) )
50 iitopon ⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) )
51 50 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) )
52 51 cnmptid ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 𝑥 ) ∈ ( II Cn II ) )
53 0elunit ⊢ 0 ∈ ( 0 [,] 1 )
54 53 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 0 ∈ ( 0 [,] 1 ) )
55 51 51 54 cnmptc ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ 0 ) ∈ ( II Cn II ) )
56 eqid ⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) )
57 eqid ⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
58 eqid ⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
59 dfii2 ⊢ II = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) )
60 0re ⊢ 0 ∈ ℝ
61 60 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 0 ∈ ℝ )
62 1re ⊢ 1 ∈ ℝ
63 62 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 1 ∈ ℝ )
64 halfre ⊢ ( 1 / 2 ) ∈ ℝ
65 halfge0 ⊢ 0 ≤ ( 1 / 2 )
66 halflt1 ⊢ ( 1 / 2 ) < 1
67 64 62 66 ltleii ⊢ ( 1 / 2 ) ≤ 1
68 elicc01 ⊢ ( ( 1 / 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 1 ) )
69 64 65 67 68 mpbir3an ⊢ ( 1 / 2 ) ∈ ( 0 [,] 1 )
70 69 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 1 / 2 ) ∈ ( 0 [,] 1 ) )
71 simprl ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → 𝑦 = ( 1 / 2 ) )
72 71 oveq2d ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = ( 2 · ( 1 / 2 ) ) )
73 2cn ⊢ 2 ∈ ℂ
74 2ne0 ⊢ 2 ≠ 0
75 73 74 recidi ⊢ ( 2 · ( 1 / 2 ) ) = 1
76 72 75 eqtrdi ⊢ ( ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑧 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑦 ) = 1 )
77 retopon ⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )
78 iccssre ⊢ ( ( 0 ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ )
79 60 64 78 mp2an ⊢ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ
80 resttopon ⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) )
81 77 79 80 mp2an ⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) )
82 81 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) )
83 82 51 cnmpt1st ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ) )
84 57 iihalf1cn ⊢ ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II )
85 84 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) )
86 oveq2 ⊢ ( 𝑥 = 𝑦 → ( 2 · 𝑥 ) = ( 2 · 𝑦 ) )
87 82 51 83 82 85 86 cnmpt21 ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ ( 2 · 𝑦 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn II ) )
88 iccssre ⊢ ( ( ( 1 / 2 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ )
89 64 62 88 mp2an ⊢ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ
90 resttopon ⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) )
91 77 89 90 mp2an ⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) )
92 91 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) )
93 8 a1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → 1 ∈ ( 0 [,] 1 ) )
94 92 51 51 93 cnmpt2c ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ 1 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn II ) )
95 56 57 58 59 61 63 70 51 76 87 94 cnmpopc ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑦 ∈ ( 0 [,] 1 ) , 𝑧 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 2 ) , ( 2 · 𝑦 ) , 1 ) ) ∈ ( ( II ×t II ) Cn II ) )
96 breq1 ⊢ ( 𝑦 = 𝑥 → ( 𝑦 ≤ ( 1 / 2 ) ↔ 𝑥 ≤ ( 1 / 2 ) ) )
97 oveq2 ⊢ ( 𝑦 = 𝑥 → ( 2 · 𝑦 ) = ( 2 · 𝑥 ) )
98 96 97 ifbieq1d ⊢ ( 𝑦 = 𝑥 → if ( 𝑦 ≤ ( 1 / 2 ) , ( 2 · 𝑦 ) , 1 ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) )
99 98 adantr ⊢ ( ( 𝑦 = 𝑥 ∧ 𝑧 = 0 ) → if ( 𝑦 ≤ ( 1 / 2 ) , ( 2 · 𝑦 ) , 1 ) = if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) )
100 51 52 55 51 51 95 99 cnmpt12 ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ∈ ( II Cn II ) )
101 id ⊢ ( 𝑥 = 0 → 𝑥 = 0 )
102 101 65 eqbrtrdi ⊢ ( 𝑥 = 0 → 𝑥 ≤ ( 1 / 2 ) )
103 102 32 syl ⊢ ( 𝑥 = 0 → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = ( 2 · 𝑥 ) )
104 oveq2 ⊢ ( 𝑥 = 0 → ( 2 · 𝑥 ) = ( 2 · 0 ) )
105 2t0e0 ⊢ ( 2 · 0 ) = 0
106 104 105 eqtrdi ⊢ ( 𝑥 = 0 → ( 2 · 𝑥 ) = 0 )
107 103 106 eqtrd ⊢ ( 𝑥 = 0 → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = 0 )
108 eqid ⊢ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) = ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) )
109 c0ex ⊢ 0 ∈ V
110 107 108 109 fvmpt ⊢ ( 0 ∈ ( 0 [,] 1 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ‘ 0 ) = 0 )
111 53 110 mp1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ‘ 0 ) = 0 )
112 64 62 ltnlei ⊢ ( ( 1 / 2 ) < 1 ↔ ¬ 1 ≤ ( 1 / 2 ) )
113 66 112 mpbi ⊢ ¬ 1 ≤ ( 1 / 2 )
114 breq1 ⊢ ( 𝑥 = 1 → ( 𝑥 ≤ ( 1 / 2 ) ↔ 1 ≤ ( 1 / 2 ) ) )
115 113 114 mtbiri ⊢ ( 𝑥 = 1 → ¬ 𝑥 ≤ ( 1 / 2 ) )
116 115 39 syl ⊢ ( 𝑥 = 1 → if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) = 1 )
117 1ex ⊢ 1 ∈ V
118 116 108 117 fvmpt ⊢ ( 1 ∈ ( 0 [,] 1 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ‘ 1 ) = 1 )
119 8 118 mp1i ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ‘ 1 ) = 1 )
120 23 100 111 119 reparpht ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ∘ ( 𝑥 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( 2 · 𝑥 ) , 1 ) ) ) ( ≃ph ‘ 𝐽 ) 𝐹 )
121 49 120 eqbrtrd ⊢ ( ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ‘ 1 ) = 𝑌 ) → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝑃 ) ( ≃ph ‘ 𝐽 ) 𝐹 )