htpycc

Description: Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009) (Revised by Mario Carneiro, 23-Feb-2015)

	Ref	Expression
Hypotheses	htpycc.1	⊢ 𝑁 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝑥 𝐿 ( 2 · 𝑦 ) ) , ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) )
	htpycc.2	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
	htpycc.4	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) )
	htpycc.5	⊢ ( 𝜑 → 𝐺 ∈ ( 𝐽 Cn 𝐾 ) )
	htpycc.6	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐽 Cn 𝐾 ) )
	htpycc.7	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐹 ( 𝐽 Htpy 𝐾 ) 𝐺 ) )
	htpycc.8	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐺 ( 𝐽 Htpy 𝐾 ) 𝐻 ) )
Assertion	htpycc	⊢ ( 𝜑 → 𝑁 ∈ ( 𝐹 ( 𝐽 Htpy 𝐾 ) 𝐻 ) )

Proof

Step	Hyp	Ref	Expression
1		htpycc.1	⊢ 𝑁 = ( 𝑥 ∈ 𝑋 , 𝑦 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝑥 𝐿 ( 2 · 𝑦 ) ) , ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) )
2		htpycc.2	⊢ ( 𝜑 → 𝐽 ∈ ( TopOn ‘ 𝑋 ) )
3		htpycc.4	⊢ ( 𝜑 → 𝐹 ∈ ( 𝐽 Cn 𝐾 ) )
4		htpycc.5	⊢ ( 𝜑 → 𝐺 ∈ ( 𝐽 Cn 𝐾 ) )
5		htpycc.6	⊢ ( 𝜑 → 𝐻 ∈ ( 𝐽 Cn 𝐾 ) )
6		htpycc.7	⊢ ( 𝜑 → 𝐿 ∈ ( 𝐹 ( 𝐽 Htpy 𝐾 ) 𝐺 ) )
7		htpycc.8	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐺 ( 𝐽 Htpy 𝐾 ) 𝐻 ) )
8		iitopon	⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) )
9	8	a1i	⊢ ( 𝜑 → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) )
10		eqid	⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) )
11		eqid	⊢ ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) )
12		eqid	⊢ ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) )
13		dfii2	⊢ II = ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] 1 ) )
14		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
15		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
16		halfre	⊢ ( 1 / 2 ) ∈ ℝ
17		halfge0	⊢ 0 ≤ ( 1 / 2 )
18		1re	⊢ 1 ∈ ℝ
19		halflt1	⊢ ( 1 / 2 ) < 1
20	16 18 19	ltleii	⊢ ( 1 / 2 ) ≤ 1
21		elicc01	⊢ ( ( 1 / 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 1 ) )
22	16 17 20 21	mpbir3an	⊢ ( 1 / 2 ) ∈ ( 0 [,] 1 )
23	22	a1i	⊢ ( 𝜑 → ( 1 / 2 ) ∈ ( 0 [,] 1 ) )
24	2 3 4 6	htpyi	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( ( 𝑠 𝐿 0 ) = ( 𝐹 ‘ 𝑠 ) ∧ ( 𝑠 𝐿 1 ) = ( 𝐺 ‘ 𝑠 ) ) )
25	24	simprd	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝐿 1 ) = ( 𝐺 ‘ 𝑠 ) )
26	2 4 5 7	htpyi	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( ( 𝑠 𝑀 0 ) = ( 𝐺 ‘ 𝑠 ) ∧ ( 𝑠 𝑀 1 ) = ( 𝐻 ‘ 𝑠 ) ) )
27	26	simpld	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝑀 0 ) = ( 𝐺 ‘ 𝑠 ) )
28	25 27	eqtr4d	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝐿 1 ) = ( 𝑠 𝑀 0 ) )
29	28	ralrimiva	⊢ ( 𝜑 → ∀ 𝑠 ∈ 𝑋 ( 𝑠 𝐿 1 ) = ( 𝑠 𝑀 0 ) )
30		oveq1	⊢ ( 𝑠 = 𝑥 → ( 𝑠 𝐿 1 ) = ( 𝑥 𝐿 1 ) )
31		oveq1	⊢ ( 𝑠 = 𝑥 → ( 𝑠 𝑀 0 ) = ( 𝑥 𝑀 0 ) )
32	30 31	eqeq12d	⊢ ( 𝑠 = 𝑥 → ( ( 𝑠 𝐿 1 ) = ( 𝑠 𝑀 0 ) ↔ ( 𝑥 𝐿 1 ) = ( 𝑥 𝑀 0 ) ) )
33	32	rspccva	⊢ ( ( ∀ 𝑠 ∈ 𝑋 ( 𝑠 𝐿 1 ) = ( 𝑠 𝑀 0 ) ∧ 𝑥 ∈ 𝑋 ) → ( 𝑥 𝐿 1 ) = ( 𝑥 𝑀 0 ) )
34	29 33	sylan	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝑋 ) → ( 𝑥 𝐿 1 ) = ( 𝑥 𝑀 0 ) )
35	34	adantrl	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝑥 𝐿 1 ) = ( 𝑥 𝑀 0 ) )
36		simprl	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → 𝑦 = ( 1 / 2 ) )
37	36	oveq2d	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → ( 2 · 𝑦 ) = ( 2 · ( 1 / 2 ) ) )
38		2cn	⊢ 2 ∈ ℂ
39		2ne0	⊢ 2 ≠ 0
40	38 39	recidi	⊢ ( 2 · ( 1 / 2 ) ) = 1
41	37 40	eqtrdi	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → ( 2 · 𝑦 ) = 1 )
42	41	oveq2d	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝑥 𝐿 ( 2 · 𝑦 ) ) = ( 𝑥 𝐿 1 ) )
43	41	oveq1d	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → ( ( 2 · 𝑦 ) − 1 ) = ( 1 − 1 ) )
44		1m1e0	⊢ ( 1 − 1 ) = 0
45	43 44	eqtrdi	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → ( ( 2 · 𝑦 ) − 1 ) = 0 )
46	45	oveq2d	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) = ( 𝑥 𝑀 0 ) )
47	35 42 46	3eqtr4d	⊢ ( ( 𝜑 ∧ ( 𝑦 = ( 1 / 2 ) ∧ 𝑥 ∈ 𝑋 ) ) → ( 𝑥 𝐿 ( 2 · 𝑦 ) ) = ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) )
48		retopon	⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )
49		0re	⊢ 0 ∈ ℝ
50		iccssre	⊢ ( ( 0 ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ )
51	49 16 50	mp2an	⊢ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ
52		resttopon	⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) )
53	48 51 52	mp2an	⊢ ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) )
54	53	a1i	⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) )
55	54 2	cnmpt2nd	⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) ×_t 𝐽 ) Cn 𝐽 ) )
56	54 2	cnmpt1st	⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑥 ∈ 𝑋 ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) ×_t 𝐽 ) Cn ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) ) )
57	11	iihalf1cn	⊢ ( 𝑧 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑧 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) Cn II )
58	57	a1i	⊢ ( 𝜑 → ( 𝑧 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑧 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) )
59		oveq2	⊢ ( 𝑧 = 𝑦 → ( 2 · 𝑧 ) = ( 2 · 𝑦 ) )
60	54 2 56 54 58 59	cnmpt21	⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑥 ∈ 𝑋 ↦ ( 2 · 𝑦 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) ×_t 𝐽 ) Cn II ) )
61	2 3 4	htpycn	⊢ ( 𝜑 → ( 𝐹 ( 𝐽 Htpy 𝐾 ) 𝐺 ) ⊆ ( ( 𝐽 ×_t II ) Cn 𝐾 ) )
62	61 6	sseldd	⊢ ( 𝜑 → 𝐿 ∈ ( ( 𝐽 ×_t II ) Cn 𝐾 ) )
63	54 2 55 60 62	cnmpt22f	⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑥 ∈ 𝑋 ↦ ( 𝑥 𝐿 ( 2 · 𝑦 ) ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾_t ( 0 [,] ( 1 / 2 ) ) ) ×_t 𝐽 ) Cn 𝐾 ) )
64		iccssre	⊢ ( ( ( 1 / 2 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ )
65	16 18 64	mp2an	⊢ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ
66		resttopon	⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) )
67	48 65 66	mp2an	⊢ ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) )
68	67	a1i	⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) )
69	68 2	cnmpt2nd	⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑥 ∈ 𝑋 ↦ 𝑥 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) ×_t 𝐽 ) Cn 𝐽 ) )
70	68 2	cnmpt1st	⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑥 ∈ 𝑋 ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) ×_t 𝐽 ) Cn ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) ) )
71	12	iihalf2cn	⊢ ( 𝑧 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 2 · 𝑧 ) − 1 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) Cn II )
72	71	a1i	⊢ ( 𝜑 → ( 𝑧 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 2 · 𝑧 ) − 1 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) )
73	59	oveq1d	⊢ ( 𝑧 = 𝑦 → ( ( 2 · 𝑧 ) − 1 ) = ( ( 2 · 𝑦 ) − 1 ) )
74	68 2 70 68 72 73	cnmpt21	⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑥 ∈ 𝑋 ↦ ( ( 2 · 𝑦 ) − 1 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) ×_t 𝐽 ) Cn II ) )
75	2 4 5	htpycn	⊢ ( 𝜑 → ( 𝐺 ( 𝐽 Htpy 𝐾 ) 𝐻 ) ⊆ ( ( 𝐽 ×_t II ) Cn 𝐾 ) )
76	75 7	sseldd	⊢ ( 𝜑 → 𝑀 ∈ ( ( 𝐽 ×_t II ) Cn 𝐾 ) )
77	68 2 69 74 76	cnmpt22f	⊢ ( 𝜑 → ( 𝑦 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑥 ∈ 𝑋 ↦ ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾_t ( ( 1 / 2 ) [,] 1 ) ) ×_t 𝐽 ) Cn 𝐾 ) )
78	10 11 12 13 14 15 23 2 47 63 77	cnmpopc	⊢ ( 𝜑 → ( 𝑦 ∈ ( 0 [,] 1 ) , 𝑥 ∈ 𝑋 ↦ if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝑥 𝐿 ( 2 · 𝑦 ) ) , ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) ) ∈ ( ( II ×_t 𝐽 ) Cn 𝐾 ) )
79	9 2 78	cnmptcom	⊢ ( 𝜑 → ( 𝑥 ∈ 𝑋 , 𝑦 ∈ ( 0 [,] 1 ) ↦ if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝑥 𝐿 ( 2 · 𝑦 ) ) , ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) ) ∈ ( ( 𝐽 ×_t II ) Cn 𝐾 ) )
80	1 79	eqeltrid	⊢ ( 𝜑 → 𝑁 ∈ ( ( 𝐽 ×_t II ) Cn 𝐾 ) )
81		simpr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → 𝑠 ∈ 𝑋 )
82		0elunit	⊢ 0 ∈ ( 0 [,] 1 )
83		simpr	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → 𝑦 = 0 )
84	83 17	eqbrtrdi	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → 𝑦 ≤ ( 1 / 2 ) )
85	84	iftrued	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝑥 𝐿 ( 2 · 𝑦 ) ) , ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) = ( 𝑥 𝐿 ( 2 · 𝑦 ) ) )
86		simpl	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → 𝑥 = 𝑠 )
87	83	oveq2d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → ( 2 · 𝑦 ) = ( 2 · 0 ) )
88		2t0e0	⊢ ( 2 · 0 ) = 0
89	87 88	eqtrdi	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → ( 2 · 𝑦 ) = 0 )
90	86 89	oveq12d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → ( 𝑥 𝐿 ( 2 · 𝑦 ) ) = ( 𝑠 𝐿 0 ) )
91	85 90	eqtrd	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝑥 𝐿 ( 2 · 𝑦 ) ) , ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) = ( 𝑠 𝐿 0 ) )
92		ovex	⊢ ( 𝑠 𝐿 0 ) ∈ V
93	91 1 92	ovmpoa	⊢ ( ( 𝑠 ∈ 𝑋 ∧ 0 ∈ ( 0 [,] 1 ) ) → ( 𝑠 𝑁 0 ) = ( 𝑠 𝐿 0 ) )
94	81 82 93	sylancl	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝑁 0 ) = ( 𝑠 𝐿 0 ) )
95	24	simpld	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝐿 0 ) = ( 𝐹 ‘ 𝑠 ) )
96	94 95	eqtrd	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝑁 0 ) = ( 𝐹 ‘ 𝑠 ) )
97		1elunit	⊢ 1 ∈ ( 0 [,] 1 )
98	16 18	ltnlei	⊢ ( ( 1 / 2 ) < 1 ↔ ¬ 1 ≤ ( 1 / 2 ) )
99	19 98	mpbi	⊢ ¬ 1 ≤ ( 1 / 2 )
100		simpr	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → 𝑦 = 1 )
101	100	breq1d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( 𝑦 ≤ ( 1 / 2 ) ↔ 1 ≤ ( 1 / 2 ) ) )
102	99 101	mtbiri	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ¬ 𝑦 ≤ ( 1 / 2 ) )
103	102	iffalsed	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝑥 𝐿 ( 2 · 𝑦 ) ) , ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) = ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) )
104		simpl	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → 𝑥 = 𝑠 )
105	100	oveq2d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( 2 · 𝑦 ) = ( 2 · 1 ) )
106		2t1e2	⊢ ( 2 · 1 ) = 2
107	105 106	eqtrdi	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( 2 · 𝑦 ) = 2 )
108	107	oveq1d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( ( 2 · 𝑦 ) − 1 ) = ( 2 − 1 ) )
109		2m1e1	⊢ ( 2 − 1 ) = 1
110	108 109	eqtrdi	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( ( 2 · 𝑦 ) − 1 ) = 1 )
111	104 110	oveq12d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) = ( 𝑠 𝑀 1 ) )
112	103 111	eqtrd	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → if ( 𝑦 ≤ ( 1 / 2 ) , ( 𝑥 𝐿 ( 2 · 𝑦 ) ) , ( 𝑥 𝑀 ( ( 2 · 𝑦 ) − 1 ) ) ) = ( 𝑠 𝑀 1 ) )
113		ovex	⊢ ( 𝑠 𝑀 1 ) ∈ V
114	112 1 113	ovmpoa	⊢ ( ( 𝑠 ∈ 𝑋 ∧ 1 ∈ ( 0 [,] 1 ) ) → ( 𝑠 𝑁 1 ) = ( 𝑠 𝑀 1 ) )
115	81 97 114	sylancl	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝑁 1 ) = ( 𝑠 𝑀 1 ) )
116	26	simprd	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝑀 1 ) = ( 𝐻 ‘ 𝑠 ) )
117	115 116	eqtrd	⊢ ( ( 𝜑 ∧ 𝑠 ∈ 𝑋 ) → ( 𝑠 𝑁 1 ) = ( 𝐻 ‘ 𝑠 ) )
118	2 3 5 80 96 117	ishtpyd	⊢ ( 𝜑 → 𝑁 ∈ ( 𝐹 ( 𝐽 Htpy 𝐾 ) 𝐻 ) )

Metamath Proof Explorer

Theorem htpycc

Proof