Theorem pcohtpylem

Description: Lemma for pcohtpy . (Contributed by Jeff Madsen, 15-Jun-2010) (Revised by Mario Carneiro, 24-Feb-2015)

	Ref	Expression
Hypotheses	pcohtpy.4	⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 0 ) )
	pcohtpy.5	⊢ ( 𝜑 → 𝐹 ( ≃ph ‘ 𝐽 ) 𝐻 )
	pcohtpy.6	⊢ ( 𝜑 → 𝐺 ( ≃ph ‘ 𝐽 ) 𝐾 )
	pcohtpylem.7	⊢ 𝑃 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) )
	pcohtpylem.8	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐻 ) )
	pcohtpylem.9	⊢ ( 𝜑 → 𝑁 ∈ ( 𝐺 ( PHtpy ‘ 𝐽 ) 𝐾 ) )
Assertion	pcohtpylem	⊢ ( 𝜑 → 𝑃 ∈ ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ( PHtpy ‘ 𝐽 ) ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pcohtpy.4	⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 0 ) )
2		pcohtpy.5	⊢ ( 𝜑 → 𝐹 ( ≃ph ‘ 𝐽 ) 𝐻 )
3		pcohtpy.6	⊢ ( 𝜑 → 𝐺 ( ≃ph ‘ 𝐽 ) 𝐾 )
4		pcohtpylem.7	⊢ 𝑃 = ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) )
5		pcohtpylem.8	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐻 ) )
6		pcohtpylem.9	⊢ ( 𝜑 → 𝑁 ∈ ( 𝐺 ( PHtpy ‘ 𝐽 ) 𝐾 ) )
7		isphtpc	⊢ ( 𝐹 ( ≃ph ‘ 𝐽 ) 𝐻 ↔ ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ 𝐻 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐻 ) ≠ ∅ ) )
8	2 7	sylib	⊢ ( 𝜑 → ( 𝐹 ∈ ( II Cn 𝐽 ) ∧ 𝐻 ∈ ( II Cn 𝐽 ) ∧ ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐻 ) ≠ ∅ ) )
9	8	simp1d	⊢ ( 𝜑 → 𝐹 ∈ ( II Cn 𝐽 ) )
10		isphtpc	⊢ ( 𝐺 ( ≃ph ‘ 𝐽 ) 𝐾 ↔ ( 𝐺 ∈ ( II Cn 𝐽 ) ∧ 𝐾 ∈ ( II Cn 𝐽 ) ∧ ( 𝐺 ( PHtpy ‘ 𝐽 ) 𝐾 ) ≠ ∅ ) )
11	3 10	sylib	⊢ ( 𝜑 → ( 𝐺 ∈ ( II Cn 𝐽 ) ∧ 𝐾 ∈ ( II Cn 𝐽 ) ∧ ( 𝐺 ( PHtpy ‘ 𝐽 ) 𝐾 ) ≠ ∅ ) )
12	11	simp1d	⊢ ( 𝜑 → 𝐺 ∈ ( II Cn 𝐽 ) )
13	9 12 1	pcocn	⊢ ( 𝜑 → ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ∈ ( II Cn 𝐽 ) )
14	8	simp2d	⊢ ( 𝜑 → 𝐻 ∈ ( II Cn 𝐽 ) )
15	11	simp2d	⊢ ( 𝜑 → 𝐾 ∈ ( II Cn 𝐽 ) )
16	9 14 5	phtpy01	⊢ ( 𝜑 → ( ( 𝐹 ‘ 0 ) = ( 𝐻 ‘ 0 ) ∧ ( 𝐹 ‘ 1 ) = ( 𝐻 ‘ 1 ) ) )
17	16	simprd	⊢ ( 𝜑 → ( 𝐹 ‘ 1 ) = ( 𝐻 ‘ 1 ) )
18	12 15 6	phtpy01	⊢ ( 𝜑 → ( ( 𝐺 ‘ 0 ) = ( 𝐾 ‘ 0 ) ∧ ( 𝐺 ‘ 1 ) = ( 𝐾 ‘ 1 ) ) )
19	18	simpld	⊢ ( 𝜑 → ( 𝐺 ‘ 0 ) = ( 𝐾 ‘ 0 ) )
20	1 17 19	3eqtr3d	⊢ ( 𝜑 → ( 𝐻 ‘ 1 ) = ( 𝐾 ‘ 0 ) )
21	14 15 20	pcocn	⊢ ( 𝜑 → ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ∈ ( II Cn 𝐽 ) )
22		eqid	⊢ ( topGen ‘ ran (,) ) = ( topGen ‘ ran (,) )
23		eqid	⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) )
24		eqid	⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) = ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) )
25		dfii2	⊢ II = ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] 1 ) )
26		0red	⊢ ( 𝜑 → 0 ∈ ℝ )
27		1red	⊢ ( 𝜑 → 1 ∈ ℝ )
28		halfre	⊢ ( 1 / 2 ) ∈ ℝ
29		halfge0	⊢ 0 ≤ ( 1 / 2 )
30		1re	⊢ 1 ∈ ℝ
31		halflt1	⊢ ( 1 / 2 ) < 1
32	28 30 31	ltleii	⊢ ( 1 / 2 ) ≤ 1
33		elicc01	⊢ ( ( 1 / 2 ) ∈ ( 0 [,] 1 ) ↔ ( ( 1 / 2 ) ∈ ℝ ∧ 0 ≤ ( 1 / 2 ) ∧ ( 1 / 2 ) ≤ 1 ) )
34	28 29 32 33	mpbir3an	⊢ ( 1 / 2 ) ∈ ( 0 [,] 1 )
35	34	a1i	⊢ ( 𝜑 → ( 1 / 2 ) ∈ ( 0 [,] 1 ) )
36		iitopon	⊢ II ∈ ( TopOn ‘ ( 0 [,] 1 ) )
37	36	a1i	⊢ ( 𝜑 → II ∈ ( TopOn ‘ ( 0 [,] 1 ) ) )
38	1	adantr	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( 𝐹 ‘ 1 ) = ( 𝐺 ‘ 0 ) )
39	9 14 5	phtpyi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝑀 𝑦 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝑀 𝑦 ) = ( 𝐹 ‘ 1 ) ) )
40	39	simprd	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 [,] 1 ) ) → ( 1 𝑀 𝑦 ) = ( 𝐹 ‘ 1 ) )
41	40	adantrl	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( 1 𝑀 𝑦 ) = ( 𝐹 ‘ 1 ) )
42	12 15 6	phtpyi	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝑁 𝑦 ) = ( 𝐺 ‘ 0 ) ∧ ( 1 𝑁 𝑦 ) = ( 𝐺 ‘ 1 ) ) )
43	42	simpld	⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 0 [,] 1 ) ) → ( 0 𝑁 𝑦 ) = ( 𝐺 ‘ 0 ) )
44	43	adantrl	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( 0 𝑁 𝑦 ) = ( 𝐺 ‘ 0 ) )
45	38 41 44	3eqtr4d	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( 1 𝑀 𝑦 ) = ( 0 𝑁 𝑦 ) )
46		simprl	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → 𝑥 = ( 1 / 2 ) )
47	46	oveq2d	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑥 ) = ( 2 · ( 1 / 2 ) ) )
48		2cn	⊢ 2 ∈ ℂ
49		2ne0	⊢ 2 ≠ 0
50	48 49	recidi	⊢ ( 2 · ( 1 / 2 ) ) = 1
51	47 50	eqtrdi	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( 2 · 𝑥 ) = 1 )
52	51	oveq1d	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( 2 · 𝑥 ) 𝑀 𝑦 ) = ( 1 𝑀 𝑦 ) )
53	51	oveq1d	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( 2 · 𝑥 ) − 1 ) = ( 1 − 1 ) )
54		1m1e0	⊢ ( 1 − 1 ) = 0
55	53 54	eqtrdi	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( 2 · 𝑥 ) − 1 ) = 0 )
56	55	oveq1d	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) = ( 0 𝑁 𝑦 ) )
57	45 52 56	3eqtr4d	⊢ ( ( 𝜑 ∧ ( 𝑥 = ( 1 / 2 ) ∧ 𝑦 ∈ ( 0 [,] 1 ) ) ) → ( ( 2 · 𝑥 ) 𝑀 𝑦 ) = ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) )
58		retopon	⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )
59		0re	⊢ 0 ∈ ℝ
60		iccssre	⊢ ( ( 0 ∈ ℝ ∧ ( 1 / 2 ) ∈ ℝ ) → ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ )
61	59 28 60	mp2an	⊢ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ
62		resttopon	⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( 0 [,] ( 1 / 2 ) ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) )
63	58 61 62	mp2an	⊢ ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) )
64	63	a1i	⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ∈ ( TopOn ‘ ( 0 [,] ( 1 / 2 ) ) ) )
65	64 37	cnmpt1st	⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ 𝑥 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ) )
66	23	iihalf1cn	⊢ ( 𝑧 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑧 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II )
67	66	a1i	⊢ ( 𝜑 → ( 𝑧 ∈ ( 0 [,] ( 1 / 2 ) ) ↦ ( 2 · 𝑧 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) Cn II ) )
68		oveq2	⊢ ( 𝑧 = 𝑥 → ( 2 · 𝑧 ) = ( 2 · 𝑥 ) )
69	64 37 65 64 67 68	cnmpt21	⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( 2 · 𝑥 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn II ) )
70	64 37	cnmpt2nd	⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn II ) )
71	9 14	phtpycn	⊢ ( 𝜑 → ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐻 ) ⊆ ( ( II ×t II ) Cn 𝐽 ) )
72	71 5	sseldd	⊢ ( 𝜑 → 𝑀 ∈ ( ( II ×t II ) Cn 𝐽 ) )
73	64 37 69 70 72	cnmpt22f	⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] ( 1 / 2 ) ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( 2 · 𝑥 ) 𝑀 𝑦 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( 0 [,] ( 1 / 2 ) ) ) ×t II ) Cn 𝐽 ) )
74		iccssre	⊢ ( ( ( 1 / 2 ) ∈ ℝ ∧ 1 ∈ ℝ ) → ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ )
75	28 30 74	mp2an	⊢ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ
76		resttopon	⊢ ( ( ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ ) ∧ ( ( 1 / 2 ) [,] 1 ) ⊆ ℝ ) → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) )
77	58 75 76	mp2an	⊢ ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) )
78	77	a1i	⊢ ( 𝜑 → ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ∈ ( TopOn ‘ ( ( 1 / 2 ) [,] 1 ) ) )
79	78 37	cnmpt1st	⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ 𝑥 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ) )
80	24	iihalf2cn	⊢ ( 𝑧 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 2 · 𝑧 ) − 1 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II )
81	80	a1i	⊢ ( 𝜑 → ( 𝑧 ∈ ( ( 1 / 2 ) [,] 1 ) ↦ ( ( 2 · 𝑧 ) − 1 ) ) ∈ ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) Cn II ) )
82	68	oveq1d	⊢ ( 𝑧 = 𝑥 → ( ( 2 · 𝑧 ) − 1 ) = ( ( 2 · 𝑥 ) − 1 ) )
83	78 37 79 78 81 82	cnmpt21	⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( 2 · 𝑥 ) − 1 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn II ) )
84	78 37	cnmpt2nd	⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ 𝑦 ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn II ) )
85	12 15	phtpycn	⊢ ( 𝜑 → ( 𝐺 ( PHtpy ‘ 𝐽 ) 𝐾 ) ⊆ ( ( II ×t II ) Cn 𝐽 ) )
86	85 6	sseldd	⊢ ( 𝜑 → 𝑁 ∈ ( ( II ×t II ) Cn 𝐽 ) )
87	78 37 83 84 86	cnmpt22f	⊢ ( 𝜑 → ( 𝑥 ∈ ( ( 1 / 2 ) [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) ∈ ( ( ( ( topGen ‘ ran (,) ) ↾t ( ( 1 / 2 ) [,] 1 ) ) ×t II ) Cn 𝐽 ) )
88	22 23 24 25 26 27 35 37 57 73 87	cnmpopc	⊢ ( 𝜑 → ( 𝑥 ∈ ( 0 [,] 1 ) , 𝑦 ∈ ( 0 [,] 1 ) ↦ if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) ) ∈ ( ( II ×t II ) Cn 𝐽 ) )
89	4 88	eqeltrid	⊢ ( 𝜑 → 𝑃 ∈ ( ( II ×t II ) Cn 𝐽 ) )
90		simpll	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ 𝑠 ≤ ( 1 / 2 ) ) → 𝜑 )
91		elii1	⊢ ( 𝑠 ∈ ( 0 [,] ( 1 / 2 ) ) ↔ ( 𝑠 ∈ ( 0 [,] 1 ) ∧ 𝑠 ≤ ( 1 / 2 ) ) )
92		iihalf1	⊢ ( 𝑠 ∈ ( 0 [,] ( 1 / 2 ) ) → ( 2 · 𝑠 ) ∈ ( 0 [,] 1 ) )
93	91 92	sylbir	⊢ ( ( 𝑠 ∈ ( 0 [,] 1 ) ∧ 𝑠 ≤ ( 1 / 2 ) ) → ( 2 · 𝑠 ) ∈ ( 0 [,] 1 ) )
94	93	adantll	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ 𝑠 ≤ ( 1 / 2 ) ) → ( 2 · 𝑠 ) ∈ ( 0 [,] 1 ) )
95	9 14	phtpyhtpy	⊢ ( 𝜑 → ( 𝐹 ( PHtpy ‘ 𝐽 ) 𝐻 ) ⊆ ( 𝐹 ( II Htpy 𝐽 ) 𝐻 ) )
96	95 5	sseldd	⊢ ( 𝜑 → 𝑀 ∈ ( 𝐹 ( II Htpy 𝐽 ) 𝐻 ) )
97	37 9 14 96	htpyi	⊢ ( ( 𝜑 ∧ ( 2 · 𝑠 ) ∈ ( 0 [,] 1 ) ) → ( ( ( 2 · 𝑠 ) 𝑀 0 ) = ( 𝐹 ‘ ( 2 · 𝑠 ) ) ∧ ( ( 2 · 𝑠 ) 𝑀 1 ) = ( 𝐻 ‘ ( 2 · 𝑠 ) ) ) )
98	90 94 97	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ 𝑠 ≤ ( 1 / 2 ) ) → ( ( ( 2 · 𝑠 ) 𝑀 0 ) = ( 𝐹 ‘ ( 2 · 𝑠 ) ) ∧ ( ( 2 · 𝑠 ) 𝑀 1 ) = ( 𝐻 ‘ ( 2 · 𝑠 ) ) ) )
99	98	simpld	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ 𝑠 ≤ ( 1 / 2 ) ) → ( ( 2 · 𝑠 ) 𝑀 0 ) = ( 𝐹 ‘ ( 2 · 𝑠 ) ) )
100		simpll	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑠 ≤ ( 1 / 2 ) ) → 𝜑 )
101		elii2	⊢ ( ( 𝑠 ∈ ( 0 [,] 1 ) ∧ ¬ 𝑠 ≤ ( 1 / 2 ) ) → 𝑠 ∈ ( ( 1 / 2 ) [,] 1 ) )
102	101	adantll	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑠 ≤ ( 1 / 2 ) ) → 𝑠 ∈ ( ( 1 / 2 ) [,] 1 ) )
103		iihalf2	⊢ ( 𝑠 ∈ ( ( 1 / 2 ) [,] 1 ) → ( ( 2 · 𝑠 ) − 1 ) ∈ ( 0 [,] 1 ) )
104	102 103	syl	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑠 ≤ ( 1 / 2 ) ) → ( ( 2 · 𝑠 ) − 1 ) ∈ ( 0 [,] 1 ) )
105	12 15	phtpyhtpy	⊢ ( 𝜑 → ( 𝐺 ( PHtpy ‘ 𝐽 ) 𝐾 ) ⊆ ( 𝐺 ( II Htpy 𝐽 ) 𝐾 ) )
106	105 6	sseldd	⊢ ( 𝜑 → 𝑁 ∈ ( 𝐺 ( II Htpy 𝐽 ) 𝐾 ) )
107	37 12 15 106	htpyi	⊢ ( ( 𝜑 ∧ ( ( 2 · 𝑠 ) − 1 ) ∈ ( 0 [,] 1 ) ) → ( ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) = ( 𝐺 ‘ ( ( 2 · 𝑠 ) − 1 ) ) ∧ ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) = ( 𝐾 ‘ ( ( 2 · 𝑠 ) − 1 ) ) ) )
108	100 104 107	syl2anc	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑠 ≤ ( 1 / 2 ) ) → ( ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) = ( 𝐺 ‘ ( ( 2 · 𝑠 ) − 1 ) ) ∧ ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) = ( 𝐾 ‘ ( ( 2 · 𝑠 ) − 1 ) ) ) )
109	108	simpld	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑠 ≤ ( 1 / 2 ) ) → ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) = ( 𝐺 ‘ ( ( 2 · 𝑠 ) − 1 ) ) )
110	99 109	ifeq12da	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 0 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑠 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑠 ) − 1 ) ) ) )
111		simpr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → 𝑠 ∈ ( 0 [,] 1 ) )
112		0elunit	⊢ 0 ∈ ( 0 [,] 1 )
113		simpl	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → 𝑥 = 𝑠 )
114	113	breq1d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → ( 𝑥 ≤ ( 1 / 2 ) ↔ 𝑠 ≤ ( 1 / 2 ) ) )
115	113	oveq2d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → ( 2 · 𝑥 ) = ( 2 · 𝑠 ) )
116		simpr	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → 𝑦 = 0 )
117	115 116	oveq12d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → ( ( 2 · 𝑥 ) 𝑀 𝑦 ) = ( ( 2 · 𝑠 ) 𝑀 0 ) )
118	115	oveq1d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → ( ( 2 · 𝑥 ) − 1 ) = ( ( 2 · 𝑠 ) − 1 ) )
119	118 116	oveq12d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) = ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) )
120	114 117 119	ifbieq12d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 0 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 0 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) ) )
121		ovex	⊢ ( ( 2 · 𝑠 ) 𝑀 0 ) ∈ V
122		ovex	⊢ ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) ∈ V
123	121 122	ifex	⊢ if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 0 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) ) ∈ V
124	120 4 123	ovmpoa	⊢ ( ( 𝑠 ∈ ( 0 [,] 1 ) ∧ 0 ∈ ( 0 [,] 1 ) ) → ( 𝑠 𝑃 0 ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 0 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) ) )
125	111 112 124	sylancl	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 𝑠 𝑃 0 ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 0 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 0 ) ) )
126	9 12	pcovalg	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 𝑠 ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( 𝐹 ‘ ( 2 · 𝑠 ) ) , ( 𝐺 ‘ ( ( 2 · 𝑠 ) − 1 ) ) ) )
127	110 125 126	3eqtr4d	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 𝑠 𝑃 0 ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 𝑠 ) )
128	98	simprd	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ 𝑠 ≤ ( 1 / 2 ) ) → ( ( 2 · 𝑠 ) 𝑀 1 ) = ( 𝐻 ‘ ( 2 · 𝑠 ) ) )
129	108	simprd	⊢ ( ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) ∧ ¬ 𝑠 ≤ ( 1 / 2 ) ) → ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) = ( 𝐾 ‘ ( ( 2 · 𝑠 ) − 1 ) ) )
130	128 129	ifeq12da	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 1 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( 𝐻 ‘ ( 2 · 𝑠 ) ) , ( 𝐾 ‘ ( ( 2 · 𝑠 ) − 1 ) ) ) )
131		1elunit	⊢ 1 ∈ ( 0 [,] 1 )
132		simpl	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → 𝑥 = 𝑠 )
133	132	breq1d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( 𝑥 ≤ ( 1 / 2 ) ↔ 𝑠 ≤ ( 1 / 2 ) ) )
134	132	oveq2d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( 2 · 𝑥 ) = ( 2 · 𝑠 ) )
135		simpr	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → 𝑦 = 1 )
136	134 135	oveq12d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( ( 2 · 𝑥 ) 𝑀 𝑦 ) = ( ( 2 · 𝑠 ) 𝑀 1 ) )
137	134	oveq1d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( ( 2 · 𝑥 ) − 1 ) = ( ( 2 · 𝑠 ) − 1 ) )
138	137 135	oveq12d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) = ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) )
139	133 136 138	ifbieq12d	⊢ ( ( 𝑥 = 𝑠 ∧ 𝑦 = 1 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 1 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) ) )
140		ovex	⊢ ( ( 2 · 𝑠 ) 𝑀 1 ) ∈ V
141		ovex	⊢ ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) ∈ V
142	140 141	ifex	⊢ if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 1 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) ) ∈ V
143	139 4 142	ovmpoa	⊢ ( ( 𝑠 ∈ ( 0 [,] 1 ) ∧ 1 ∈ ( 0 [,] 1 ) ) → ( 𝑠 𝑃 1 ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 1 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) ) )
144	111 131 143	sylancl	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 𝑠 𝑃 1 ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( ( 2 · 𝑠 ) 𝑀 1 ) , ( ( ( 2 · 𝑠 ) − 1 ) 𝑁 1 ) ) )
145	14 15	pcovalg	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ‘ 𝑠 ) = if ( 𝑠 ≤ ( 1 / 2 ) , ( 𝐻 ‘ ( 2 · 𝑠 ) ) , ( 𝐾 ‘ ( ( 2 · 𝑠 ) − 1 ) ) ) )
146	130 144 145	3eqtr4d	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 𝑠 𝑃 1 ) = ( ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ‘ 𝑠 ) )
147	9 14 5	phtpyi	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝑀 𝑠 ) = ( 𝐹 ‘ 0 ) ∧ ( 1 𝑀 𝑠 ) = ( 𝐹 ‘ 1 ) ) )
148	147	simpld	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 0 𝑀 𝑠 ) = ( 𝐹 ‘ 0 ) )
149		simpl	⊢ ( ( 𝑥 = 0 ∧ 𝑦 = 𝑠 ) → 𝑥 = 0 )
150	149 29	eqbrtrdi	⊢ ( ( 𝑥 = 0 ∧ 𝑦 = 𝑠 ) → 𝑥 ≤ ( 1 / 2 ) )
151	150	iftrued	⊢ ( ( 𝑥 = 0 ∧ 𝑦 = 𝑠 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) = ( ( 2 · 𝑥 ) 𝑀 𝑦 ) )
152	149	oveq2d	⊢ ( ( 𝑥 = 0 ∧ 𝑦 = 𝑠 ) → ( 2 · 𝑥 ) = ( 2 · 0 ) )
153		2t0e0	⊢ ( 2 · 0 ) = 0
154	152 153	eqtrdi	⊢ ( ( 𝑥 = 0 ∧ 𝑦 = 𝑠 ) → ( 2 · 𝑥 ) = 0 )
155		simpr	⊢ ( ( 𝑥 = 0 ∧ 𝑦 = 𝑠 ) → 𝑦 = 𝑠 )
156	154 155	oveq12d	⊢ ( ( 𝑥 = 0 ∧ 𝑦 = 𝑠 ) → ( ( 2 · 𝑥 ) 𝑀 𝑦 ) = ( 0 𝑀 𝑠 ) )
157	151 156	eqtrd	⊢ ( ( 𝑥 = 0 ∧ 𝑦 = 𝑠 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) = ( 0 𝑀 𝑠 ) )
158		ovex	⊢ ( 0 𝑀 𝑠 ) ∈ V
159	157 4 158	ovmpoa	⊢ ( ( 0 ∈ ( 0 [,] 1 ) ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 0 𝑃 𝑠 ) = ( 0 𝑀 𝑠 ) )
160	112 111 159	sylancr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 0 𝑃 𝑠 ) = ( 0 𝑀 𝑠 ) )
161	9 12	pco0	⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 0 ) = ( 𝐹 ‘ 0 ) )
162	161	adantr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 0 ) = ( 𝐹 ‘ 0 ) )
163	148 160 162	3eqtr4d	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 0 𝑃 𝑠 ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 0 ) )
164	12 15 6	phtpyi	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( ( 0 𝑁 𝑠 ) = ( 𝐺 ‘ 0 ) ∧ ( 1 𝑁 𝑠 ) = ( 𝐺 ‘ 1 ) ) )
165	164	simprd	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 1 𝑁 𝑠 ) = ( 𝐺 ‘ 1 ) )
166	28 30	ltnlei	⊢ ( ( 1 / 2 ) < 1 ↔ ¬ 1 ≤ ( 1 / 2 ) )
167	31 166	mpbi	⊢ ¬ 1 ≤ ( 1 / 2 )
168		simpl	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → 𝑥 = 1 )
169	168	breq1d	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → ( 𝑥 ≤ ( 1 / 2 ) ↔ 1 ≤ ( 1 / 2 ) ) )
170	167 169	mtbiri	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → ¬ 𝑥 ≤ ( 1 / 2 ) )
171	170	iffalsed	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) = ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) )
172	168	oveq2d	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → ( 2 · 𝑥 ) = ( 2 · 1 ) )
173		2t1e2	⊢ ( 2 · 1 ) = 2
174	172 173	eqtrdi	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → ( 2 · 𝑥 ) = 2 )
175	174	oveq1d	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → ( ( 2 · 𝑥 ) − 1 ) = ( 2 − 1 ) )
176		2m1e1	⊢ ( 2 − 1 ) = 1
177	175 176	eqtrdi	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → ( ( 2 · 𝑥 ) − 1 ) = 1 )
178		simpr	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → 𝑦 = 𝑠 )
179	177 178	oveq12d	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) = ( 1 𝑁 𝑠 ) )
180	171 179	eqtrd	⊢ ( ( 𝑥 = 1 ∧ 𝑦 = 𝑠 ) → if ( 𝑥 ≤ ( 1 / 2 ) , ( ( 2 · 𝑥 ) 𝑀 𝑦 ) , ( ( ( 2 · 𝑥 ) − 1 ) 𝑁 𝑦 ) ) = ( 1 𝑁 𝑠 ) )
181		ovex	⊢ ( 1 𝑁 𝑠 ) ∈ V
182	180 4 181	ovmpoa	⊢ ( ( 1 ∈ ( 0 [,] 1 ) ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 1 𝑃 𝑠 ) = ( 1 𝑁 𝑠 ) )
183	131 111 182	sylancr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 1 𝑃 𝑠 ) = ( 1 𝑁 𝑠 ) )
184	9 12	pco1	⊢ ( 𝜑 → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 1 ) = ( 𝐺 ‘ 1 ) )
185	184	adantr	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 1 ) = ( 𝐺 ‘ 1 ) )
186	165 183 185	3eqtr4d	⊢ ( ( 𝜑 ∧ 𝑠 ∈ ( 0 [,] 1 ) ) → ( 1 𝑃 𝑠 ) = ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ‘ 1 ) )
187	13 21 89 127 146 163 186	isphtpy2d	⊢ ( 𝜑 → 𝑃 ∈ ( ( 𝐹 ( *𝑝 ‘ 𝐽 ) 𝐺 ) ( PHtpy ‘ 𝐽 ) ( 𝐻 ( *𝑝 ‘ 𝐽 ) 𝐾 ) ) )