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Theorem pcohtpylem 23624

Description: Lemma for pcohtpy 23625. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)

**Hypotheses**
Ref	Expression
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)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)))
pcohtpylem.8	⊢ (𝜑 → 𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))
pcohtpylem.9	⊢ (𝜑 → 𝑁 ∈ (𝐺(PHtpy‘𝐽)𝐾))

**Assertion**
Ref	Expression
pcohtpylem	⊢ (𝜑 → 𝑃 ∈ ((𝐹(_𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(_𝑝‘𝐽)𝐾)))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝑀,𝑦 𝑥,𝑁,𝑦 𝜑,𝑥,𝑦 𝑥,𝐺,𝑦 𝑥,𝐻,𝑦 𝑥,𝐽,𝑦 𝑥,𝐾,𝑦 Allowed substitution hints: 𝑃(𝑥,𝑦)

Proof of Theorem pcohtpylem Dummy variables 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcohtpy.5	. . . . 5 ⊢ (𝜑 → 𝐹( ≃_ph‘𝐽)𝐻)
2		isphtpc 23599	. . . . 5 ⊢ (𝐹( ≃_ph‘𝐽)𝐻 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
3	1, 2	sylib 221	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
4	3	simp1d 1139	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
5		pcohtpy.6	. . . . 5 ⊢ (𝜑 → 𝐺( ≃_ph‘𝐽)𝐾)
6		isphtpc 23599	. . . . 5 ⊢ (𝐺( ≃_ph‘𝐽)𝐾 ↔ (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
7	5, 6	sylib 221	. . . 4 ⊢ (𝜑 → (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
8	7	simp1d 1139	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
9		pcohtpy.4	. . 3 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))
10	4, 8, 9	pcocn 23622	. 2 ⊢ (𝜑 → (𝐹(*_𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽))
11	3	simp2d 1140	. . 3 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
12	7	simp2d 1140	. . 3 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐽))
13		pcohtpylem.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))
14	4, 11, 13	phtpy01 23590	. . . . 5 ⊢ (𝜑 → ((𝐹‘0) = (𝐻‘0) ∧ (𝐹‘1) = (𝐻‘1)))
15	14	simprd 499	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐻‘1))
16		pcohtpylem.9	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝐺(PHtpy‘𝐽)𝐾))
17	8, 12, 16	phtpy01 23590	. . . . 5 ⊢ (𝜑 → ((𝐺‘0) = (𝐾‘0) ∧ (𝐺‘1) = (𝐾‘1)))
18	17	simpld 498	. . . 4 ⊢ (𝜑 → (𝐺‘0) = (𝐾‘0))
19	9, 15, 18	3eqtr3d 2841	. . 3 ⊢ (𝜑 → (𝐻‘1) = (𝐾‘0))
20	11, 12, 19	pcocn 23622	. 2 ⊢ (𝜑 → (𝐻(*_𝑝‘𝐽)𝐾) ∈ (II Cn 𝐽))
21		pcohtpylem.7	. . 3 ⊢ 𝑃 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)))
22		eqid 2798	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
23		eqid 2798	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
24		eqid 2798	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
25		dfii2 23487	. . . 4 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
26		0red 10633	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
27		1red 10631	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
28		halfre 11839	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
29		halfge0 11842	. . . . . 6 ⊢ 0 ≤ (1 / 2)
30		1re 10630	. . . . . . 7 ⊢ 1 ∈ ℝ
31		halflt1 11843	. . . . . . 7 ⊢ (1 / 2) < 1
32	28, 30, 31	ltleii 10752	. . . . . 6 ⊢ (1 / 2) ≤ 1
33		elicc01 12844	. . . . . 6 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
34	28, 29, 32, 33	mpbir3an 1338	. . . . 5 ⊢ (1 / 2) ∈ (0[,]1)
35	34	a1i 11	. . . 4 ⊢ (𝜑 → (1 / 2) ∈ (0[,]1))
36		iitopon 23484	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
37	36	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
38	9	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (𝐹‘1) = (𝐺‘0))
39	4, 11, 13	phtpyi 23589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → ((0𝑀𝑦) = (𝐹‘0) ∧ (1𝑀𝑦) = (𝐹‘1)))
40	39	simprd 499	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → (1𝑀𝑦) = (𝐹‘1))
41	40	adantrl 715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (1𝑀𝑦) = (𝐹‘1))
42	8, 12, 16	phtpyi 23589	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → ((0𝑁𝑦) = (𝐺‘0) ∧ (1𝑁𝑦) = (𝐺‘1)))
43	42	simpld 498	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → (0𝑁𝑦) = (𝐺‘0))
44	43	adantrl 715	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (0𝑁𝑦) = (𝐺‘0))
45	38, 41, 44	3eqtr4d 2843	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (1𝑀𝑦) = (0𝑁𝑦))
46		simprl 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 = (1 / 2))
47	46	oveq2d 7151	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = (2 · (1 / 2)))
48		2cn 11700	. . . . . . . 8 ⊢ 2 ∈ ℂ
49		2ne0 11729	. . . . . . . 8 ⊢ 2 ≠ 0
50	48, 49	recidi 11360	. . . . . . 7 ⊢ (2 · (1 / 2)) = 1
51	47, 50	eqtrdi 2849	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = 1)
52	51	oveq1d 7150	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (1𝑀𝑦))
53	51	oveq1d 7150	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = (1 − 1))
54		1m1e0 11697	. . . . . . 7 ⊢ (1 − 1) = 0
55	53, 54	eqtrdi 2849	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = 0)
56	55	oveq1d 7150	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (((2 · 𝑥) − 1)𝑁𝑦) = (0𝑁𝑦))
57	45, 52, 56	3eqtr4d 2843	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (((2 · 𝑥) − 1)𝑁𝑦))
58		retopon 23369	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
59		0re 10632	. . . . . . . 8 ⊢ 0 ∈ ℝ
60		iccssre 12807	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
61	59, 28, 60	mp2an 691	. . . . . . 7 ⊢ (0[,](1 / 2)) ⊆ ℝ
62		resttopon 21766	. . . . . . 7 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
63	58, 61, 62	mp2an 691	. . . . . 6 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
64	63	a1i 11	. . . . 5 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
65	64, 37	cnmpt1st 22273	. . . . . 6 ⊢ (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 23537	. . . . . . 7 ⊢ (𝑧 ∈ (0[,](1 / 2)) ↦ (2 · 𝑧)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II)
67	66	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ (0[,](1 / 2)) ↦ (2 · 𝑧)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II))
68		oveq2 7143	. . . . . 6 ⊢ (𝑧 = 𝑥 → (2 · 𝑧) = (2 · 𝑥))
69	64, 37, 65, 64, 67, 68	cnmpt21 22276	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ (2 · 𝑥)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
70	64, 37	cnmpt2nd 22274	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
71	4, 11	phtpycn 23588	. . . . . 6 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
72	71, 13	sseldd 3916	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((II ×_t II) Cn 𝐽))
73	64, 37, 69, 70, 72	cnmpt22f 22280	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ ((2 · 𝑥)𝑀𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn 𝐽))
74		iccssre 12807	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
75	28, 30, 74	mp2an 691	. . . . . . 7 ⊢ ((1 / 2)[,]1) ⊆ ℝ
76		resttopon 21766	. . . . . . 7 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
77	58, 75, 76	mp2an 691	. . . . . 6 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
78	77	a1i 11	. . . . 5 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
79	78, 37	cnmpt1st 22273	. . . . . 6 ⊢ (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 23539	. . . . . . 7 ⊢ (𝑧 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑧) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II)
81	80	a1i 11	. . . . . 6 ⊢ (𝜑 → (𝑧 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑧) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II))
82	68	oveq1d 7150	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((2 · 𝑧) − 1) = ((2 · 𝑥) − 1))
83	78, 37, 79, 78, 81, 82	cnmpt21 22276	. . . . 5 ⊢ (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 22274	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((1 / 2)[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn II))
85	8, 12	phtpycn 23588	. . . . . 6 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ ((II ×_t II) Cn 𝐽))
86	85, 16	sseldd 3916	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ((II ×_t II) Cn 𝐽))
87	78, 37, 83, 84, 86	cnmpt22f 22280	. . . 4 ⊢ (𝜑 → (𝑥 ∈ ((1 / 2)[,]1), 𝑦 ∈ (0[,]1) ↦ (((2 · 𝑥) − 1)𝑁𝑦)) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn 𝐽))
88	22, 23, 24, 25, 26, 27, 35, 37, 57, 73, 87	cnmpopc 23533	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦))) ∈ ((II ×_t II) Cn 𝐽))
89	21, 88	eqeltrid 2894	. 2 ⊢ (𝜑 → 𝑃 ∈ ((II ×_t II) Cn 𝐽))
90		simpll 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → 𝜑)
91		elii1 23540	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) ↔ (𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)))
92		iihalf1 23536	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
93	91, 92	sylbir 238	. . . . . . 7 ⊢ ((𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
94	93	adantll 713	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
95	4, 11	phtpyhtpy 23587	. . . . . . . 8 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ (𝐹(II Htpy 𝐽)𝐻))
96	95, 13	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝐹(II Htpy 𝐽)𝐻))
97	37, 4, 11, 96	htpyi 23579	. . . . . 6 ⊢ ((𝜑 ∧ (2 · 𝑠) ∈ (0[,]1)) → (((2 · 𝑠)𝑀0) = (𝐹‘(2 · 𝑠)) ∧ ((2 · 𝑠)𝑀1) = (𝐻‘(2 · 𝑠))))
98	90, 94, 97	syl2anc 587	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → (((2 · 𝑠)𝑀0) = (𝐹‘(2 · 𝑠)) ∧ ((2 · 𝑠)𝑀1) = (𝐻‘(2 · 𝑠))))
99	98	simpld 498	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → ((2 · 𝑠)𝑀0) = (𝐹‘(2 · 𝑠)))
100		simpll 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝜑)
101		elii2 23541	. . . . . . . 8 ⊢ ((𝑠 ∈ (0[,]1) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝑠 ∈ ((1 / 2)[,]1))
102	101	adantll 713	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝑠 ∈ ((1 / 2)[,]1))
103		iihalf2 23538	. . . . . . 7 ⊢ (𝑠 ∈ ((1 / 2)[,]1) → ((2 · 𝑠) − 1) ∈ (0[,]1))
104	102, 103	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → ((2 · 𝑠) − 1) ∈ (0[,]1))
105	8, 12	phtpyhtpy 23587	. . . . . . . 8 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ (𝐺(II Htpy 𝐽)𝐾))
106	105, 16	sseldd 3916	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝐺(II Htpy 𝐽)𝐾))
107	37, 8, 12, 106	htpyi 23579	. . . . . 6 ⊢ ((𝜑 ∧ ((2 · 𝑠) − 1) ∈ (0[,]1)) → ((((2 · 𝑠) − 1)𝑁0) = (𝐺‘((2 · 𝑠) − 1)) ∧ (((2 · 𝑠) − 1)𝑁1) = (𝐾‘((2 · 𝑠) − 1))))
108	100, 104, 107	syl2anc 587	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → ((((2 · 𝑠) − 1)𝑁0) = (𝐺‘((2 · 𝑠) − 1)) ∧ (((2 · 𝑠) − 1)𝑁1) = (𝐾‘((2 · 𝑠) − 1))))
109	108	simpld 498	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → (((2 · 𝑠) − 1)𝑁0) = (𝐺‘((2 · 𝑠) − 1)))
110	99, 109	ifeq12da 4457	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)) = if(𝑠 ≤ (1 / 2), (𝐹‘(2 · 𝑠)), (𝐺‘((2 · 𝑠) − 1))))
111		simpr 488	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
112		0elunit 12847	. . . 4 ⊢ 0 ∈ (0[,]1)
113		simpl 486	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
114	113	breq1d 5040	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
115	113	oveq2d 7151	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑥) = (2 · 𝑠))
116		simpr 488	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
117	115, 116	oveq12d 7153	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀0))
118	115	oveq1d 7150	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
119	118, 116	oveq12d 7153	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁0))
120	114, 117, 119	ifbieq12d 4452	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)))
121		ovex 7168	. . . . . 6 ⊢ ((2 · 𝑠)𝑀0) ∈ V
122		ovex 7168	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁0) ∈ V
123	121, 122	ifex 4473	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)) ∈ V
124	120, 21, 123	ovmpoa 7284	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝑃0) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)))
125	111, 112, 124	sylancl 589	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃0) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)))
126	4, 8	pcovalg 23617	. . 3 ⊢ ((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 2843	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃0) = ((𝐹(*_𝑝‘𝐽)𝐺)‘𝑠))
128	98	simprd 499	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → ((2 · 𝑠)𝑀1) = (𝐻‘(2 · 𝑠)))
129	108	simprd 499	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → (((2 · 𝑠) − 1)𝑁1) = (𝐾‘((2 · 𝑠) − 1)))
130	128, 129	ifeq12da 4457	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)) = if(𝑠 ≤ (1 / 2), (𝐻‘(2 · 𝑠)), (𝐾‘((2 · 𝑠) − 1))))
131		1elunit 12848	. . . 4 ⊢ 1 ∈ (0[,]1)
132		simpl 486	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
133	132	breq1d 5040	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
134	132	oveq2d 7151	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑥) = (2 · 𝑠))
135		simpr 488	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
136	134, 135	oveq12d 7153	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀1))
137	134	oveq1d 7150	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
138	137, 135	oveq12d 7153	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁1))
139	133, 136, 138	ifbieq12d 4452	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)))
140		ovex 7168	. . . . . 6 ⊢ ((2 · 𝑠)𝑀1) ∈ V
141		ovex 7168	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁1) ∈ V
142	140, 141	ifex 4473	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)) ∈ V
143	139, 21, 142	ovmpoa 7284	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝑃1) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)))
144	111, 131, 143	sylancl 589	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃1) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)))
145	11, 12	pcovalg 23617	. . 3 ⊢ ((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 2843	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃1) = ((𝐻(*_𝑝‘𝐽)𝐾)‘𝑠))
147	4, 11, 13	phtpyi 23589	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝑀𝑠) = (𝐹‘0) ∧ (1𝑀𝑠) = (𝐹‘1)))
148	147	simpld 498	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑀𝑠) = (𝐹‘0))
149		simpl 486	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 = 0)
150	149, 29	eqbrtrdi 5069	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 ≤ (1 / 2))
151	150	iftrued 4433	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = ((2 · 𝑥)𝑀𝑦))
152	149	oveq2d 7151	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 0))
153		2t0e0 11794	. . . . . . . 8 ⊢ (2 · 0) = 0
154	152, 153	eqtrdi 2849	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 0)
155		simpr 488	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
156	154, 155	oveq12d 7153	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((2 · 𝑥)𝑀𝑦) = (0𝑀𝑠))
157	151, 156	eqtrd 2833	. . . . 5 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (0𝑀𝑠))
158		ovex 7168	. . . . 5 ⊢ (0𝑀𝑠) ∈ V
159	157, 21, 158	ovmpoa 7284	. . . 4 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = (0𝑀𝑠))
160	112, 111, 159	sylancr 590	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = (0𝑀𝑠))
161	4, 8	pco0 23619	. . . 4 ⊢ (𝜑 → ((𝐹(*_𝑝‘𝐽)𝐺)‘0) = (𝐹‘0))
162	161	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹(*_𝑝‘𝐽)𝐺)‘0) = (𝐹‘0))
163	148, 160, 162	3eqtr4d 2843	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘0))
164	8, 12, 16	phtpyi 23589	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝑁𝑠) = (𝐺‘0) ∧ (1𝑁𝑠) = (𝐺‘1)))
165	164	simprd 499	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑁𝑠) = (𝐺‘1))
166	28, 30	ltnlei 10750	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
167	31, 166	mpbi 233	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
168		simpl 486	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1)
169	168	breq1d 5040	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
170	167, 169	mtbiri 330	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ¬ 𝑥 ≤ (1 / 2))
171	170	iffalsed 4436	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (((2 · 𝑥) − 1)𝑁𝑦))
172	168	oveq2d 7151	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 1))
173		2t1e2 11788	. . . . . . . . . 10 ⊢ (2 · 1) = 2
174	172, 173	eqtrdi 2849	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 2)
175	174	oveq1d 7150	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = (2 − 1))
176		2m1e1 11751	. . . . . . . 8 ⊢ (2 − 1) = 1
177	175, 176	eqtrdi 2849	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = 1)
178		simpr 488	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
179	177, 178	oveq12d 7153	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((2 · 𝑥) − 1)𝑁𝑦) = (1𝑁𝑠))
180	171, 179	eqtrd 2833	. . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (1𝑁𝑠))
181		ovex 7168	. . . . 5 ⊢ (1𝑁𝑠) ∈ V
182	180, 21, 181	ovmpoa 7284	. . . 4 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = (1𝑁𝑠))
183	131, 111, 182	sylancr 590	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = (1𝑁𝑠))
184	4, 8	pco1 23620	. . . 4 ⊢ (𝜑 → ((𝐹(*_𝑝‘𝐽)𝐺)‘1) = (𝐺‘1))
185	184	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹(*_𝑝‘𝐽)𝐺)‘1) = (𝐺‘1))
186	165, 183, 185	3eqtr4d 2843	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘1))
187	10, 20, 89, 127, 146, 163, 186	isphtpy2d 23592	1 ⊢ (𝜑 → 𝑃 ∈ ((𝐹(_𝑝‘𝐽)𝐺)(PHtpy‘𝐽)(𝐻(_𝑝‘𝐽)𝐾)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 ifcif 4425 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ℝcr 10525 0cc0 10526 1c1 10527 · cmul 10531 < clt 10664 ≤ cle 10665 − cmin 10859 / cdiv 11286 2c2 11680 (,)cioo 12726 [,]cicc 12729 ↾_t crest 16686 topGenctg 16703 TopOnctopon 21515 Cn ccn 21829 ×_t ctx 22165 IIcii 23480 Htpy chtpy 23572 PHtpycphtpy 23573 ≃_phcphtpc 23574 *_𝑝cpco 23605

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-cn 21832 df-cnp 21833 df-tx 22167 df-hmeo 22360 df-xms 22927 df-ms 22928 df-tms 22929 df-ii 23482 df-htpy 23575 df-phtpy 23576 df-phtpc 23597 df-pco 23610

This theorem is referenced by: pcohtpy 23625

W3C validator