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Theorem pcohtpylem 25065

Description: Lemma for pcohtpy 25066. (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 25039	. . . . 5 ⊢ (𝐹( ≃_ph‘𝐽)𝐻 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
3	1, 2	sylib 218	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
4	3	simp1d 1141	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
5		pcohtpy.6	. . . . 5 ⊢ (𝜑 → 𝐺( ≃_ph‘𝐽)𝐾)
6		isphtpc 25039	. . . . 5 ⊢ (𝐺( ≃_ph‘𝐽)𝐾 ↔ (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
7	5, 6	sylib 218	. . . 4 ⊢ (𝜑 → (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
8	7	simp1d 1141	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
9		pcohtpy.4	. . 3 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))
10	4, 8, 9	pcocn 25063	. 2 ⊢ (𝜑 → (𝐹(*_𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽))
11	3	simp2d 1142	. . 3 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
12	7	simp2d 1142	. . 3 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐽))
13		pcohtpylem.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))
14	4, 11, 13	phtpy01 25030	. . . . 5 ⊢ (𝜑 → ((𝐹‘0) = (𝐻‘0) ∧ (𝐹‘1) = (𝐻‘1)))
15	14	simprd 495	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐻‘1))
16		pcohtpylem.9	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝐺(PHtpy‘𝐽)𝐾))
17	8, 12, 16	phtpy01 25030	. . . . 5 ⊢ (𝜑 → ((𝐺‘0) = (𝐾‘0) ∧ (𝐺‘1) = (𝐾‘1)))
18	17	simpld 494	. . . 4 ⊢ (𝜑 → (𝐺‘0) = (𝐾‘0))
19	9, 15, 18	3eqtr3d 2782	. . 3 ⊢ (𝜑 → (𝐻‘1) = (𝐾‘0))
20	11, 12, 19	pcocn 25063	. 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 2734	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
23		eqid 2734	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
24		eqid 2734	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
25		dfii2 24921	. . . 4 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
26		0red 11261	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
27		1red 11259	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
28		halfre 12477	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
29		halfge0 12480	. . . . . 6 ⊢ 0 ≤ (1 / 2)
30		1re 11258	. . . . . . 7 ⊢ 1 ∈ ℝ
31		halflt1 12481	. . . . . . 7 ⊢ (1 / 2) < 1
32	28, 30, 31	ltleii 11381	. . . . . 6 ⊢ (1 / 2) ≤ 1
33		elicc01 13502	. . . . . 6 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
34	28, 29, 32, 33	mpbir3an 1340	. . . . 5 ⊢ (1 / 2) ∈ (0[,]1)
35	34	a1i 11	. . . 4 ⊢ (𝜑 → (1 / 2) ∈ (0[,]1))
36		iitopon 24918	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
37	36	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
38	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (𝐹‘1) = (𝐺‘0))
39	4, 11, 13	phtpyi 25029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → ((0𝑀𝑦) = (𝐹‘0) ∧ (1𝑀𝑦) = (𝐹‘1)))
40	39	simprd 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → (1𝑀𝑦) = (𝐹‘1))
41	40	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (1𝑀𝑦) = (𝐹‘1))
42	8, 12, 16	phtpyi 25029	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → ((0𝑁𝑦) = (𝐺‘0) ∧ (1𝑁𝑦) = (𝐺‘1)))
43	42	simpld 494	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → (0𝑁𝑦) = (𝐺‘0))
44	43	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (0𝑁𝑦) = (𝐺‘0))
45	38, 41, 44	3eqtr4d 2784	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (1𝑀𝑦) = (0𝑁𝑦))
46		simprl 771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 = (1 / 2))
47	46	oveq2d 7446	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = (2 · (1 / 2)))
48		2cn 12338	. . . . . . . 8 ⊢ 2 ∈ ℂ
49		2ne0 12367	. . . . . . . 8 ⊢ 2 ≠ 0
50	48, 49	recidi 11995	. . . . . . 7 ⊢ (2 · (1 / 2)) = 1
51	47, 50	eqtrdi 2790	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = 1)
52	51	oveq1d 7445	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (1𝑀𝑦))
53	51	oveq1d 7445	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = (1 − 1))
54		1m1e0 12335	. . . . . . 7 ⊢ (1 − 1) = 0
55	53, 54	eqtrdi 2790	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = 0)
56	55	oveq1d 7445	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (((2 · 𝑥) − 1)𝑁𝑦) = (0𝑁𝑦))
57	45, 52, 56	3eqtr4d 2784	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (((2 · 𝑥) − 1)𝑁𝑦))
58		retopon 24799	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
59		0re 11260	. . . . . . . 8 ⊢ 0 ∈ ℝ
60		iccssre 13465	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
61	59, 28, 60	mp2an 692	. . . . . . 7 ⊢ (0[,](1 / 2)) ⊆ ℝ
62		resttopon 23184	. . . . . . 7 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
63	58, 61, 62	mp2an 692	. . . . . 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 23691	. . . . . 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 24972	. . . . . . 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 7438	. . . . . 6 ⊢ (𝑧 = 𝑥 → (2 · 𝑧) = (2 · 𝑥))
69	64, 37, 65, 64, 67, 68	cnmpt21 23694	. . . . 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 23692	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
71	4, 11	phtpycn 25028	. . . . . 6 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
72	71, 13	sseldd 3995	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((II ×_t II) Cn 𝐽))
73	64, 37, 69, 70, 72	cnmpt22f 23698	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ ((2 · 𝑥)𝑀𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn 𝐽))
74		iccssre 13465	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
75	28, 30, 74	mp2an 692	. . . . . . 7 ⊢ ((1 / 2)[,]1) ⊆ ℝ
76		resttopon 23184	. . . . . . 7 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
77	58, 75, 76	mp2an 692	. . . . . 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 23691	. . . . . 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 24975	. . . . . . 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 7445	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((2 · 𝑧) − 1) = ((2 · 𝑥) − 1))
83	78, 37, 79, 78, 81, 82	cnmpt21 23694	. . . . 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 23692	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((1 / 2)[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn II))
85	8, 12	phtpycn 25028	. . . . . 6 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ ((II ×_t II) Cn 𝐽))
86	85, 16	sseldd 3995	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ((II ×_t II) Cn 𝐽))
87	78, 37, 83, 84, 86	cnmpt22f 23698	. . . 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 24968	. . 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 2842	. 2 ⊢ (𝜑 → 𝑃 ∈ ((II ×_t II) Cn 𝐽))
90		simpll 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → 𝜑)
91		elii1 24977	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) ↔ (𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)))
92		iihalf1 24971	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
93	91, 92	sylbir 235	. . . . . . 7 ⊢ ((𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
94	93	adantll 714	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
95	4, 11	phtpyhtpy 25027	. . . . . . . 8 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ (𝐹(II Htpy 𝐽)𝐻))
96	95, 13	sseldd 3995	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝐹(II Htpy 𝐽)𝐻))
97	37, 4, 11, 96	htpyi 25019	. . . . . 6 ⊢ ((𝜑 ∧ (2 · 𝑠) ∈ (0[,]1)) → (((2 · 𝑠)𝑀0) = (𝐹‘(2 · 𝑠)) ∧ ((2 · 𝑠)𝑀1) = (𝐻‘(2 · 𝑠))))
98	90, 94, 97	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → (((2 · 𝑠)𝑀0) = (𝐹‘(2 · 𝑠)) ∧ ((2 · 𝑠)𝑀1) = (𝐻‘(2 · 𝑠))))
99	98	simpld 494	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → ((2 · 𝑠)𝑀0) = (𝐹‘(2 · 𝑠)))
100		simpll 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝜑)
101		elii2 24978	. . . . . . . 8 ⊢ ((𝑠 ∈ (0[,]1) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝑠 ∈ ((1 / 2)[,]1))
102	101	adantll 714	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝑠 ∈ ((1 / 2)[,]1))
103		iihalf2 24974	. . . . . . 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 25027	. . . . . . . 8 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ (𝐺(II Htpy 𝐽)𝐾))
106	105, 16	sseldd 3995	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝐺(II Htpy 𝐽)𝐾))
107	37, 8, 12, 106	htpyi 25019	. . . . . 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 584	. . . . 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 494	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → (((2 · 𝑠) − 1)𝑁0) = (𝐺‘((2 · 𝑠) − 1)))
110	99, 109	ifeq12da 4563	. . 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 484	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
112		0elunit 13505	. . . 4 ⊢ 0 ∈ (0[,]1)
113		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
114	113	breq1d 5157	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
115	113	oveq2d 7446	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑥) = (2 · 𝑠))
116		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
117	115, 116	oveq12d 7448	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀0))
118	115	oveq1d 7445	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
119	118, 116	oveq12d 7448	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁0))
120	114, 117, 119	ifbieq12d 4558	. . . . 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 7463	. . . . . 6 ⊢ ((2 · 𝑠)𝑀0) ∈ V
122		ovex 7463	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁0) ∈ V
123	121, 122	ifex 4580	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)) ∈ V
124	120, 21, 123	ovmpoa 7587	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝑃0) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)))
125	111, 112, 124	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃0) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)))
126	4, 8	pcovalg 25058	. . 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 2784	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃0) = ((𝐹(*_𝑝‘𝐽)𝐺)‘𝑠))
128	98	simprd 495	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → ((2 · 𝑠)𝑀1) = (𝐻‘(2 · 𝑠)))
129	108	simprd 495	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → (((2 · 𝑠) − 1)𝑁1) = (𝐾‘((2 · 𝑠) − 1)))
130	128, 129	ifeq12da 4563	. . 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 13506	. . . 4 ⊢ 1 ∈ (0[,]1)
132		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
133	132	breq1d 5157	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
134	132	oveq2d 7446	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑥) = (2 · 𝑠))
135		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
136	134, 135	oveq12d 7448	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀1))
137	134	oveq1d 7445	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
138	137, 135	oveq12d 7448	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁1))
139	133, 136, 138	ifbieq12d 4558	. . . . 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 7463	. . . . . 6 ⊢ ((2 · 𝑠)𝑀1) ∈ V
141		ovex 7463	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁1) ∈ V
142	140, 141	ifex 4580	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)) ∈ V
143	139, 21, 142	ovmpoa 7587	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝑃1) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)))
144	111, 131, 143	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃1) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)))
145	11, 12	pcovalg 25058	. . 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 2784	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃1) = ((𝐻(*_𝑝‘𝐽)𝐾)‘𝑠))
147	4, 11, 13	phtpyi 25029	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝑀𝑠) = (𝐹‘0) ∧ (1𝑀𝑠) = (𝐹‘1)))
148	147	simpld 494	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑀𝑠) = (𝐹‘0))
149		simpl 482	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 = 0)
150	149, 29	eqbrtrdi 5186	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 ≤ (1 / 2))
151	150	iftrued 4538	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = ((2 · 𝑥)𝑀𝑦))
152	149	oveq2d 7446	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 0))
153		2t0e0 12432	. . . . . . . 8 ⊢ (2 · 0) = 0
154	152, 153	eqtrdi 2790	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 0)
155		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
156	154, 155	oveq12d 7448	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((2 · 𝑥)𝑀𝑦) = (0𝑀𝑠))
157	151, 156	eqtrd 2774	. . . . 5 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (0𝑀𝑠))
158		ovex 7463	. . . . 5 ⊢ (0𝑀𝑠) ∈ V
159	157, 21, 158	ovmpoa 7587	. . . 4 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = (0𝑀𝑠))
160	112, 111, 159	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = (0𝑀𝑠))
161	4, 8	pco0 25060	. . . 4 ⊢ (𝜑 → ((𝐹(*_𝑝‘𝐽)𝐺)‘0) = (𝐹‘0))
162	161	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹(*_𝑝‘𝐽)𝐺)‘0) = (𝐹‘0))
163	148, 160, 162	3eqtr4d 2784	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘0))
164	8, 12, 16	phtpyi 25029	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝑁𝑠) = (𝐺‘0) ∧ (1𝑁𝑠) = (𝐺‘1)))
165	164	simprd 495	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑁𝑠) = (𝐺‘1))
166	28, 30	ltnlei 11379	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
167	31, 166	mpbi 230	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
168		simpl 482	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1)
169	168	breq1d 5157	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
170	167, 169	mtbiri 327	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ¬ 𝑥 ≤ (1 / 2))
171	170	iffalsed 4541	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (((2 · 𝑥) − 1)𝑁𝑦))
172	168	oveq2d 7446	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 1))
173		2t1e2 12426	. . . . . . . . . 10 ⊢ (2 · 1) = 2
174	172, 173	eqtrdi 2790	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 2)
175	174	oveq1d 7445	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = (2 − 1))
176		2m1e1 12389	. . . . . . . 8 ⊢ (2 − 1) = 1
177	175, 176	eqtrdi 2790	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = 1)
178		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
179	177, 178	oveq12d 7448	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((2 · 𝑥) − 1)𝑁𝑦) = (1𝑁𝑠))
180	171, 179	eqtrd 2774	. . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (1𝑁𝑠))
181		ovex 7463	. . . . 5 ⊢ (1𝑁𝑠) ∈ V
182	180, 21, 181	ovmpoa 7587	. . . 4 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = (1𝑁𝑠))
183	131, 111, 182	sylancr 587	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = (1𝑁𝑠))
184	4, 8	pco1 25061	. . . 4 ⊢ (𝜑 → ((𝐹(*_𝑝‘𝐽)𝐺)‘1) = (𝐺‘1))
185	184	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹(*_𝑝‘𝐽)𝐺)‘1) = (𝐺‘1))
186	165, 183, 185	3eqtr4d 2784	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘1))
187	10, 20, 89, 127, 146, 163, 186	isphtpy2d 25032	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 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ⊆ wss 3962 ∅c0 4338 ifcif 4530 class class class wbr 5147 ↦ cmpt 5230 ran crn 5689 ‘cfv 6562 (class class class)co 7430 ∈ cmpo 7432 ℝcr 11151 0cc0 11152 1c1 11153 · cmul 11157 < clt 11292 ≤ cle 11293 − cmin 11489 / cdiv 11917 2c2 12318 (,)cioo 13383 [,]cicc 13386 ↾_t crest 17466 topGenctg 17483 TopOnctopon 22931 Cn ccn 23247 ×_t ctx 23583 IIcii 24914 Htpy chtpy 25012 PHtpycphtpy 25013 ≃_phcphtpc 25014 *_𝑝cpco 25046

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-cn 23250 df-cnp 23251 df-tx 23585 df-hmeo 23778 df-xms 24345 df-ms 24346 df-tms 24347 df-ii 24916 df-htpy 25015 df-phtpy 25016 df-phtpc 25037 df-pco 25051

This theorem is referenced by: pcohtpy 25066

W3C validator