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Theorem pcohtpylem 24382

Description: Lemma for pcohtpy 24383. (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 24357	. . . . 5 ⊢ (𝐹( ≃_ph‘𝐽)𝐻 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
3	1, 2	sylib 217	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
4	3	simp1d 1142	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
5		pcohtpy.6	. . . . 5 ⊢ (𝜑 → 𝐺( ≃_ph‘𝐽)𝐾)
6		isphtpc 24357	. . . . 5 ⊢ (𝐺( ≃_ph‘𝐽)𝐾 ↔ (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
7	5, 6	sylib 217	. . . 4 ⊢ (𝜑 → (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
8	7	simp1d 1142	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
9		pcohtpy.4	. . 3 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))
10	4, 8, 9	pcocn 24380	. 2 ⊢ (𝜑 → (𝐹(*_𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽))
11	3	simp2d 1143	. . 3 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
12	7	simp2d 1143	. . 3 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐽))
13		pcohtpylem.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))
14	4, 11, 13	phtpy01 24348	. . . . 5 ⊢ (𝜑 → ((𝐹‘0) = (𝐻‘0) ∧ (𝐹‘1) = (𝐻‘1)))
15	14	simprd 496	. . . 4 ⊢ (𝜑 → (𝐹‘1) = (𝐻‘1))
16		pcohtpylem.9	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (𝐺(PHtpy‘𝐽)𝐾))
17	8, 12, 16	phtpy01 24348	. . . . 5 ⊢ (𝜑 → ((𝐺‘0) = (𝐾‘0) ∧ (𝐺‘1) = (𝐾‘1)))
18	17	simpld 495	. . . 4 ⊢ (𝜑 → (𝐺‘0) = (𝐾‘0))
19	9, 15, 18	3eqtr3d 2784	. . 3 ⊢ (𝜑 → (𝐻‘1) = (𝐾‘0))
20	11, 12, 19	pcocn 24380	. 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 2736	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
23		eqid 2736	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
24		eqid 2736	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
25		dfii2 24245	. . . 4 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
26		0red 11158	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
27		1red 11156	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
28		halfre 12367	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
29		halfge0 12370	. . . . . 6 ⊢ 0 ≤ (1 / 2)
30		1re 11155	. . . . . . 7 ⊢ 1 ∈ ℝ
31		halflt1 12371	. . . . . . 7 ⊢ (1 / 2) < 1
32	28, 30, 31	ltleii 11278	. . . . . 6 ⊢ (1 / 2) ≤ 1
33		elicc01 13383	. . . . . 6 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
34	28, 29, 32, 33	mpbir3an 1341	. . . . 5 ⊢ (1 / 2) ∈ (0[,]1)
35	34	a1i 11	. . . 4 ⊢ (𝜑 → (1 / 2) ∈ (0[,]1))
36		iitopon 24242	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
37	36	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
38	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (𝐹‘1) = (𝐺‘0))
39	4, 11, 13	phtpyi 24347	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → ((0𝑀𝑦) = (𝐹‘0) ∧ (1𝑀𝑦) = (𝐹‘1)))
40	39	simprd 496	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → (1𝑀𝑦) = (𝐹‘1))
41	40	adantrl 714	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (1𝑀𝑦) = (𝐹‘1))
42	8, 12, 16	phtpyi 24347	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → ((0𝑁𝑦) = (𝐺‘0) ∧ (1𝑁𝑦) = (𝐺‘1)))
43	42	simpld 495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (0[,]1)) → (0𝑁𝑦) = (𝐺‘0))
44	43	adantrl 714	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (0𝑁𝑦) = (𝐺‘0))
45	38, 41, 44	3eqtr4d 2786	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (1𝑀𝑦) = (0𝑁𝑦))
46		simprl 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 = (1 / 2))
47	46	oveq2d 7373	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = (2 · (1 / 2)))
48		2cn 12228	. . . . . . . 8 ⊢ 2 ∈ ℂ
49		2ne0 12257	. . . . . . . 8 ⊢ 2 ≠ 0
50	48, 49	recidi 11886	. . . . . . 7 ⊢ (2 · (1 / 2)) = 1
51	47, 50	eqtrdi 2792	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = 1)
52	51	oveq1d 7372	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (1𝑀𝑦))
53	51	oveq1d 7372	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = (1 − 1))
54		1m1e0 12225	. . . . . . 7 ⊢ (1 − 1) = 0
55	53, 54	eqtrdi 2792	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = 0)
56	55	oveq1d 7372	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (((2 · 𝑥) − 1)𝑁𝑦) = (0𝑁𝑦))
57	45, 52, 56	3eqtr4d 2786	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (((2 · 𝑥) − 1)𝑁𝑦))
58		retopon 24127	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
59		0re 11157	. . . . . . . 8 ⊢ 0 ∈ ℝ
60		iccssre 13346	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
61	59, 28, 60	mp2an 690	. . . . . . 7 ⊢ (0[,](1 / 2)) ⊆ ℝ
62		resttopon 22512	. . . . . . 7 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
63	58, 61, 62	mp2an 690	. . . . . 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 23019	. . . . . 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 24295	. . . . . . 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 7365	. . . . . 6 ⊢ (𝑧 = 𝑥 → (2 · 𝑧) = (2 · 𝑥))
69	64, 37, 65, 64, 67, 68	cnmpt21 23022	. . . . 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 23020	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
71	4, 11	phtpycn 24346	. . . . . 6 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
72	71, 13	sseldd 3945	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((II ×_t II) Cn 𝐽))
73	64, 37, 69, 70, 72	cnmpt22f 23026	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ ((2 · 𝑥)𝑀𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn 𝐽))
74		iccssre 13346	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
75	28, 30, 74	mp2an 690	. . . . . . 7 ⊢ ((1 / 2)[,]1) ⊆ ℝ
76		resttopon 22512	. . . . . . 7 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
77	58, 75, 76	mp2an 690	. . . . . 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 23019	. . . . . 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 24297	. . . . . . 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 7372	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((2 · 𝑧) − 1) = ((2 · 𝑥) − 1))
83	78, 37, 79, 78, 81, 82	cnmpt21 23022	. . . . 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 23020	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((1 / 2)[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn II))
85	8, 12	phtpycn 24346	. . . . . 6 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ ((II ×_t II) Cn 𝐽))
86	85, 16	sseldd 3945	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ((II ×_t II) Cn 𝐽))
87	78, 37, 83, 84, 86	cnmpt22f 23026	. . . 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 24291	. . 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 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → 𝜑)
91		elii1 24298	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) ↔ (𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)))
92		iihalf1 24294	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
93	91, 92	sylbir 234	. . . . . . 7 ⊢ ((𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
94	93	adantll 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
95	4, 11	phtpyhtpy 24345	. . . . . . . 8 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ (𝐹(II Htpy 𝐽)𝐻))
96	95, 13	sseldd 3945	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝐹(II Htpy 𝐽)𝐻))
97	37, 4, 11, 96	htpyi 24337	. . . . . 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 495	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → ((2 · 𝑠)𝑀0) = (𝐹‘(2 · 𝑠)))
100		simpll 765	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝜑)
101		elii2 24299	. . . . . . . 8 ⊢ ((𝑠 ∈ (0[,]1) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝑠 ∈ ((1 / 2)[,]1))
102	101	adantll 712	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝑠 ∈ ((1 / 2)[,]1))
103		iihalf2 24296	. . . . . . 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 24345	. . . . . . . 8 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ (𝐺(II Htpy 𝐽)𝐾))
106	105, 16	sseldd 3945	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝐺(II Htpy 𝐽)𝐾))
107	37, 8, 12, 106	htpyi 24337	. . . . . 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 495	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → (((2 · 𝑠) − 1)𝑁0) = (𝐺‘((2 · 𝑠) − 1)))
110	99, 109	ifeq12da 4519	. . 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 485	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
112		0elunit 13386	. . . 4 ⊢ 0 ∈ (0[,]1)
113		simpl 483	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
114	113	breq1d 5115	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
115	113	oveq2d 7373	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑥) = (2 · 𝑠))
116		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
117	115, 116	oveq12d 7375	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀0))
118	115	oveq1d 7372	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
119	118, 116	oveq12d 7375	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁0))
120	114, 117, 119	ifbieq12d 4514	. . . . 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 7390	. . . . . 6 ⊢ ((2 · 𝑠)𝑀0) ∈ V
122		ovex 7390	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁0) ∈ V
123	121, 122	ifex 4536	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)) ∈ V
124	120, 21, 123	ovmpoa 7510	. . . 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 24375	. . 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 2786	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃0) = ((𝐹(*_𝑝‘𝐽)𝐺)‘𝑠))
128	98	simprd 496	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → ((2 · 𝑠)𝑀1) = (𝐻‘(2 · 𝑠)))
129	108	simprd 496	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → (((2 · 𝑠) − 1)𝑁1) = (𝐾‘((2 · 𝑠) − 1)))
130	128, 129	ifeq12da 4519	. . 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 13387	. . . 4 ⊢ 1 ∈ (0[,]1)
132		simpl 483	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
133	132	breq1d 5115	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
134	132	oveq2d 7373	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑥) = (2 · 𝑠))
135		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
136	134, 135	oveq12d 7375	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀1))
137	134	oveq1d 7372	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
138	137, 135	oveq12d 7375	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁1))
139	133, 136, 138	ifbieq12d 4514	. . . . 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 7390	. . . . . 6 ⊢ ((2 · 𝑠)𝑀1) ∈ V
141		ovex 7390	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁1) ∈ V
142	140, 141	ifex 4536	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)) ∈ V
143	139, 21, 142	ovmpoa 7510	. . . 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 24375	. . 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 2786	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃1) = ((𝐻(*_𝑝‘𝐽)𝐾)‘𝑠))
147	4, 11, 13	phtpyi 24347	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝑀𝑠) = (𝐹‘0) ∧ (1𝑀𝑠) = (𝐹‘1)))
148	147	simpld 495	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑀𝑠) = (𝐹‘0))
149		simpl 483	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 = 0)
150	149, 29	eqbrtrdi 5144	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 ≤ (1 / 2))
151	150	iftrued 4494	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = ((2 · 𝑥)𝑀𝑦))
152	149	oveq2d 7373	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 0))
153		2t0e0 12322	. . . . . . . 8 ⊢ (2 · 0) = 0
154	152, 153	eqtrdi 2792	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 0)
155		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
156	154, 155	oveq12d 7375	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((2 · 𝑥)𝑀𝑦) = (0𝑀𝑠))
157	151, 156	eqtrd 2776	. . . . 5 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (0𝑀𝑠))
158		ovex 7390	. . . . 5 ⊢ (0𝑀𝑠) ∈ V
159	157, 21, 158	ovmpoa 7510	. . . 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 24377	. . . 4 ⊢ (𝜑 → ((𝐹(*_𝑝‘𝐽)𝐺)‘0) = (𝐹‘0))
162	161	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹(*_𝑝‘𝐽)𝐺)‘0) = (𝐹‘0))
163	148, 160, 162	3eqtr4d 2786	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘0))
164	8, 12, 16	phtpyi 24347	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0𝑁𝑠) = (𝐺‘0) ∧ (1𝑁𝑠) = (𝐺‘1)))
165	164	simprd 496	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑁𝑠) = (𝐺‘1))
166	28, 30	ltnlei 11276	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
167	31, 166	mpbi 229	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
168		simpl 483	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1)
169	168	breq1d 5115	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
170	167, 169	mtbiri 326	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ¬ 𝑥 ≤ (1 / 2))
171	170	iffalsed 4497	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (((2 · 𝑥) − 1)𝑁𝑦))
172	168	oveq2d 7373	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 1))
173		2t1e2 12316	. . . . . . . . . 10 ⊢ (2 · 1) = 2
174	172, 173	eqtrdi 2792	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 2)
175	174	oveq1d 7372	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = (2 − 1))
176		2m1e1 12279	. . . . . . . 8 ⊢ (2 − 1) = 1
177	175, 176	eqtrdi 2792	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = 1)
178		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
179	177, 178	oveq12d 7375	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((2 · 𝑥) − 1)𝑁𝑦) = (1𝑁𝑠))
180	171, 179	eqtrd 2776	. . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (1𝑁𝑠))
181		ovex 7390	. . . . 5 ⊢ (1𝑁𝑠) ∈ V
182	180, 21, 181	ovmpoa 7510	. . . 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 24378	. . . 4 ⊢ (𝜑 → ((𝐹(*_𝑝‘𝐽)𝐺)‘1) = (𝐺‘1))
185	184	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹(*_𝑝‘𝐽)𝐺)‘1) = (𝐺‘1))
186	165, 183, 185	3eqtr4d 2786	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘1))
187	10, 20, 89, 127, 146, 163, 186	isphtpy2d 24350	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3910 ∅c0 4282 ifcif 4486 class class class wbr 5105 ↦ cmpt 5188 ran crn 5634 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 ℝcr 11050 0cc0 11051 1c1 11052 · cmul 11056 < clt 11189 ≤ cle 11190 − cmin 11385 / cdiv 11812 2c2 12208 (,)cioo 13264 [,]cicc 13267 ↾_t crest 17302 topGenctg 17319 TopOnctopon 22259 Cn ccn 22575 ×_t ctx 22911 IIcii 24238 Htpy chtpy 24330 PHtpycphtpy 24331 ≃_phcphtpc 24332 *_𝑝cpco 24363

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-mulf 11131

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13268 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-starv 17148 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-unif 17156 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-cnfld 20797 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cld 22370 df-cn 22578 df-cnp 22579 df-tx 22913 df-hmeo 23106 df-xms 23673 df-ms 23674 df-tms 23675 df-ii 24240 df-htpy 24333 df-phtpy 24334 df-phtpc 24355 df-pco 24368

This theorem is referenced by: pcohtpy 24383

W3C validator