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Theorem pcohtpylem 24947

Description: Lemma for pcohtpy 24948. (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 24921	. . . . 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 1142	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
5		pcohtpy.6	. . . . 5 ⊢ (𝜑 → 𝐺( ≃_ph‘𝐽)𝐾)
6		isphtpc 24921	. . . . 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 1142	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
9		pcohtpy.4	. . 3 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))
10	4, 8, 9	pcocn 24945	. 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 24912	. . . . 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 24912	. . . . 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 2776	. . 3 ⊢ (𝜑 → (𝐻‘1) = (𝐾‘0))
20	11, 12, 19	pcocn 24945	. 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 2733	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
23		eqid 2733	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
24		eqid 2733	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
25		dfii2 24803	. . . 4 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
26		0red 11122	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
27		1red 11120	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
28		halfre 12341	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
29		halfge0 12344	. . . . . 6 ⊢ 0 ≤ (1 / 2)
30		1re 11119	. . . . . . 7 ⊢ 1 ∈ ℝ
31		halflt1 12345	. . . . . . 7 ⊢ (1 / 2) < 1
32	28, 30, 31	ltleii 11243	. . . . . 6 ⊢ (1 / 2) ≤ 1
33		elicc01 13368	. . . . . 6 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
34	28, 29, 32, 33	mpbir3an 1342	. . . . 5 ⊢ (1 / 2) ∈ (0[,]1)
35	34	a1i 11	. . . 4 ⊢ (𝜑 → (1 / 2) ∈ (0[,]1))
36		iitopon 24800	. . . . 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 24911	. . . . . . . 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 24911	. . . . . . . 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 2778	. . . . 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 7368	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = (2 · (1 / 2)))
48		2cn 12207	. . . . . . . 8 ⊢ 2 ∈ ℂ
49		2ne0 12236	. . . . . . . 8 ⊢ 2 ≠ 0
50	48, 49	recidi 11859	. . . . . . 7 ⊢ (2 · (1 / 2)) = 1
51	47, 50	eqtrdi 2784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = 1)
52	51	oveq1d 7367	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (1𝑀𝑦))
53	51	oveq1d 7367	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = (1 − 1))
54		1m1e0 12204	. . . . . . 7 ⊢ (1 − 1) = 0
55	53, 54	eqtrdi 2784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = 0)
56	55	oveq1d 7367	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (((2 · 𝑥) − 1)𝑁𝑦) = (0𝑁𝑦))
57	45, 52, 56	3eqtr4d 2778	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (((2 · 𝑥) − 1)𝑁𝑦))
58		retopon 24679	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
59		0re 11121	. . . . . . . 8 ⊢ 0 ∈ ℝ
60		iccssre 13331	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
61	59, 28, 60	mp2an 692	. . . . . . 7 ⊢ (0[,](1 / 2)) ⊆ ℝ
62		resttopon 23077	. . . . . . 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 23584	. . . . . 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 24854	. . . . . . 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 7360	. . . . . 6 ⊢ (𝑧 = 𝑥 → (2 · 𝑧) = (2 · 𝑥))
69	64, 37, 65, 64, 67, 68	cnmpt21 23587	. . . . 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 23585	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
71	4, 11	phtpycn 24910	. . . . . 6 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
72	71, 13	sseldd 3931	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((II ×_t II) Cn 𝐽))
73	64, 37, 69, 70, 72	cnmpt22f 23591	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ ((2 · 𝑥)𝑀𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn 𝐽))
74		iccssre 13331	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
75	28, 30, 74	mp2an 692	. . . . . . 7 ⊢ ((1 / 2)[,]1) ⊆ ℝ
76		resttopon 23077	. . . . . . 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 23584	. . . . . 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 24857	. . . . . . 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 7367	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((2 · 𝑧) − 1) = ((2 · 𝑥) − 1))
83	78, 37, 79, 78, 81, 82	cnmpt21 23587	. . . . 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 23585	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((1 / 2)[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn II))
85	8, 12	phtpycn 24910	. . . . . 6 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ ((II ×_t II) Cn 𝐽))
86	85, 16	sseldd 3931	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ((II ×_t II) Cn 𝐽))
87	78, 37, 83, 84, 86	cnmpt22f 23591	. . . 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 24850	. . 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 2837	. 2 ⊢ (𝜑 → 𝑃 ∈ ((II ×_t II) Cn 𝐽))
90		simpll 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → 𝜑)
91		elii1 24859	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) ↔ (𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)))
92		iihalf1 24853	. . . . . . . 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 24909	. . . . . . . 8 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ (𝐹(II Htpy 𝐽)𝐻))
96	95, 13	sseldd 3931	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝐹(II Htpy 𝐽)𝐻))
97	37, 4, 11, 96	htpyi 24901	. . . . . 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 766	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝜑)
101		elii2 24860	. . . . . . . 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 24856	. . . . . . 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 24909	. . . . . . . 8 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ (𝐺(II Htpy 𝐽)𝐾))
106	105, 16	sseldd 3931	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝐺(II Htpy 𝐽)𝐾))
107	37, 8, 12, 106	htpyi 24901	. . . . . 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 4508	. . 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 13371	. . . 4 ⊢ 0 ∈ (0[,]1)
113		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
114	113	breq1d 5103	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
115	113	oveq2d 7368	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑥) = (2 · 𝑠))
116		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
117	115, 116	oveq12d 7370	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀0))
118	115	oveq1d 7367	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
119	118, 116	oveq12d 7370	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁0))
120	114, 117, 119	ifbieq12d 4503	. . . . 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 7385	. . . . . 6 ⊢ ((2 · 𝑠)𝑀0) ∈ V
122		ovex 7385	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁0) ∈ V
123	121, 122	ifex 4525	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)) ∈ V
124	120, 21, 123	ovmpoa 7507	. . . 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 24940	. . 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 2778	. 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 4508	. . 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 13372	. . . 4 ⊢ 1 ∈ (0[,]1)
132		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
133	132	breq1d 5103	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
134	132	oveq2d 7368	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑥) = (2 · 𝑠))
135		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
136	134, 135	oveq12d 7370	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀1))
137	134	oveq1d 7367	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
138	137, 135	oveq12d 7370	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁1))
139	133, 136, 138	ifbieq12d 4503	. . . . 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 7385	. . . . . 6 ⊢ ((2 · 𝑠)𝑀1) ∈ V
141		ovex 7385	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁1) ∈ V
142	140, 141	ifex 4525	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)) ∈ V
143	139, 21, 142	ovmpoa 7507	. . . 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 24940	. . 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 2778	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃1) = ((𝐻(*_𝑝‘𝐽)𝐾)‘𝑠))
147	4, 11, 13	phtpyi 24911	. . . 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 5132	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 ≤ (1 / 2))
151	150	iftrued 4482	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = ((2 · 𝑥)𝑀𝑦))
152	149	oveq2d 7368	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 0))
153		2t0e0 12296	. . . . . . . 8 ⊢ (2 · 0) = 0
154	152, 153	eqtrdi 2784	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 0)
155		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
156	154, 155	oveq12d 7370	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((2 · 𝑥)𝑀𝑦) = (0𝑀𝑠))
157	151, 156	eqtrd 2768	. . . . 5 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (0𝑀𝑠))
158		ovex 7385	. . . . 5 ⊢ (0𝑀𝑠) ∈ V
159	157, 21, 158	ovmpoa 7507	. . . 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 24942	. . . 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 2778	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘0))
164	8, 12, 16	phtpyi 24911	. . . 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 11241	. . . . . . . . 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 5103	. . . . . . . 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 4485	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (((2 · 𝑥) − 1)𝑁𝑦))
172	168	oveq2d 7368	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 1))
173		2t1e2 12290	. . . . . . . . . 10 ⊢ (2 · 1) = 2
174	172, 173	eqtrdi 2784	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 2)
175	174	oveq1d 7367	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = (2 − 1))
176		2m1e1 12253	. . . . . . . 8 ⊢ (2 − 1) = 1
177	175, 176	eqtrdi 2784	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = 1)
178		simpr 484	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
179	177, 178	oveq12d 7370	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((2 · 𝑥) − 1)𝑁𝑦) = (1𝑁𝑠))
180	171, 179	eqtrd 2768	. . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (1𝑁𝑠))
181		ovex 7385	. . . . 5 ⊢ (1𝑁𝑠) ∈ V
182	180, 21, 181	ovmpoa 7507	. . . 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 24943	. . . 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 2778	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘1))
187	10, 20, 89, 127, 146, 163, 186	isphtpy2d 24914	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 1541 ∈ wcel 2113 ≠ wne 2929 ⊆ wss 3898 ∅c0 4282 ifcif 4474 class class class wbr 5093 ↦ cmpt 5174 ran crn 5620 ‘cfv 6486 (class class class)co 7352 ∈ cmpo 7354 ℝcr 11012 0cc0 11013 1c1 11014 · cmul 11018 < clt 11153 ≤ cle 11154 − cmin 11351 / cdiv 11781 2c2 12187 (,)cioo 13247 [,]cicc 13250 ↾_t crest 17326 topGenctg 17343 TopOnctopon 22826 Cn ccn 23140 ×_t ctx 23476 IIcii 24796 Htpy chtpy 24894 PHtpycphtpy 24895 ≃_phcphtpc 24896 *_𝑝cpco 24928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-icc 13254 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-cn 23143 df-cnp 23144 df-tx 23478 df-hmeo 23671 df-xms 24236 df-ms 24237 df-tms 24238 df-ii 24798 df-htpy 24897 df-phtpy 24898 df-phtpc 24919 df-pco 24933

This theorem is referenced by: pcohtpy 24948

W3C validator