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Theorem pcohtpylem 25011

Description: Lemma for pcohtpy 25012. (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 24986	. . . . 5 ⊢ (𝐹( ≃_ph‘𝐽)𝐻 ↔ (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
3	1, 2	sylib 219	. . . 4 ⊢ (𝜑 → (𝐹 ∈ (II Cn 𝐽) ∧ 𝐻 ∈ (II Cn 𝐽) ∧ (𝐹(PHtpy‘𝐽)𝐻) ≠ ∅))
4	3	simp1d 1148	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
5		pcohtpy.6	. . . . 5 ⊢ (𝜑 → 𝐺( ≃_ph‘𝐽)𝐾)
6		isphtpc 24986	. . . . 5 ⊢ (𝐺( ≃_ph‘𝐽)𝐾 ↔ (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
7	5, 6	sylib 219	. . . 4 ⊢ (𝜑 → (𝐺 ∈ (II Cn 𝐽) ∧ 𝐾 ∈ (II Cn 𝐽) ∧ (𝐺(PHtpy‘𝐽)𝐾) ≠ ∅))
8	7	simp1d 1148	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn 𝐽))
9		pcohtpy.4	. . 3 ⊢ (𝜑 → (𝐹‘1) = (𝐺‘0))
10	4, 8, 9	pcocn 25009	. 2 ⊢ (𝜑 → (𝐹(*_𝑝‘𝐽)𝐺) ∈ (II Cn 𝐽))
11	3	simp2d 1149	. . 3 ⊢ (𝜑 → 𝐻 ∈ (II Cn 𝐽))
12	7	simp2d 1149	. . 3 ⊢ (𝜑 → 𝐾 ∈ (II Cn 𝐽))
13		pcohtpylem.8	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ (𝐹(PHtpy‘𝐽)𝐻))
14	4, 11, 13	phtpy01 24977	. . . . 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 24977	. . . . 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 2783	. . 3 ⊢ (𝜑 → (𝐻‘1) = (𝐾‘0))
20	11, 12, 19	pcocn 25009	. 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 2740	. . . 4 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
23		eqid 2740	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
24		eqid 2740	. . . 4 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
25		dfii2 24874	. . . 4 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
26		0red 11145	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ)
27		1red 11143	. . . 4 ⊢ (𝜑 → 1 ∈ ℝ)
28		halfre 12388	. . . . . 6 ⊢ (1 / 2) ∈ ℝ
29		halfge0 12391	. . . . . 6 ⊢ 0 ≤ (1 / 2)
30		1re 11142	. . . . . . 7 ⊢ 1 ∈ ℝ
31		halflt1 12392	. . . . . . 7 ⊢ (1 / 2) < 1
32	28, 30, 31	ltleii 11267	. . . . . 6 ⊢ (1 / 2) ≤ 1
33		elicc01 13417	. . . . . 6 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
34	28, 29, 32, 33	mpbir3an 1348	. . . . 5 ⊢ (1 / 2) ∈ (0[,]1)
35	34	a1i 11	. . . 4 ⊢ (𝜑 → (1 / 2) ∈ (0[,]1))
36		iitopon 24871	. . . . 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 24976	. . . . . . . 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 722	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (1𝑀𝑦) = (𝐹‘1))
42	8, 12, 16	phtpyi 24976	. . . . . . . 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 722	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (0𝑁𝑦) = (𝐺‘0))
45	38, 41, 44	3eqtr4d 2785	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (1𝑀𝑦) = (0𝑁𝑦))
46		simprl 776	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 = (1 / 2))
47	46	oveq2d 7379	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = (2 · (1 / 2)))
48		2cn 12254	. . . . . . . 8 ⊢ 2 ∈ ℂ
49		2ne0 12283	. . . . . . . 8 ⊢ 2 ≠ 0
50	48, 49	recidi 11884	. . . . . . 7 ⊢ (2 · (1 / 2)) = 1
51	47, 50	eqtrdi 2791	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (2 · 𝑥) = 1)
52	51	oveq1d 7378	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (1𝑀𝑦))
53	51	oveq1d 7378	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = (1 − 1))
54		1m1e0 12251	. . . . . . 7 ⊢ (1 − 1) = 0
55	53, 54	eqtrdi 2791	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥) − 1) = 0)
56	55	oveq1d 7378	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → (((2 · 𝑥) − 1)𝑁𝑦) = (0𝑁𝑦))
57	45, 52, 56	3eqtr4d 2785	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = (1 / 2) ∧ 𝑦 ∈ (0[,]1))) → ((2 · 𝑥)𝑀𝑦) = (((2 · 𝑥) − 1)𝑁𝑦))
58		retopon 24753	. . . . . . 7 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
59		0re 11144	. . . . . . . 8 ⊢ 0 ∈ ℝ
60		iccssre 13380	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
61	59, 28, 60	mp2an 698	. . . . . . 7 ⊢ (0[,](1 / 2)) ⊆ ℝ
62		resttopon 23151	. . . . . . 7 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
63	58, 61, 62	mp2an 698	. . . . . 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 23658	. . . . . 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 24924	. . . . . . 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 7371	. . . . . 6 ⊢ (𝑧 = 𝑥 → (2 · 𝑧) = (2 · 𝑥))
69	64, 37, 65, 64, 67, 68	cnmpt21 23661	. . . . 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 23659	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
71	4, 11	phtpycn 24975	. . . . . 6 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ ((II ×_t II) Cn 𝐽))
72	71, 13	sseldd 3923	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ((II ×_t II) Cn 𝐽))
73	64, 37, 69, 70, 72	cnmpt22f 23665	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,](1 / 2)), 𝑦 ∈ (0[,]1) ↦ ((2 · 𝑥)𝑀𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn 𝐽))
74		iccssre 13380	. . . . . . . 8 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
75	28, 30, 74	mp2an 698	. . . . . . 7 ⊢ ((1 / 2)[,]1) ⊆ ℝ
76		resttopon 23151	. . . . . . 7 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
77	58, 75, 76	mp2an 698	. . . . . 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 23658	. . . . . 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 24926	. . . . . . 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 7378	. . . . . 6 ⊢ (𝑧 = 𝑥 → ((2 · 𝑧) − 1) = ((2 · 𝑥) − 1))
83	78, 37, 79, 78, 81, 82	cnmpt21 23661	. . . . 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 23659	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ ((1 / 2)[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn II))
85	8, 12	phtpycn 24975	. . . . . 6 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ ((II ×_t II) Cn 𝐽))
86	85, 16	sseldd 3923	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ((II ×_t II) Cn 𝐽))
87	78, 37, 83, 84, 86	cnmpt22f 23665	. . . 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 24920	. . 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 2844	. 2 ⊢ (𝜑 → 𝑃 ∈ ((II ×_t II) Cn 𝐽))
90		simpll 772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → 𝜑)
91		elii1 24927	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) ↔ (𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)))
92		iihalf1 24923	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,](1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
93	91, 92	sylbir 236	. . . . . . 7 ⊢ ((𝑠 ∈ (0[,]1) ∧ 𝑠 ≤ (1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
94	93	adantll 720	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ 𝑠 ≤ (1 / 2)) → (2 · 𝑠) ∈ (0[,]1))
95	4, 11	phtpyhtpy 24974	. . . . . . . 8 ⊢ (𝜑 → (𝐹(PHtpy‘𝐽)𝐻) ⊆ (𝐹(II Htpy 𝐽)𝐻))
96	95, 13	sseldd 3923	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ (𝐹(II Htpy 𝐽)𝐻))
97	37, 4, 11, 96	htpyi 24966	. . . . . 6 ⊢ ((𝜑 ∧ (2 · 𝑠) ∈ (0[,]1)) → (((2 · 𝑠)𝑀0) = (𝐹‘(2 · 𝑠)) ∧ ((2 · 𝑠)𝑀1) = (𝐻‘(2 · 𝑠))))
98	90, 94, 97	syl2anc 590	. . . . 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 772	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝜑)
101		elii2 24928	. . . . . . . 8 ⊢ ((𝑠 ∈ (0[,]1) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝑠 ∈ ((1 / 2)[,]1))
102	101	adantll 720	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ (0[,]1)) ∧ ¬ 𝑠 ≤ (1 / 2)) → 𝑠 ∈ ((1 / 2)[,]1))
103		iihalf2 24925	. . . . . . 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 24974	. . . . . . . 8 ⊢ (𝜑 → (𝐺(PHtpy‘𝐽)𝐾) ⊆ (𝐺(II Htpy 𝐽)𝐾))
106	105, 16	sseldd 3923	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ (𝐺(II Htpy 𝐽)𝐾))
107	37, 8, 12, 106	htpyi 24966	. . . . . 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 590	. . . . 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 4495	. . 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 13420	. . . 4 ⊢ 0 ∈ (0[,]1)
113		simpl 483	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
114	113	breq1d 5089	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
115	113	oveq2d 7379	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑥) = (2 · 𝑠))
116		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
117	115, 116	oveq12d 7381	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀0))
118	115	oveq1d 7378	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
119	118, 116	oveq12d 7381	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁0))
120	114, 117, 119	ifbieq12d 4490	. . . . 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 7396	. . . . . 6 ⊢ ((2 · 𝑠)𝑀0) ∈ V
122		ovex 7396	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁0) ∈ V
123	121, 122	ifex 4512	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)) ∈ V
124	120, 21, 123	ovmpoa 7518	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝑃0) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)))
125	111, 112, 124	sylancl 592	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃0) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀0), (((2 · 𝑠) − 1)𝑁0)))
126	4, 8	pcovalg 25004	. . 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 2785	. 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 4495	. . 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 13421	. . . 4 ⊢ 1 ∈ (0[,]1)
132		simpl 483	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
133	132	breq1d 5089	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥 ≤ (1 / 2) ↔ 𝑠 ≤ (1 / 2)))
134	132	oveq2d 7379	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑥) = (2 · 𝑠))
135		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
136	134, 135	oveq12d 7381	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥)𝑀𝑦) = ((2 · 𝑠)𝑀1))
137	134	oveq1d 7378	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑥) − 1) = ((2 · 𝑠) − 1))
138	137, 135	oveq12d 7381	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((2 · 𝑥) − 1)𝑁𝑦) = (((2 · 𝑠) − 1)𝑁1))
139	133, 136, 138	ifbieq12d 4490	. . . . 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 7396	. . . . . 6 ⊢ ((2 · 𝑠)𝑀1) ∈ V
141		ovex 7396	. . . . . 6 ⊢ (((2 · 𝑠) − 1)𝑁1) ∈ V
142	140, 141	ifex 4512	. . . . 5 ⊢ if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)) ∈ V
143	139, 21, 142	ovmpoa 7518	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝑃1) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)))
144	111, 131, 143	sylancl 592	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃1) = if(𝑠 ≤ (1 / 2), ((2 · 𝑠)𝑀1), (((2 · 𝑠) − 1)𝑁1)))
145	11, 12	pcovalg 25004	. . 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 2785	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝑃1) = ((𝐻(*_𝑝‘𝐽)𝐾)‘𝑠))
147	4, 11, 13	phtpyi 24976	. . . 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 5118	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 ≤ (1 / 2))
151	150	iftrued 4469	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = ((2 · 𝑥)𝑀𝑦))
152	149	oveq2d 7379	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 0))
153		2t0e0 12343	. . . . . . . 8 ⊢ (2 · 0) = 0
154	152, 153	eqtrdi 2791	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 0)
155		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
156	154, 155	oveq12d 7381	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((2 · 𝑥)𝑀𝑦) = (0𝑀𝑠))
157	151, 156	eqtrd 2775	. . . . 5 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (0𝑀𝑠))
158		ovex 7396	. . . . 5 ⊢ (0𝑀𝑠) ∈ V
159	157, 21, 158	ovmpoa 7518	. . . 4 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = (0𝑀𝑠))
160	112, 111, 159	sylancr 593	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = (0𝑀𝑠))
161	4, 8	pco0 25006	. . . 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 2785	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘0))
164	8, 12, 16	phtpyi 24976	. . . 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 11265	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
167	31, 166	mpbi 231	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
168		simpl 483	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1)
169	168	breq1d 5089	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
170	167, 169	mtbiri 328	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ¬ 𝑥 ≤ (1 / 2))
171	170	iffalsed 4472	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (((2 · 𝑥) − 1)𝑁𝑦))
172	168	oveq2d 7379	. . . . . . . . . 10 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = (2 · 1))
173		2t1e2 12337	. . . . . . . . . 10 ⊢ (2 · 1) = 2
174	172, 173	eqtrdi 2791	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (2 · 𝑥) = 2)
175	174	oveq1d 7378	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = (2 − 1))
176		2m1e1 12300	. . . . . . . 8 ⊢ (2 − 1) = 1
177	175, 176	eqtrdi 2791	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((2 · 𝑥) − 1) = 1)
178		simpr 485	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
179	177, 178	oveq12d 7381	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((2 · 𝑥) − 1)𝑁𝑦) = (1𝑁𝑠))
180	171, 179	eqtrd 2775	. . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → if(𝑥 ≤ (1 / 2), ((2 · 𝑥)𝑀𝑦), (((2 · 𝑥) − 1)𝑁𝑦)) = (1𝑁𝑠))
181		ovex 7396	. . . . 5 ⊢ (1𝑁𝑠) ∈ V
182	180, 21, 181	ovmpoa 7518	. . . 4 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = (1𝑁𝑠))
183	131, 111, 182	sylancr 593	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = (1𝑁𝑠))
184	4, 8	pco1 25007	. . . 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 2785	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝑃𝑠) = ((𝐹(*_𝑝‘𝐽)𝐺)‘1))
187	10, 20, 89, 127, 146, 163, 186	isphtpy2d 24979	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 1092 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ⊆ wss 3890 ∅c0 4268 ifcif 4461 class class class wbr 5079 ↦ cmpt 5160 ran crn 5626 ‘cfv 6492 (class class class)co 7363 ∈ cmpo 7365 ℝcr 11035 0cc0 11036 1c1 11037 · cmul 11041 < clt 11177 ≤ cle 11178 − cmin 11375 / cdiv 11805 2c2 12234 (,)cioo 13296 [,]cicc 13299 ↾_t crest 17381 topGenctg 17398 TopOnctopon 22900 Cn ccn 23214 ×_t ctx 23550 IIcii 24867 Htpy chtpy 24959 PHtpycphtpy 24960 ≃_phcphtpc 24961 *_𝑝cpco 24992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-icc 13303 df-fz 13460 df-fzo 13607 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-cn 23217 df-cnp 23218 df-tx 23552 df-hmeo 23745 df-xms 24310 df-ms 24311 df-tms 24312 df-ii 24869 df-htpy 24962 df-phtpy 24963 df-phtpc 24984 df-pco 24997

This theorem is referenced by: pcohtpy 25012

W3C validator