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Theorem pcopt2 24186

Description: Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)

**Hypothesis**
Ref	Expression
pcopt.1	⊢ 𝑃 = ((0[,]1) × {𝑌})

**Assertion**
Ref	Expression
pcopt2	⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃)( ≃_ph‘𝐽)𝐹)

Proof of Theorem pcopt2 Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcopt.1	. . . . . . . . 9 ⊢ 𝑃 = ((0[,]1) × {𝑌})
2	1	fveq1i 6775	. . . . . . . 8 ⊢ (𝑃‘((2 · 𝑥) − 1)) = (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1))
3		simpr 485	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹‘1) = 𝑌)
4		iiuni 24044	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
5		eqid 2738	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 22397	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽)
7	6	adantr 481	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹:(0[,]1)⟶∪ 𝐽)
8		1elunit 13202	. . . . . . . . . . 11 ⊢ 1 ∈ (0[,]1)
9		ffvelrn 6959	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹‘1) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2840	. . . . . . . . 9 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii2 24099	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
13		iihalf2 24096	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((1 / 2)[,]1) → ((2 · 𝑥) − 1) ∈ (0[,]1))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → ((2 · 𝑥) − 1) ∈ (0[,]1))
15		fvconst2g 7077	. . . . . . . . 9 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ ((2 · 𝑥) − 1) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1)) = 𝑌)
16	11, 14, 15	syl2an 596	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1)) = 𝑌)
17	2, 16	eqtrid 2790	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝑃‘((2 · 𝑥) − 1)) = 𝑌)
18		simplr 766	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝐹‘1) = 𝑌)
19	17, 18	eqtr4d 2781	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝑃‘((2 · 𝑥) − 1)) = (𝐹‘1))
20	19	anassrs 468	. . . . 5 ⊢ ((((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) ∧ ¬ 𝑥 ≤ (1 / 2)) → (𝑃‘((2 · 𝑥) − 1)) = (𝐹‘1))
21	20	ifeq2da 4491	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝑃‘((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1)))
22	21	mpteq2dva 5174	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝑃‘((2 · 𝑥) − 1)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))))
23		simpl 483	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
24		cntop2 22392	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
25	24	adantr 481	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ Top)
26		toptopon2 22067	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
27	25, 26	sylib 217	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
28	1	pcoptcl 24184	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
29	27, 11, 28	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
30	29	simp1d 1141	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
31	23, 30	pcoval 24174	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝑃‘((2 · 𝑥) − 1)))))
32		iftrue 4465	. . . . . . . . 9 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
33	32	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
34		elii1 24098	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
35		iihalf1 24094	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
36	34, 35	sylbir 234	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
37	33, 36	eqeltrd 2839	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
38	37	ex 413	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1)))
39		iffalse 4468	. . . . . . 7 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
40	39, 8	eqeltrdi 2847	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
41	38, 40	pm2.61d1 180	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
42	41	adantl 482	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
43		eqidd 2739	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))
44	7	feqmptd 6837	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
45		fveq2 6774	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))
46		fvif 6790	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))
47	45, 46	eqtrdi 2794	. . . 4 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) → (𝐹‘𝑦) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1)))
48	42, 43, 44, 47	fmptco 7001	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))))
49	22, 31, 48	3eqtr4d 2788	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃) = (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))))
50		iitopon 24042	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
51	50	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
52	51	cnmptid 22812	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
53		0elunit 13201	. . . . . 6 ⊢ 0 ∈ (0[,]1)
54	53	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ (0[,]1))
55	51, 51, 54	cnmptc 22813	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
56		eqid 2738	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		eqid 2738	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
58		eqid 2738	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
59		dfii2 24045	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
60		0re 10977	. . . . . 6 ⊢ 0 ∈ ℝ
61	60	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ ℝ)
62		1re 10975	. . . . . 6 ⊢ 1 ∈ ℝ
63	62	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ ℝ)
64		halfre 12187	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
65		halfge0 12190	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
66		halflt1 12191	. . . . . . . 8 ⊢ (1 / 2) < 1
67	64, 62, 66	ltleii 11098	. . . . . . 7 ⊢ (1 / 2) ≤ 1
68		elicc01 13198	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
69	64, 65, 67, 68	mpbir3an 1340	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
70	69	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (1 / 2) ∈ (0[,]1))
71		simprl 768	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
72	71	oveq2d 7291	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
73		2cn 12048	. . . . . . 7 ⊢ 2 ∈ ℂ
74		2ne0 12077	. . . . . . 7 ⊢ 2 ≠ 0
75	73, 74	recidi 11706	. . . . . 6 ⊢ (2 · (1 / 2)) = 1
76	72, 75	eqtrdi 2794	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
77		retopon 23927	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
78		iccssre 13161	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
79	60, 64, 78	mp2an 689	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
80		resttopon 22312	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
81	77, 79, 80	mp2an 689	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
82	81	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
83	82, 51	cnmpt1st 22819	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑦 ∈ (0[,](1 / 2)), 𝑧 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))))
84	57	iihalf1cn 24095	. . . . . . 7 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II)
85	84	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II))
86		oveq2 7283	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
87	82, 51, 83, 82, 85, 86	cnmpt21 22822	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑦 ∈ (0[,](1 / 2)), 𝑧 ∈ (0[,]1) ↦ (2 · 𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
88		iccssre 13161	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
89	64, 62, 88	mp2an 689	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
90		resttopon 22312	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
91	77, 89, 90	mp2an 689	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
92	91	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
93	8	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ (0[,]1))
94	92, 51, 51, 93	cnmpt2c 22821	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑦 ∈ ((1 / 2)[,]1), 𝑧 ∈ (0[,]1) ↦ 1) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn II))
95	56, 57, 58, 59, 61, 63, 70, 51, 76, 87, 94	cnmpopc 24091	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑦 ∈ (0[,]1), 𝑧 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1)) ∈ ((II ×_t II) Cn II))
96		breq1 5077	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
97		oveq2 7283	. . . . . 6 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
98	96, 97	ifbieq1d 4483	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
99	98	adantr 481	. . . 4 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 0) → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
100	51, 52, 55, 51, 51, 95, 99	cnmpt12 22818	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) ∈ (II Cn II))
101		id 22	. . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 = 0)
102	101, 65	eqbrtrdi 5113	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 ≤ (1 / 2))
103	102, 32	syl 17	. . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
104		oveq2 7283	. . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0))
105		2t0e0 12142	. . . . . . 7 ⊢ (2 · 0) = 0
106	104, 105	eqtrdi 2794	. . . . . 6 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0)
107	103, 106	eqtrd 2778	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 0)
108		eqid 2738	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
109		c0ex 10969	. . . . 5 ⊢ 0 ∈ V
110	107, 108, 109	fvmpt 6875	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
111	53, 110	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
112	64, 62	ltnlei 11096	. . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
113	66, 112	mpbi 229	. . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2)
114		breq1 5077	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
115	113, 114	mtbiri 327	. . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2))
116	115, 39	syl 17	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
117		1ex 10971	. . . . 5 ⊢ 1 ∈ V
118	116, 108, 117	fvmpt 6875	. . . 4 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
119	8, 118	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
120	23, 100, 111, 119	reparpht 24161	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))( ≃_ph‘𝐽)𝐹)
121	49, 120	eqbrtrd 5096	1 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃)( ≃_ph‘𝐽)𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ⊆ wss 3887 ifcif 4459 {csn 4561 ∪ cuni 4839 class class class wbr 5074 ↦ cmpt 5157 × cxp 5587 ran crn 5590 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ℝcr 10870 0cc0 10871 1c1 10872 · cmul 10876 < clt 11009 ≤ cle 11010 − cmin 11205 / cdiv 11632 2c2 12028 (,)cioo 13079 [,]cicc 13082 ↾_t crest 17131 topGenctg 17148 Topctop 22042 TopOnctopon 22059 Cn ccn 22375 IIcii 24038 ≃_phcphtpc 24132 *_𝑝cpco 24163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-cn 22378 df-cnp 22379 df-tx 22713 df-hmeo 22906 df-xms 23473 df-ms 23474 df-tms 23475 df-ii 24040 df-htpy 24133 df-phtpy 24134 df-phtpc 24155 df-pco 24168

This theorem is referenced by: pcophtb 24192 pi1xfrcnvlem 24219

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