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Theorem pcopt2 25191

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 6882	. . . . . . . 8 ⊢ (𝑃‘((2 · 𝑥) − 1)) = (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1))
3		simpr 489	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹‘1) = 𝑌)
4		iiuni 25049	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
5		eqid 2763	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 23412	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽)
7	6	adantr 485	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹:(0[,]1)⟶∪ 𝐽)
8		1elunit 13501	. . . . . . . . . . 11 ⊢ 1 ∈ (0[,]1)
9		ffvelcdm 7076	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 597	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹‘1) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2864	. . . . . . . . 9 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii2 25104	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
13		iihalf2 25101	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((1 / 2)[,]1) → ((2 · 𝑥) − 1) ∈ (0[,]1))
14	12, 13	syl 18	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → ((2 · 𝑥) − 1) ∈ (0[,]1))
15		fvconst2g 7200	. . . . . . . . 9 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ ((2 · 𝑥) − 1) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1)) = 𝑌)
16	11, 14, 15	syl2an 607	. . . . . . . 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 2810	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝑃‘((2 · 𝑥) − 1)) = 𝑌)
18		simplr 780	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝐹‘1) = 𝑌)
19	17, 18	eqtr4d 2801	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝑃‘((2 · 𝑥) − 1)) = (𝐹‘1))
20	19	anassrs 472	. . . . 5 ⊢ ((((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) ∧ ¬ 𝑥 ≤ (1 / 2)) → (𝑃‘((2 · 𝑥) − 1)) = (𝐹‘1))
21	20	ifeq2da 4520	. . . 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 5204	. . 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 487	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
24		cntop2 23407	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
25	24	adantr 485	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ Top)
26		toptopon2 23084	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
27	25, 26	sylib 221	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
28	1	pcoptcl 25189	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
29	27, 11, 28	syl2anc 595	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
30	29	simp1d 1160	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
31	23, 30	pcoval 25179	. . 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 4493	. . . . . . . . 9 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
33	32	adantl 486	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
34		elii1 25103	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
35		iihalf1 25099	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
36	34, 35	sylbir 238	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
37	33, 36	eqeltrd 2863	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
38	37	ex 417	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1)))
39		iffalse 4496	. . . . . . 7 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
40	39, 8	eqeltrdi 2871	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
41	38, 40	pm2.61d1 182	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
42	41	adantl 486	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
43		eqidd 2764	. . . 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 6949	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
45		fveq2 6881	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))
46		fvif 6897	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))
47	45, 46	eqtrdi 2814	. . . 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 7125	. . 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 2808	. 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 25047	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
51	50	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
52	51	cnmptid 23827	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
53		0elunit 13500	. . . . . 6 ⊢ 0 ∈ (0[,]1)
54	53	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ (0[,]1))
55	51, 51, 54	cnmptc 23828	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
56		eqid 2763	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		eqid 2763	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
58		eqid 2763	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
59		dfii2 25050	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
60		0re 11214	. . . . . 6 ⊢ 0 ∈ ℝ
61	60	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ ℝ)
62		1re 11212	. . . . . 6 ⊢ 1 ∈ ℝ
63	62	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ ℝ)
64		halfre 12461	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
65		halfge0 12464	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
66		halflt1 12465	. . . . . . . 8 ⊢ (1 / 2) < 1
67	64, 62, 66	ltleii 11337	. . . . . . 7 ⊢ (1 / 2) ≤ 1
68		elicc01 13497	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
69	64, 65, 67, 68	mpbir3an 1360	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
70	69	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (1 / 2) ∈ (0[,]1))
71		simprl 782	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
72	71	oveq2d 7426	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
73		2thalfe1 12352	. . . . . 6 ⊢ (2 · (1 / 2)) = 1
74	72, 73	eqtrdi 2814	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
75		retopon 24929	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
76		iccssre 13460	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
77	60, 64, 76	mp2an 704	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
78		resttopon 23327	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
79	75, 77, 78	mp2an 704	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
80	79	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
81	80, 51	cnmpt1st 23834	. . . . . 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)))))
82	57	iihalf1cn 25100	. . . . . . 7 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II)
83	82	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II))
84		oveq2 7418	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
85	80, 51, 81, 80, 83, 84	cnmpt21 23837	. . . . 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))
86		iccssre 13460	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
87	64, 62, 86	mp2an 704	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
88		resttopon 23327	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
89	75, 87, 88	mp2an 704	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
90	89	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
91	8	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ (0[,]1))
92	90, 51, 51, 91	cnmpt2c 23836	. . . . 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))
93	56, 57, 58, 59, 61, 63, 70, 51, 74, 85, 92	cnmpopc 25096	. . . 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))
94		breq1 5112	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
95		oveq2 7418	. . . . . 6 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
96	94, 95	ifbieq1d 4512	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
97	96	adantr 485	. . . 4 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 0) → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
98	51, 52, 55, 51, 51, 93, 97	cnmpt12 23833	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) ∈ (II Cn II))
99		id 23	. . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 = 0)
100	99, 65	eqbrtrdi 5150	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 ≤ (1 / 2))
101	100, 32	syl 18	. . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
102		oveq2 7418	. . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0))
103		2t0e0 12415	. . . . . . 7 ⊢ (2 · 0) = 0
104	102, 103	eqtrdi 2814	. . . . . 6 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0)
105	101, 104	eqtrd 2798	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 0)
106		eqid 2763	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
107		c0ex 11204	. . . . 5 ⊢ 0 ∈ V
108	105, 106, 107	fvmpt 6989	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
109	53, 108	mp1i 14	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
110	64, 62	ltnlei 11335	. . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
111	66, 110	mpbi 233	. . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2)
112		breq1 5112	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
113	111, 112	mtbiri 330	. . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2))
114	113, 39	syl 18	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
115		1ex 11207	. . . . 5 ⊢ 1 ∈ V
116	114, 106, 115	fvmpt 6989	. . . 4 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
117	8, 116	mp1i 14	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
118	23, 98, 109, 117	reparpht 25166	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))( ≃_ph‘𝐽)𝐹)
119	49, 118	eqbrtrd 5133	1 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃)( ≃_ph‘𝐽)𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ⊆ wss 3905 ifcif 4487 {csn 4589 ∪ cuni 4872 class class class wbr 5109 ↦ cmpt 5192 × cxp 5659 ran crn 5662 ∘ ccom 5665 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ℝcr 11103 0cc0 11104 1c1 11105 · cmul 11109 < clt 11247 ≤ cle 11248 − cmin 11445 / cdiv 11875 2c2 12299 (,)cioo 13376 [,]cicc 13379 ↾_t crest 17477 topGenctg 17494 Topctop 23059 TopOnctopon 23076 Cn ccn 23390 IIcii 25043 ≃_phcphtpc 25137 *_𝑝cpco 25168

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-pre-sup 11182 ax-addf 11183

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-supp 8153 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9318 df-fi 9367 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9930 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-div 11876 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-7 12312 df-8 12313 df-9 12314 df-n0 12509 df-z 12596 df-dec 12716 df-uz 12867 df-q 12977 df-rp 13021 df-xneg 13141 df-xadd 13142 df-xmul 13143 df-ioo 13380 df-icc 13383 df-fz 13540 df-fzo 13688 df-seq 14043 df-exp 14103 df-hash 14372 df-cj 15155 df-re 15156 df-im 15157 df-sqrt 15291 df-abs 15292 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-starv 17329 df-sca 17330 df-vsca 17331 df-ip 17332 df-tset 17333 df-ple 17334 df-ds 17336 df-unif 17337 df-hom 17338 df-cco 17339 df-rest 17479 df-topn 17480 df-0g 17498 df-gsum 17499 df-topgen 17500 df-pt 17501 df-prds 17504 df-xrs 17560 df-qtop 17565 df-imas 17566 df-xps 17568 df-mre 17642 df-mrc 17643 df-acs 17645 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-mulg 19138 df-cntz 19391 df-cmn 19856 df-psmet 21523 df-xmet 21524 df-met 21525 df-bl 21526 df-mopn 21527 df-cnfld 21532 df-top 23060 df-topon 23077 df-topsp 23099 df-bases 23112 df-cld 23185 df-cn 23393 df-cnp 23394 df-tx 23728 df-hmeo 23921 df-xms 24486 df-ms 24487 df-tms 24488 df-ii 25045 df-htpy 25138 df-phtpy 25139 df-phtpc 25160 df-pco 25173

This theorem is used by: pcophtb 25197 pi1xfrcnvlem 25224

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