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Theorem pcopt2 25147

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 6880	. . . . . . . 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 25005	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
5		eqid 2769	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 23368	. . . . . . . . . . . 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 13493	. . . . . . . . . . 11 ⊢ 1 ∈ (0[,]1)
9		ffvelcdm 7074	. . . . . . . . . . 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 2870	. . . . . . . . 9 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii2 25060	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
13		iihalf2 25057	. . . . . . . . . 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 7198	. . . . . . . . 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 2816	. . . . . . 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 2807	. . . . . 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 4522	. . . 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 5205	. . 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 23363	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
25	24	adantr 485	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ Top)
26		toptopon2 23040	. . . . . . 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 25145	. . . . . 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 1158	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
31	23, 30	pcoval 25135	. . 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 4495	. . . . . . . . 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 25059	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
35		iihalf1 25055	. . . . . . . . 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 2869	. . . . . . 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 4498	. . . . . . 7 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
40	39, 8	eqeltrdi 2877	. . . . . 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 2770	. . . 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 6947	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
45		fveq2 6879	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))
46		fvif 6895	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))
47	45, 46	eqtrdi 2820	. . . 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 7123	. . 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 2814	. 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 25003	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
51	50	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
52	51	cnmptid 23783	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
53		0elunit 13492	. . . . . 6 ⊢ 0 ∈ (0[,]1)
54	53	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ (0[,]1))
55	51, 51, 54	cnmptc 23784	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
56		eqid 2769	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		eqid 2769	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
58		eqid 2769	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
59		dfii2 25006	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
60		0re 11206	. . . . . 6 ⊢ 0 ∈ ℝ
61	60	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ ℝ)
62		1re 11204	. . . . . 6 ⊢ 1 ∈ ℝ
63	62	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ ℝ)
64		halfre 12453	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
65		halfge0 12456	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
66		halflt1 12457	. . . . . . . 8 ⊢ (1 / 2) < 1
67	64, 62, 66	ltleii 11329	. . . . . . 7 ⊢ (1 / 2) ≤ 1
68		elicc01 13489	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
69	64, 65, 67, 68	mpbir3an 1358	. . . . . 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 7424	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
73		2cn 12312	. . . . . . 7 ⊢ 2 ∈ ℂ
74		2ne0 12343	. . . . . . 7 ⊢ 2 ≠ 0
75	73, 74	recidi 11942	. . . . . 6 ⊢ (2 · (1 / 2)) = 1
76	72, 75	eqtrdi 2820	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
77		retopon 24885	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
78		iccssre 13452	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
79	60, 64, 78	mp2an 704	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
80		resttopon 23283	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
81	77, 79, 80	mp2an 704	. . . . . . 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 23790	. . . . . 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 25056	. . . . . . 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 7416	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
87	82, 51, 83, 82, 85, 86	cnmpt21 23793	. . . . 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 13452	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
89	64, 62, 88	mp2an 704	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
90		resttopon 23283	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
91	77, 89, 90	mp2an 704	. . . . . . 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 23792	. . . . 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 25052	. . . 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 5113	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
97		oveq2 7416	. . . . . 6 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
98	96, 97	ifbieq1d 4514	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
99	98	adantr 485	. . . 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 23789	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) ∈ (II Cn II))
101		id 23	. . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 = 0)
102	101, 65	eqbrtrdi 5151	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 ≤ (1 / 2))
103	102, 32	syl 18	. . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
104		oveq2 7416	. . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0))
105		2t0e0 12407	. . . . . . 7 ⊢ (2 · 0) = 0
106	104, 105	eqtrdi 2820	. . . . . 6 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0)
107	103, 106	eqtrd 2804	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 0)
108		eqid 2769	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
109		c0ex 11196	. . . . 5 ⊢ 0 ∈ V
110	107, 108, 109	fvmpt 6987	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
111	53, 110	mp1i 14	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
112	64, 62	ltnlei 11327	. . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
113	66, 112	mpbi 233	. . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2)
114		breq1 5113	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
115	113, 114	mtbiri 330	. . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2))
116	115, 39	syl 18	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
117		1ex 11199	. . . . 5 ⊢ 1 ∈ V
118	116, 108, 117	fvmpt 6987	. . . 4 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
119	8, 118	mp1i 14	. . 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 25122	. 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 5134	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 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ⊆ wss 3913 ifcif 4489 {csn 4591 ∪ cuni 4873 class class class wbr 5110 ↦ cmpt 5193 × cxp 5657 ran crn 5660 ∘ ccom 5663 ⟶wf 6529 ‘cfv 6533 (class class class)co 7408 ℝcr 11095 0cc0 11096 1c1 11097 · cmul 11101 < clt 11239 ≤ cle 11240 − cmin 11437 / cdiv 11867 2c2 12291 (,)cioo 13368 [,]cicc 13371 ↾_t crest 17469 topGenctg 17486 Topctop 23015 TopOnctopon 23032 Cn ccn 23346 IIcii 24999 ≃_phcphtpc 25093 *_𝑝cpco 25124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174 ax-addf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 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 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-div 11868 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12501 df-z 12588 df-dec 12708 df-uz 12859 df-q 12969 df-rp 13013 df-xneg 13133 df-xadd 13134 df-xmul 13135 df-ioo 13372 df-icc 13375 df-fz 13532 df-fzo 13679 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-starv 17321 df-sca 17322 df-vsca 17323 df-ip 17324 df-tset 17325 df-ple 17326 df-ds 17328 df-unif 17329 df-hom 17330 df-cco 17331 df-rest 17471 df-topn 17472 df-0g 17490 df-gsum 17491 df-topgen 17492 df-pt 17493 df-prds 17496 df-xrs 17552 df-qtop 17557 df-imas 17558 df-xps 17560 df-mre 17634 df-mrc 17635 df-acs 17637 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-mulg 19130 df-cntz 19383 df-cmn 19848 df-psmet 21479 df-xmet 21480 df-met 21481 df-bl 21482 df-mopn 21483 df-cnfld 21488 df-top 23016 df-topon 23033 df-topsp 23055 df-bases 23068 df-cld 23141 df-cn 23349 df-cnp 23350 df-tx 23684 df-hmeo 23877 df-xms 24442 df-ms 24443 df-tms 24444 df-ii 25001 df-htpy 25094 df-phtpy 25095 df-phtpc 25116 df-pco 25129

This theorem is referenced by: pcophtb 25153 pi1xfrcnvlem 25180

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