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Theorem pcopt 23626

Description: Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)

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

**Assertion**
Ref	Expression
pcopt	⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)𝐹)

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

Step	Hyp	Ref	Expression
1		pcopt.1	. . . . . . . . . 10 ⊢ 𝑃 = ((0[,]1) × {𝑌})
2	1	fveq1i 6671	. . . . . . . . 9 ⊢ (𝑃‘(2 · 𝑥)) = (((0[,]1) × {𝑌})‘(2 · 𝑥))
3		simpr 487	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) = 𝑌)
4		iiuni 23489	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
5		eqid 2821	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 21854	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽)
7	6	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹:(0[,]1)⟶∪ 𝐽)
8		0elunit 12856	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
9		ffvelrn 6849	. . . . . . . . . . . 12 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 588	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2914	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii1 23539	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
13		iihalf1 23535	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
14	12, 13	sylbir 237	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
15		fvconst2g 6964	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ (2 · 𝑥) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
16	11, 14, 15	syl2an 597	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
17	2, 16	syl5eq 2868	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = 𝑌)
18		simplr 767	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝐹‘0) = 𝑌)
19	17, 18	eqtr4d 2859	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = (𝐹‘0))
20	19	ifeq1d 4485	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))))
21	20	expr 459	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1)))))
22		iffalse 4476	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
23		iffalse 4476	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
24	22, 23	eqtr4d 2859	. . . . 5 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))))
25	21, 24	pm2.61d1 182	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))))
26	25	mpteq2dva 5161	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1)))))
27		cntop2 21849	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
28	27	adantr 483	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ Top)
29		toptopon2 21526	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
30	28, 29	sylib 220	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
31	1	pcoptcl 23625	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
32	30, 11, 31	syl2anc 586	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
33	32	simp1d 1138	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
34		simpl 485	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
35	33, 34	pcoval 23615	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃(*_𝑝‘𝐽)𝐹) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1)))))
36		iffalse 4476	. . . . . . . . 9 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
37	36	adantl 484	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
38		elii2 23540	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
39		iihalf2 23537	. . . . . . . . 9 ⊢ (𝑥 ∈ ((1 / 2)[,]1) → ((2 · 𝑥) − 1) ∈ (0[,]1))
40	38, 39	syl 17	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → ((2 · 𝑥) − 1) ∈ (0[,]1))
41	37, 40	eqeltrd 2913	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
42	41	ex 415	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1)))
43		iftrue 4473	. . . . . . 7 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 0)
44	43, 8	eqeltrdi 2921	. . . . . 6 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
45	42, 44	pm2.61d2 183	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
46	45	adantl 484	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
47		eqid 2821	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
48	47	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))
49	7	feqmptd 6733	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
50		fveq2 6670	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))
51		fvif 6686	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1)))
52	50, 51	syl6eq 2872	. . . 4 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) → (𝐹‘𝑦) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))))
53	46, 48, 49, 52	fmptco 6891	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1)))))
54	26, 35, 53	3eqtr4d 2866	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃(*_𝑝‘𝐽)𝐹) = (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))))
55		iitopon 23487	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
56	55	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
57	56	cnmptid 22269	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
58	8	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 0 ∈ (0[,]1))
59	56, 56, 58	cnmptc 22270	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
60		eqid 2821	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
61		eqid 2821	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
62		eqid 2821	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
63		dfii2 23490	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
64		0re 10643	. . . . . 6 ⊢ 0 ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 0 ∈ ℝ)
66		1re 10641	. . . . . 6 ⊢ 1 ∈ ℝ
67	66	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 1 ∈ ℝ)
68		halfre 11852	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
69		halfge0 11855	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
70		halflt1 11856	. . . . . . . 8 ⊢ (1 / 2) < 1
71	68, 66, 70	ltleii 10763	. . . . . . 7 ⊢ (1 / 2) ≤ 1
72		elicc01 12855	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
73	68, 69, 71, 72	mpbir3an 1337	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
74	73	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (1 / 2) ∈ (0[,]1))
75		simprl 769	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
76	75	oveq2d 7172	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
77		2cn 11713	. . . . . . . . 9 ⊢ 2 ∈ ℂ
78		2ne0 11742	. . . . . . . . 9 ⊢ 2 ≠ 0
79	77, 78	recidi 11371	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
80	76, 79	syl6eq 2872	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
81	80	oveq1d 7171	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → ((2 · 𝑦) − 1) = (1 − 1))
82		1m1e0 11710	. . . . . 6 ⊢ (1 − 1) = 0
83	81, 82	syl6req 2873	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 0 = ((2 · 𝑦) − 1))
84		retopon 23372	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
85		iccssre 12819	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
86	64, 68, 85	mp2an 690	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
87		resttopon 21769	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
88	84, 86, 87	mp2an 690	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
89	88	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
90	89, 56, 56, 58	cnmpt2c 22278	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑦 ∈ (0[,](1 / 2)), 𝑧 ∈ (0[,]1) ↦ 0) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
91		iccssre 12819	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
92	68, 66, 91	mp2an 690	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
93		resttopon 21769	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
94	84, 92, 93	mp2an 690	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
95	94	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
96	95, 56	cnmpt1st 22276	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑦 ∈ ((1 / 2)[,]1), 𝑧 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))))
97	62	iihalf2cn 23538	. . . . . . 7 ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II)
98	97	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II))
99		oveq2 7164	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
100	99	oveq1d 7171	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2 · 𝑥) − 1) = ((2 · 𝑦) − 1))
101	95, 56, 96, 95, 98, 100	cnmpt21 22279	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑦 ∈ ((1 / 2)[,]1), 𝑧 ∈ (0[,]1) ↦ ((2 · 𝑦) − 1)) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn II))
102	60, 61, 62, 63, 65, 67, 74, 56, 83, 90, 101	cnmpopc 23532	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑦 ∈ (0[,]1), 𝑧 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1))) ∈ ((II ×_t II) Cn II))
103		breq1 5069	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
104		oveq2 7164	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
105	104	oveq1d 7171	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((2 · 𝑦) − 1) = ((2 · 𝑥) − 1))
106	103, 105	ifbieq2d 4492	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1)) = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
107	106	adantr 483	. . . 4 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 0) → if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1)) = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
108	56, 57, 59, 56, 56, 102, 107	cnmpt12 22275	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) ∈ (II Cn II))
109		id 22	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 = 0)
110	109, 69	eqbrtrdi 5105	. . . . . 6 ⊢ (𝑥 = 0 → 𝑥 ≤ (1 / 2))
111	110, 43	syl 17	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 0)
112		c0ex 10635	. . . . 5 ⊢ 0 ∈ V
113	111, 47, 112	fvmpt 6768	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘0) = 0)
114	8, 113	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘0) = 0)
115		1elunit 12857	. . . 4 ⊢ 1 ∈ (0[,]1)
116	68, 66	ltnlei 10761	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
117	70, 116	mpbi 232	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
118		breq1 5069	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
119	117, 118	mtbiri 329	. . . . . . 7 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2))
120	119, 36	syl 17	. . . . . 6 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
121		oveq2 7164	. . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1))
122		2t1e2 11801	. . . . . . . . 9 ⊢ (2 · 1) = 2
123	121, 122	syl6eq 2872	. . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2)
124	123	oveq1d 7171	. . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1))
125		2m1e1 11764	. . . . . . 7 ⊢ (2 − 1) = 1
126	124, 125	syl6eq 2872	. . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1)
127	120, 126	eqtrd 2856	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 1)
128		1ex 10637	. . . . 5 ⊢ 1 ∈ V
129	127, 47, 128	fvmpt 6768	. . . 4 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘1) = 1)
130	115, 129	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘1) = 1)
131	34, 108, 114, 130	reparpht 23602	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))( ≃_ph‘𝐽)𝐹)
132	54, 131	eqbrtrd 5088	1 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 ifcif 4467 {csn 4567 ∪ cuni 4838 class class class wbr 5066 ↦ cmpt 5146 × cxp 5553 ran crn 5556 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℝcr 10536 0cc0 10537 1c1 10538 · cmul 10542 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 2c2 11693 (,)cioo 12739 [,]cicc 12742 ↾_t crest 16694 topGenctg 16711 Topctop 21501 TopOnctopon 21518 Cn ccn 21832 IIcii 23483 ≃_phcphtpc 23573 *_𝑝cpco 23604

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-cn 21835 df-cnp 21836 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-ii 23485 df-htpy 23574 df-phtpy 23575 df-phtpc 23596 df-pco 23609

This theorem is referenced by: pcophtb 23633 pi1grplem 23653 pi1xfr 23659 pi1xfrcnvlem 23660

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