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Theorem pcopt 24971

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 6876	. . . . . . . . 9 ⊢ (𝑃‘(2 · 𝑥)) = (((0[,]1) × {𝑌})‘(2 · 𝑥))
3		simpr 484	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) = 𝑌)
4		iiuni 24823	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
5		eqid 2735	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 23182	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽)
7	6	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹:(0[,]1)⟶∪ 𝐽)
8		0elunit 13484	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
9		ffvelcdm 7070	. . . . . . . . . . . 12 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2835	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii1 24880	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
13		iihalf1 24874	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
14	12, 13	sylbir 235	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
15		fvconst2g 7193	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ (2 · 𝑥) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
16	11, 14, 15	syl2an 596	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
17	2, 16	eqtrid 2782	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = 𝑌)
18		simplr 768	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝐹‘0) = 𝑌)
19	17, 18	eqtr4d 2773	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = (𝐹‘0))
20	19	ifeq1d 4520	. . . . . 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 456	. . . . 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 4509	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
23		iffalse 4509	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
24	22, 23	eqtr4d 2773	. . . . 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 180	. . . 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 5214	. . 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 23177	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
28	27	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ Top)
29		toptopon2 22854	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
30	28, 29	sylib 218	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
31	1	pcoptcl 24970	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
32	30, 11, 31	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
33	32	simp1d 1142	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
34		simpl 482	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
35	33, 34	pcoval 24960	. . 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 4509	. . . . . . . . 9 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
37	36	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
38		elii2 24881	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
39		iihalf2 24877	. . . . . . . . 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 2834	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
42	41	ex 412	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1)))
43		iftrue 4506	. . . . . . 7 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 0)
44	43, 8	eqeltrdi 2842	. . . . . 6 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
45	42, 44	pm2.61d2 181	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
46	45	adantl 481	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
47		eqid 2735	. . . . 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 6946	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
50		fveq2 6875	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))
51		fvif 6891	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1)))
52	50, 51	eqtrdi 2786	. . . 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 7118	. . 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 2780	. 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 24821	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
56	55	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
57	56	cnmptid 23597	. . . 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 23598	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
60		eqid 2735	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
61		eqid 2735	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
62		eqid 2735	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
63		dfii2 24824	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
64		0re 11235	. . . . . 6 ⊢ 0 ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 0 ∈ ℝ)
66		1re 11233	. . . . . 6 ⊢ 1 ∈ ℝ
67	66	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 1 ∈ ℝ)
68		halfre 12452	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
69		halfge0 12455	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
70		halflt1 12456	. . . . . . . 8 ⊢ (1 / 2) < 1
71	68, 66, 70	ltleii 11356	. . . . . . 7 ⊢ (1 / 2) ≤ 1
72		elicc01 13481	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
73	68, 69, 71, 72	mpbir3an 1342	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
74	73	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (1 / 2) ∈ (0[,]1))
75		simprl 770	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
76	75	oveq2d 7419	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
77		2cn 12313	. . . . . . . . 9 ⊢ 2 ∈ ℂ
78		2ne0 12342	. . . . . . . . 9 ⊢ 2 ≠ 0
79	77, 78	recidi 11970	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
80	76, 79	eqtrdi 2786	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
81	80	oveq1d 7418	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → ((2 · 𝑦) − 1) = (1 − 1))
82		1m1e0 12310	. . . . . 6 ⊢ (1 − 1) = 0
83	81, 82	eqtr2di 2787	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 0 = ((2 · 𝑦) − 1))
84		retopon 24700	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
85		iccssre 13444	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
86	64, 68, 85	mp2an 692	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
87		resttopon 23097	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
88	84, 86, 87	mp2an 692	. . . . . . 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 23606	. . . . 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 13444	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
92	68, 66, 91	mp2an 692	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
93		resttopon 23097	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
94	84, 92, 93	mp2an 692	. . . . . . 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 23604	. . . . . 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 24878	. . . . . . 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 7411	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
100	99	oveq1d 7418	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2 · 𝑥) − 1) = ((2 · 𝑦) − 1))
101	95, 56, 96, 95, 98, 100	cnmpt21 23607	. . . . 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 24871	. . . 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 5122	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
104		oveq2 7411	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
105	104	oveq1d 7418	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((2 · 𝑦) − 1) = ((2 · 𝑥) − 1))
106	103, 105	ifbieq2d 4527	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1)) = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
107	106	adantr 480	. . . 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 23603	. . 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 5158	. . . . . 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 11227	. . . . 5 ⊢ 0 ∈ V
113	111, 47, 112	fvmpt 6985	. . . 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 13485	. . . 4 ⊢ 1 ∈ (0[,]1)
116	68, 66	ltnlei 11354	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
117	70, 116	mpbi 230	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
118		breq1 5122	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
119	117, 118	mtbiri 327	. . . . . . 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 7411	. . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1))
122		2t1e2 12401	. . . . . . . . 9 ⊢ (2 · 1) = 2
123	121, 122	eqtrdi 2786	. . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2)
124	123	oveq1d 7418	. . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1))
125		2m1e1 12364	. . . . . . 7 ⊢ (2 − 1) = 1
126	124, 125	eqtrdi 2786	. . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1)
127	120, 126	eqtrd 2770	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 1)
128		1ex 11229	. . . . 5 ⊢ 1 ∈ V
129	127, 47, 128	fvmpt 6985	. . . 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 24947	. 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 5141	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 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ⊆ wss 3926 ifcif 4500 {csn 4601 ∪ cuni 4883 class class class wbr 5119 ↦ cmpt 5201 × cxp 5652 ran crn 5655 ∘ ccom 5658 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 ℝcr 11126 0cc0 11127 1c1 11128 · cmul 11132 < clt 11267 ≤ cle 11268 − cmin 11464 / cdiv 11892 2c2 12293 (,)cioo 13360 [,]cicc 13363 ↾_t crest 17432 topGenctg 17449 Topctop 22829 TopOnctopon 22846 Cn ccn 23160 IIcii 24817 ≃_phcphtpc 24917 *_𝑝cpco 24949

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-icc 13367 df-fz 13523 df-fzo 13670 df-seq 14018 df-exp 14078 df-hash 14347 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-cn 23163 df-cnp 23164 df-tx 23498 df-hmeo 23691 df-xms 24257 df-ms 24258 df-tms 24259 df-ii 24819 df-htpy 24918 df-phtpy 24919 df-phtpc 24940 df-pco 24954

This theorem is referenced by: pcophtb 24978 pi1grplem 24998 pi1xfr 25004 pi1xfrcnvlem 25005

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