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Theorem pcopt 25014

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 6835	. . . . . . . . 9 ⊢ (𝑃‘(2 · 𝑥)) = (((0[,]1) × {𝑌})‘(2 · 𝑥))
3		simpr 485	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) = 𝑌)
4		iiuni 24873	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
5		eqid 2740	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 23236	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽)
7	6	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹:(0[,]1)⟶∪ 𝐽)
8		0elunit 13420	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
9		ffvelcdm 7029	. . . . . . . . . . . 12 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 592	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2841	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii1 24927	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
13		iihalf1 24923	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
14	12, 13	sylbir 236	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
15		fvconst2g 7153	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ (2 · 𝑥) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
16	11, 14, 15	syl2an 602	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
17	2, 16	eqtrid 2787	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = 𝑌)
18		simplr 774	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝐹‘0) = 𝑌)
19	17, 18	eqtr4d 2778	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = (𝐹‘0))
20	19	ifeq1d 4481	. . . . . 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 457	. . . . 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 4470	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
23		iffalse 4470	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
24	22, 23	eqtr4d 2778	. . . . 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 181	. . . 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 5172	. . 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 23231	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
28	27	adantr 481	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ Top)
29		toptopon2 22908	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
30	28, 29	sylib 219	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
31	1	pcoptcl 25013	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
32	30, 11, 31	syl2anc 590	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
33	32	simp1d 1148	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
34		simpl 483	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
35	33, 34	pcoval 25003	. . 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 4470	. . . . . . . . 9 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
37	36	adantl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
38		elii2 24928	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
39		iihalf2 24925	. . . . . . . . 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 2840	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
42	41	ex 413	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1)))
43		iftrue 4467	. . . . . . 7 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 0)
44	43, 8	eqeltrdi 2848	. . . . . 6 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
45	42, 44	pm2.61d2 182	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
46	45	adantl 482	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
47		eqid 2740	. . . . 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 6902	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
50		fveq2 6834	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))
51		fvif 6850	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1)))
52	50, 51	eqtrdi 2791	. . . 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 7078	. . 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 2785	. 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 24871	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
56	55	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
57	56	cnmptid 23651	. . . 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 23652	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
60		eqid 2740	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
61		eqid 2740	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
62		eqid 2740	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
63		dfii2 24874	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
64		0re 11144	. . . . . 6 ⊢ 0 ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 0 ∈ ℝ)
66		1re 11142	. . . . . 6 ⊢ 1 ∈ ℝ
67	66	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 1 ∈ ℝ)
68		halfre 12388	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
69		halfge0 12391	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
70		halflt1 12392	. . . . . . . 8 ⊢ (1 / 2) < 1
71	68, 66, 70	ltleii 11267	. . . . . . 7 ⊢ (1 / 2) ≤ 1
72		elicc01 13417	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
73	68, 69, 71, 72	mpbir3an 1348	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
74	73	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (1 / 2) ∈ (0[,]1))
75		simprl 776	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
76	75	oveq2d 7379	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
77		2cn 12254	. . . . . . . . 9 ⊢ 2 ∈ ℂ
78		2ne0 12283	. . . . . . . . 9 ⊢ 2 ≠ 0
79	77, 78	recidi 11884	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
80	76, 79	eqtrdi 2791	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
81	80	oveq1d 7378	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → ((2 · 𝑦) − 1) = (1 − 1))
82		1m1e0 12251	. . . . . 6 ⊢ (1 − 1) = 0
83	81, 82	eqtr2di 2792	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 0 = ((2 · 𝑦) − 1))
84		retopon 24753	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
85		iccssre 13380	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
86	64, 68, 85	mp2an 698	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
87		resttopon 23151	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
88	84, 86, 87	mp2an 698	. . . . . . 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 23660	. . . . 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 13380	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
92	68, 66, 91	mp2an 698	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
93		resttopon 23151	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
94	84, 92, 93	mp2an 698	. . . . . . 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 23658	. . . . . 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 24926	. . . . . . 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 7371	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
100	99	oveq1d 7378	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2 · 𝑥) − 1) = ((2 · 𝑦) − 1))
101	95, 56, 96, 95, 98, 100	cnmpt21 23661	. . . . 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 24920	. . . 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 5082	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
104		oveq2 7371	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
105	104	oveq1d 7378	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((2 · 𝑦) − 1) = ((2 · 𝑥) − 1))
106	103, 105	ifbieq2d 4488	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1)) = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
107	106	adantr 481	. . . 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 23657	. . 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 5118	. . . . . 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 11136	. . . . 5 ⊢ 0 ∈ V
113	111, 47, 112	fvmpt 6942	. . . 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 13421	. . . 4 ⊢ 1 ∈ (0[,]1)
116	68, 66	ltnlei 11265	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
117	70, 116	mpbi 231	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
118		breq1 5082	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
119	117, 118	mtbiri 328	. . . . . . 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 7371	. . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1))
122		2t1e2 12337	. . . . . . . . 9 ⊢ (2 · 1) = 2
123	121, 122	eqtrdi 2791	. . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2)
124	123	oveq1d 7378	. . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1))
125		2m1e1 12300	. . . . . . 7 ⊢ (2 − 1) = 1
126	124, 125	eqtrdi 2791	. . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1)
127	120, 126	eqtrd 2775	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 1)
128		1ex 11138	. . . . 5 ⊢ 1 ∈ V
129	127, 47, 128	fvmpt 6942	. . . 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 24990	. 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 5101	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 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 ⊆ wss 3890 ifcif 4461 {csn 4562 ∪ cuni 4845 class class class wbr 5079 ↦ cmpt 5160 × cxp 5623 ran crn 5626 ∘ ccom 5629 ⟶wf 6488 ‘cfv 6492 (class class class)co 7363 ℝcr 11035 0cc0 11036 1c1 11037 · cmul 11041 < clt 11177 ≤ cle 11178 − cmin 11375 / cdiv 11805 2c2 12234 (,)cioo 13296 [,]cicc 13299 ↾_t crest 17381 topGenctg 17398 Topctop 22883 TopOnctopon 22900 Cn ccn 23214 IIcii 24867 ≃_phcphtpc 24961 *_𝑝cpco 24992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-pre-sup 11114 ax-addf 11115

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-se 5579 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-supp 8108 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-ixp 8843 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-fsupp 9272 df-fi 9321 df-sup 9352 df-inf 9353 df-oi 9422 df-card 9861 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-div 11806 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-7 12247 df-8 12248 df-9 12249 df-n0 12436 df-z 12523 df-dec 12643 df-uz 12787 df-q 12897 df-rp 12941 df-xneg 13061 df-xadd 13062 df-xmul 13063 df-ioo 13300 df-icc 13303 df-fz 13460 df-fzo 13607 df-seq 13962 df-exp 14022 df-hash 14291 df-cj 15059 df-re 15060 df-im 15061 df-sqrt 15195 df-abs 15196 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17383 df-topn 17384 df-0g 17402 df-gsum 17403 df-topgen 17404 df-pt 17405 df-prds 17408 df-xrs 17464 df-qtop 17469 df-imas 17470 df-xps 17472 df-mre 17546 df-mrc 17547 df-acs 17549 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-mulg 19042 df-cntz 19290 df-cmn 19755 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-cnfld 21355 df-top 22884 df-topon 22901 df-topsp 22923 df-bases 22936 df-cld 23009 df-cn 23217 df-cnp 23218 df-tx 23552 df-hmeo 23745 df-xms 24310 df-ms 24311 df-tms 24312 df-ii 24869 df-htpy 24962 df-phtpy 24963 df-phtpc 24984 df-pco 24997

This theorem is referenced by: pcophtb 25021 pi1grplem 25041 pi1xfr 25047 pi1xfrcnvlem 25048

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