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Theorem pcopt 25376

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 6912	. . . . . . . . 9 ⊢ (𝑃‘(2 · 𝑥)) = (((0[,]1) × {𝑌})‘(2 · 𝑥))
3		simpr 490	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) = 𝑌)
4		iiuni 25235	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
5		eqid 2763	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 23597	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽)
7	6	adantr 486	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹:(0[,]1)⟶∪ 𝐽)
8		0elunit 13630	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
9		ffvelcdm 7108	. . . . . . . . . . . 12 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2865	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii1 25289	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
13		iihalf1 25285	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
14	12, 13	sylbir 238	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
15		fvconst2g 7234	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ (2 · 𝑥) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
16	11, 14, 15	syl2an 608	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
17	2, 16	eqtrid 2810	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = 𝑌)
18		simplr 781	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝐹‘0) = 𝑌)
19	17, 18	eqtr4d 2801	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = (𝐹‘0))
20	19	ifeq1d 4505	. . . . . 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 462	. . . . 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 4494	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
23		iffalse 4494	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
24	22, 23	eqtr4d 2801	. . . . 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 5202	. . 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 23592	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
28	27	adantr 486	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ Top)
29		toptopon2 23269	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
30	28, 29	sylib 221	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
31	1	pcoptcl 25375	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
32	30, 11, 31	syl2anc 596	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
33	32	simp1d 1160	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
34		simpl 488	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
35	33, 34	pcoval 25365	. . 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 4494	. . . . . . . . 9 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
37	36	adantl 487	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
38		elii2 25290	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
39		iihalf2 25287	. . . . . . . . 9 ⊢ (𝑥 ∈ ((1 / 2)[,]1) → ((2 · 𝑥) − 1) ∈ (0[,]1))
40	38, 39	syl 18	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → ((2 · 𝑥) − 1) ∈ (0[,]1))
41	37, 40	eqeltrd 2864	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
42	41	ex 418	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1)))
43		iftrue 4491	. . . . . . 7 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 0)
44	43, 8	eqeltrdi 2872	. . . . . 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 487	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
47		eqid 2763	. . . . 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 6979	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
50		fveq2 6911	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))
51		fvif 6927	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1)))
52	50, 51	eqtrdi 2814	. . . 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 7157	. . 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 2808	. 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 25233	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
56	55	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
57	56	cnmptid 24013	. . . 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 24014	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
60		eqid 2763	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
61		eqid 2763	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
62		eqid 2763	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
63		dfii2 25236	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
64		0re 11340	. . . . . 6 ⊢ 0 ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 0 ∈ ℝ)
66		1re 11338	. . . . . 6 ⊢ 1 ∈ ℝ
67	66	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 1 ∈ ℝ)
68		halfre 12589	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
69		halfge0 12592	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
70		halflt1 12593	. . . . . . . 8 ⊢ (1 / 2) < 1
71	68, 66, 70	ltleii 11463	. . . . . . 7 ⊢ (1 / 2) ≤ 1
72		elicc01 13627	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
73	68, 69, 71, 72	mpbir3an 1360	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
74	73	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (1 / 2) ∈ (0[,]1))
75		simprl 783	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
76	75	oveq2d 7462	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
77		2thalfe1 12480	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
78	76, 77	eqtrdi 2814	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
79	78	oveq1d 7461	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → ((2 · 𝑦) − 1) = (1 − 1))
80		1m1e0 12445	. . . . . 6 ⊢ (1 − 1) = 0
81	79, 80	eqtr2di 2815	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 0 = ((2 · 𝑦) − 1))
82		retopon 25115	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
83		iccssre 13590	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
84	64, 68, 83	mp2an 705	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
85		resttopon 23512	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
86	82, 84, 85	mp2an 705	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
87	86	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
88	87, 56, 56, 58	cnmpt2c 24022	. . . . 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))
89		iccssre 13590	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
90	68, 66, 89	mp2an 705	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
91		resttopon 23512	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
92	82, 90, 91	mp2an 705	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
93	92	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
94	93, 56	cnmpt1st 24020	. . . . . 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))))
95	62	iihalf2cn 25288	. . . . . . 7 ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II)
96	95	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))
97		oveq2 7454	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
98	97	oveq1d 7461	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2 · 𝑥) − 1) = ((2 · 𝑦) − 1))
99	93, 56, 94, 93, 96, 98	cnmpt21 24023	. . . . 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))
100	60, 61, 62, 63, 65, 67, 74, 56, 81, 88, 99	cnmpopc 25282	. . . 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))
101		breq1 5109	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
102		oveq2 7454	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
103	102	oveq1d 7461	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((2 · 𝑦) − 1) = ((2 · 𝑥) − 1))
104	101, 103	ifbieq2d 4512	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1)) = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
105	104	adantr 486	. . . 4 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 0) → if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1)) = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
106	56, 57, 59, 56, 56, 100, 105	cnmpt12 24019	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) ∈ (II Cn II))
107		id 23	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 = 0)
108	107, 69	eqbrtrdi 5148	. . . . . 6 ⊢ (𝑥 = 0 → 𝑥 ≤ (1 / 2))
109	108, 43	syl 18	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 0)
110		c0ex 11330	. . . . 5 ⊢ 0 ∈ V
111	109, 47, 110	fvmpt 7019	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘0) = 0)
112	8, 111	mp1i 14	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘0) = 0)
113		1elunit 13631	. . . 4 ⊢ 1 ∈ (0[,]1)
114	68, 66	ltnlei 11461	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
115	70, 114	mpbi 233	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
116		breq1 5109	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
117	115, 116	mtbiri 330	. . . . . . 7 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2))
118	117, 36	syl 18	. . . . . 6 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
119		oveq2 7454	. . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1))
120		2t1e2 12535	. . . . . . . . 9 ⊢ (2 · 1) = 2
121	119, 120	eqtrdi 2814	. . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2)
122	121	oveq1d 7461	. . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1))
123		2m1e1 12497	. . . . . . 7 ⊢ (2 − 1) = 1
124	122, 123	eqtrdi 2814	. . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1)
125	118, 124	eqtrd 2798	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 1)
126		1ex 11333	. . . . 5 ⊢ 1 ∈ V
127	125, 47, 126	fvmpt 7019	. . . 4 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘1) = 1)
128	113, 127	mp1i 14	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘1) = 1)
129	34, 106, 112, 128	reparpht 25352	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))( ≃_ph‘𝐽)𝐹)
130	54, 129	eqbrtrd 5131	1 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃(*_𝑝‘𝐽)𝐹)( ≃_ph‘𝐽)𝐹)

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ⊆ wss 3900 ifcif 4485 {csn 4588 ∪ cuni 4870 class class class wbr 5106 ↦ cmpt 5190 × cxp 5654 ran crn 5657 ∘ ccom 5660 ⟶wf 6557 ‘cfv 6561 (class class class)co 7446 ℝcr 11229 0cc0 11230 1c1 11231 · cmul 11235 < clt 11373 ≤ cle 11374 − cmin 11571 / cdiv 12003 2c2 12427 (,)cioo 13506 [,]cicc 13509 ↾_t crest 17623 topGenctg 17640 Topctop 23244 TopOnctopon 23261 Cn ccn 23575 IIcii 25229 ≃_phcphtpc 25323 *_𝑝cpco 25354

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5236 ax-sep 5253 ax-nul 5264 ax-pow 5332 ax-pr 5396 ax-un 7778 ax-cnex 11286 ax-resscn 11287 ax-1cn 11288 ax-icn 11289 ax-addcl 11290 ax-addrcl 11291 ax-mulcl 11292 ax-mulrcl 11293 ax-mulcom 11294 ax-addass 11295 ax-mulass 11296 ax-distr 11297 ax-i2m1 11298 ax-1ne0 11299 ax-1rid 11300 ax-rnegex 11301 ax-rrecex 11302 ax-cnre 11303 ax-pre-lttri 11304 ax-pre-lttrn 11305 ax-pre-ltadd 11306 ax-pre-mulgt0 11307 ax-pre-sup 11308 ax-addf 11309

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5551 df-eprel 5556 df-po 5564 df-so 5565 df-fr 5609 df-se 5610 df-we 5611 df-xp 5662 df-rel 5663 df-cnv 5664 df-co 5665 df-dm 5666 df-rn 5667 df-res 5668 df-ima 5669 df-pred 6325 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6515 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7403 df-ov 7449 df-oprab 7450 df-mpo 7451 df-of 7720 df-om 7905 df-1st 8028 df-2nd 8029 df-supp 8208 df-frecs 8330 df-wrecs 8361 df-recs 8410 df-rdg 8449 df-1o 8507 df-2o 8508 df-er 8748 df-map 8880 df-ixp 8957 df-en 9005 df-dom 9006 df-sdom 9007 df-fin 9008 df-fsupp 9385 df-fi 9433 df-sup 9464 df-inf 9465 df-oi 9534 df-card 10050 df-pnf 11375 df-mnf 11376 df-xr 11377 df-ltxr 11378 df-le 11379 df-sub 11573 df-neg 11574 df-div 12004 df-nn 12366 df-2 12435 df-3 12436 df-4 12437 df-5 12438 df-6 12439 df-7 12440 df-8 12441 df-9 12442 df-n0 12637 df-z 12724 df-dec 12845 df-uz 12996 df-q 13106 df-rp 13151 df-xneg 13271 df-xadd 13272 df-xmul 13273 df-ioo 13510 df-icc 13513 df-fz 13670 df-fzo 13819 df-seq 14175 df-exp 14235 df-hash 14505 df-cj 15298 df-re 15299 df-im 15300 df-sqrt 15434 df-abs 15435 df-struct 17357 df-sets 17374 df-slot 17392 df-ndx 17404 df-base 17420 df-ress 17441 df-plusg 17473 df-mulr 17474 df-starv 17475 df-sca 17476 df-vsca 17477 df-ip 17478 df-tset 17479 df-ple 17480 df-ds 17482 df-unif 17483 df-hom 17484 df-cco 17485 df-rest 17625 df-topn 17626 df-0g 17644 df-gsum 17645 df-topgen 17646 df-pt 17647 df-prds 17650 df-xrs 17706 df-qtop 17711 df-imas 17712 df-xps 17714 df-mre 17788 df-mrc 17789 df-acs 17791 df-mgm 18848 df-sgrp 18940 df-mnd 18956 df-submnd 19011 df-mulg 19310 df-cntz 19564 df-cmn 20029 df-psmet 21703 df-xmet 21704 df-met 21705 df-bl 21706 df-mopn 21707 df-cnfld 21712 df-top 23245 df-topon 23262 df-topsp 23284 df-bases 23297 df-cld 23370 df-cn 23578 df-cnp 23579 df-tx 23914 df-hmeo 24107 df-xms 24672 df-ms 24673 df-tms 24674 df-ii 25231 df-htpy 25324 df-phtpy 25325 df-phtpc 25346 df-pco 25359

This theorem is used by: pcophtb 25383 pi1grplem 25403 pi1xfr 25409 pi1xfrcnvlem 25410

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