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Theorem pcopt 22822

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 6192	. . . . . . . . 9 ⊢ (𝑃‘(2 · 𝑥)) = (((0[,]1) × {𝑌})‘(2 · 𝑥))
3		simpr 477	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) = 𝑌)
4		iiuni 22684	. . . . . . . . . . . . . 14 ⊢ (0[,]1) = ∪ II
5		eqid 2622	. . . . . . . . . . . . . 14 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 21050	. . . . . . . . . . . . 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 12290	. . . . . . . . . . . 12 ⊢ 0 ∈ (0[,]1)
9		ffvelrn 6357	. . . . . . . . . . . 12 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 0 ∈ (0[,]1)) → (𝐹‘0) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 694	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹‘0) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2702	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii1 22734	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
13		iihalf1 22730	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
14	12, 13	sylbir 225	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
15		fvconst2g 6467	. . . . . . . . . 10 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ (2 · 𝑥) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
16	11, 14, 15	syl2an 494	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (((0[,]1) × {𝑌})‘(2 · 𝑥)) = 𝑌)
17	2, 16	syl5eq 2668	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = 𝑌)
18		simplr 792	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝐹‘0) = 𝑌)
19	17, 18	eqtr4d 2659	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2))) → (𝑃‘(2 · 𝑥)) = (𝐹‘0))
20	19	ifeq1d 4104	. . . . . 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 643	. . . . 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 4095	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝑃‘(2 · 𝑥)), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
23		iffalse 4095	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1))) = (𝐹‘((2 · 𝑥) − 1)))
24	22, 23	eqtr4d 2659	. . . . 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 171	. . . 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 4744	. . 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 21045	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
28	27	adantr 481	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ Top)
29	5	toptopon 20722	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
30	28, 29	sylib 208	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
31	1	pcoptcl 22821	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
32	30, 11, 31	syl2anc 693	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
33	32	simp1d 1073	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
34		simpl 473	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
35	33, 34	pcoval 22811	. . 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 4095	. . . . . . . . 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 22735	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
39		iihalf2 22732	. . . . . . . . 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 2701	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
42	41	ex 450	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1)))
43		iftrue 4092	. . . . . . 7 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 0)
44	43, 8	syl6eqel 2709	. . . . . 6 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) ∈ (0[,]1))
45	42, 44	pm2.61d2 172	. . . . 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 2622	. . . . 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 6249	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
50		fveq2 6191	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))
51		fvif 6204	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘0), (𝐹‘((2 · 𝑥) − 1)))
52	50, 51	syl6eq 2672	. . . 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 6396	. . 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 2666	. 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 22682	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
56	55	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
57	56	cnmptid 21464	. . . 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 21465	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
60		eqid 2622	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
61		eqid 2622	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
62		eqid 2622	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
63		dfii2 22685	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
64		0re 10040	. . . . . 6 ⊢ 0 ∈ ℝ
65	64	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 0 ∈ ℝ)
66		1re 10039	. . . . . 6 ⊢ 1 ∈ ℝ
67	66	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → 1 ∈ ℝ)
68		halfre 11246	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
69		halfgt0 11248	. . . . . . . 8 ⊢ 0 < (1 / 2)
70	64, 68, 69	ltleii 10160	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
71		halflt1 11250	. . . . . . . 8 ⊢ (1 / 2) < 1
72	68, 66, 71	ltleii 10160	. . . . . . 7 ⊢ (1 / 2) ≤ 1
73	64, 66	elicc2i 12239	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
74	68, 70, 72, 73	mpbir3an 1244	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
75	74	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (1 / 2) ∈ (0[,]1))
76		simprl 794	. . . . . . . . 9 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
77	76	oveq2d 6666	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
78		2cn 11091	. . . . . . . . 9 ⊢ 2 ∈ ℂ
79		2ne0 11113	. . . . . . . . 9 ⊢ 2 ≠ 0
80	78, 79	recidi 10756	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
81	77, 80	syl6eq 2672	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
82	81	oveq1d 6665	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → ((2 · 𝑦) − 1) = (1 − 1))
83		1m1e0 11089	. . . . . 6 ⊢ (1 − 1) = 0
84	82, 83	syl6req 2673	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 0 = ((2 · 𝑦) − 1))
85		retopon 22567	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
86		iccssre 12255	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
87	64, 68, 86	mp2an 708	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
88		resttopon 20965	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
89	85, 87, 88	mp2an 708	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
90	89	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
91	90, 56, 56, 58	cnmpt2c 21473	. . . . 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))
92		iccssre 12255	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
93	68, 66, 92	mp2an 708	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
94		resttopon 20965	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
95	85, 93, 94	mp2an 708	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
96	95	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
97	96, 56	cnmpt1st 21471	. . . . . 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))))
98	62	iihalf2cn 22733	. . . . . . 7 ⊢ (𝑥 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑥) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II)
99	98	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))
100		oveq2 6658	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
101	100	oveq1d 6665	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((2 · 𝑥) − 1) = ((2 · 𝑦) − 1))
102	96, 56, 97, 96, 99, 101	cnmpt21 21474	. . . . 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))
103	60, 61, 62, 63, 65, 67, 75, 56, 84, 91, 102	cnmpt2pc 22727	. . . 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))
104		breq1 4656	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
105		oveq2 6658	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
106	105	oveq1d 6665	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((2 · 𝑦) − 1) = ((2 · 𝑥) − 1))
107	104, 106	ifbieq2d 4111	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1)) = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
108	107	adantr 481	. . . 4 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 0) → if(𝑦 ≤ (1 / 2), 0, ((2 · 𝑦) − 1)) = if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))
109	56, 57, 59, 56, 56, 103, 108	cnmpt12 21470	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))) ∈ (II Cn II))
110		id 22	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 = 0)
111	110, 70	syl6eqbr 4692	. . . . . 6 ⊢ (𝑥 = 0 → 𝑥 ≤ (1 / 2))
112	111, 43	syl 17	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 0)
113		c0ex 10034	. . . . 5 ⊢ 0 ∈ V
114	112, 47, 113	fvmpt 6282	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘0) = 0)
115	8, 114	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘0) = 0)
116		1elunit 12291	. . . 4 ⊢ 1 ∈ (0[,]1)
117	68, 66	ltnlei 10158	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
118	71, 117	mpbi 220	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
119		breq1 4656	. . . . . . . 8 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
120	118, 119	mtbiri 317	. . . . . . 7 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2))
121	120, 36	syl 17	. . . . . 6 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = ((2 · 𝑥) − 1))
122		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 1 → (2 · 𝑥) = (2 · 1))
123		2t1e2 11176	. . . . . . . . 9 ⊢ (2 · 1) = 2
124	122, 123	syl6eq 2672	. . . . . . . 8 ⊢ (𝑥 = 1 → (2 · 𝑥) = 2)
125	124	oveq1d 6665	. . . . . . 7 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = (2 − 1))
126		2m1e1 11135	. . . . . . 7 ⊢ (2 − 1) = 1
127	125, 126	syl6eq 2672	. . . . . 6 ⊢ (𝑥 = 1 → ((2 · 𝑥) − 1) = 1)
128	121, 127	eqtrd 2656	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)) = 1)
129		1ex 10035	. . . . 5 ⊢ 1 ∈ V
130	128, 47, 129	fvmpt 6282	. . . 4 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘1) = 1)
131	116, 130	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1)))‘1) = 1)
132	34, 109, 115, 131	reparpht 22798	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘0) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), 0, ((2 · 𝑥) − 1))))( ≃_ph‘𝐽)𝐹)
133	54, 132	eqbrtrd 4675	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 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ifcif 4086 {csn 4177 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 ran crn 5115 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 0cc0 9936 1c1 9937 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 2c2 11070 (,)cioo 12175 [,]cicc 12178 ↾_t crest 16081 topGenctg 16098 Topctop 20698 TopOnctopon 20715 Cn ccn 21028 IIcii 22678 ≃_phcphtpc 22768 *_𝑝cpco 22800

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-cn 21031 df-cnp 21032 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-ii 22680 df-htpy 22769 df-phtpy 22770 df-phtpc 22791 df-pco 22805

This theorem is referenced by: pcophtb 22829 pi1grplem 22849 pi1xfr 22855 pi1xfrcnvlem 22856

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