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Theorem pcopt2 25069

Description: Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)

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

**Assertion**
Ref	Expression
pcopt2	⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃)( ≃_ph‘𝐽)𝐹)

Proof of Theorem pcopt2 Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pcopt.1	. . . . . . . . 9 ⊢ 𝑃 = ((0[,]1) × {𝑌})
2	1	fveq1i 6907	. . . . . . . 8 ⊢ (𝑃‘((2 · 𝑥) − 1)) = (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1))
3		simpr 484	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹‘1) = 𝑌)
4		iiuni 24920	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
5		eqid 2734	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 23269	. . . . . . . . . . . 12 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐹:(0[,]1)⟶∪ 𝐽)
7	6	adantr 480	. . . . . . . . . . 11 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹:(0[,]1)⟶∪ 𝐽)
8		1elunit 13506	. . . . . . . . . . 11 ⊢ 1 ∈ (0[,]1)
9		ffvelcdm 7100	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 586	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹‘1) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2839	. . . . . . . . 9 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii2 24978	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
13		iihalf2 24974	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((1 / 2)[,]1) → ((2 · 𝑥) − 1) ∈ (0[,]1))
14	12, 13	syl 17	. . . . . . . . 9 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → ((2 · 𝑥) − 1) ∈ (0[,]1))
15		fvconst2g 7221	. . . . . . . . 9 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ ((2 · 𝑥) − 1) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1)) = 𝑌)
16	11, 14, 15	syl2an 596	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1)) = 𝑌)
17	2, 16	eqtrid 2786	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝑃‘((2 · 𝑥) − 1)) = 𝑌)
18		simplr 769	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝐹‘1) = 𝑌)
19	17, 18	eqtr4d 2777	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝑃‘((2 · 𝑥) − 1)) = (𝐹‘1))
20	19	anassrs 467	. . . . 5 ⊢ ((((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) ∧ ¬ 𝑥 ≤ (1 / 2)) → (𝑃‘((2 · 𝑥) − 1)) = (𝐹‘1))
21	20	ifeq2da 4562	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝑃‘((2 · 𝑥) − 1))) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1)))
22	21	mpteq2dva 5247	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝑃‘((2 · 𝑥) − 1)))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))))
23		simpl 482	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
24		cntop2 23264	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ Top)
26		toptopon2 22939	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
27	25, 26	sylib 218	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
28	1	pcoptcl 25067	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
29	27, 11, 28	syl2anc 584	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
30	29	simp1d 1141	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
31	23, 30	pcoval 25057	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝑃‘((2 · 𝑥) − 1)))))
32		iftrue 4536	. . . . . . . . 9 ⊢ (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
33	32	adantl 481	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
34		elii1 24977	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
35		iihalf1 24971	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
36	34, 35	sylbir 235	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
37	33, 36	eqeltrd 2838	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
38	37	ex 412	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1)))
39		iffalse 4539	. . . . . . 7 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
40	39, 8	eqeltrdi 2846	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
41	38, 40	pm2.61d1 180	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
42	41	adantl 481	. . . 4 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
43		eqidd 2735	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))
44	7	feqmptd 6976	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
45		fveq2 6906	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))
46		fvif 6922	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))
47	45, 46	eqtrdi 2790	. . . 4 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) → (𝐹‘𝑦) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1)))
48	42, 43, 44, 47	fmptco 7148	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))))
49	22, 31, 48	3eqtr4d 2784	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃) = (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))))
50		iitopon 24918	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
51	50	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
52	51	cnmptid 23684	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
53		0elunit 13505	. . . . . 6 ⊢ 0 ∈ (0[,]1)
54	53	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ (0[,]1))
55	51, 51, 54	cnmptc 23685	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
56		eqid 2734	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		eqid 2734	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
58		eqid 2734	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
59		dfii2 24921	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
60		0re 11260	. . . . . 6 ⊢ 0 ∈ ℝ
61	60	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ ℝ)
62		1re 11258	. . . . . 6 ⊢ 1 ∈ ℝ
63	62	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ ℝ)
64		halfre 12477	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
65		halfge0 12480	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
66		halflt1 12481	. . . . . . . 8 ⊢ (1 / 2) < 1
67	64, 62, 66	ltleii 11381	. . . . . . 7 ⊢ (1 / 2) ≤ 1
68		elicc01 13502	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
69	64, 65, 67, 68	mpbir3an 1340	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
70	69	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (1 / 2) ∈ (0[,]1))
71		simprl 771	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
72	71	oveq2d 7446	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
73		2cn 12338	. . . . . . 7 ⊢ 2 ∈ ℂ
74		2ne0 12367	. . . . . . 7 ⊢ 2 ≠ 0
75	73, 74	recidi 11995	. . . . . 6 ⊢ (2 · (1 / 2)) = 1
76	72, 75	eqtrdi 2790	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
77		retopon 24799	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
78		iccssre 13465	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
79	60, 64, 78	mp2an 692	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
80		resttopon 23184	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
81	77, 79, 80	mp2an 692	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
82	81	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
83	82, 51	cnmpt1st 23691	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑦 ∈ (0[,](1 / 2)), 𝑧 ∈ (0[,]1) ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))))
84	57	iihalf1cn 24972	. . . . . . 7 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II)
85	84	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II))
86		oveq2 7438	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
87	82, 51, 83, 82, 85, 86	cnmpt21 23694	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑦 ∈ (0[,](1 / 2)), 𝑧 ∈ (0[,]1) ↦ (2 · 𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t II) Cn II))
88		iccssre 13465	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
89	64, 62, 88	mp2an 692	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
90		resttopon 23184	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
91	77, 89, 90	mp2an 692	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
92	91	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
93	8	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ (0[,]1))
94	92, 51, 51, 93	cnmpt2c 23693	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑦 ∈ ((1 / 2)[,]1), 𝑧 ∈ (0[,]1) ↦ 1) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t II) Cn II))
95	56, 57, 58, 59, 61, 63, 70, 51, 76, 87, 94	cnmpopc 24968	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑦 ∈ (0[,]1), 𝑧 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1)) ∈ ((II ×_t II) Cn II))
96		breq1 5150	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
97		oveq2 7438	. . . . . 6 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
98	96, 97	ifbieq1d 4554	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
99	98	adantr 480	. . . 4 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 0) → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
100	51, 52, 55, 51, 51, 95, 99	cnmpt12 23690	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) ∈ (II Cn II))
101		id 22	. . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 = 0)
102	101, 65	eqbrtrdi 5186	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 ≤ (1 / 2))
103	102, 32	syl 17	. . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
104		oveq2 7438	. . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0))
105		2t0e0 12432	. . . . . . 7 ⊢ (2 · 0) = 0
106	104, 105	eqtrdi 2790	. . . . . 6 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0)
107	103, 106	eqtrd 2774	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 0)
108		eqid 2734	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
109		c0ex 11252	. . . . 5 ⊢ 0 ∈ V
110	107, 108, 109	fvmpt 7015	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
111	53, 110	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
112	64, 62	ltnlei 11379	. . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
113	66, 112	mpbi 230	. . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2)
114		breq1 5150	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
115	113, 114	mtbiri 327	. . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2))
116	115, 39	syl 17	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
117		1ex 11254	. . . . 5 ⊢ 1 ∈ V
118	116, 108, 117	fvmpt 7015	. . . 4 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
119	8, 118	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
120	23, 100, 111, 119	reparpht 25044	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))( ≃_ph‘𝐽)𝐹)
121	49, 120	eqbrtrd 5169	1 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹(*_𝑝‘𝐽)𝑃)( ≃_ph‘𝐽)𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 ifcif 4530 {csn 4630 ∪ cuni 4911 class class class wbr 5147 ↦ cmpt 5230 × cxp 5686 ran crn 5689 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℝcr 11151 0cc0 11152 1c1 11153 · cmul 11157 < clt 11292 ≤ cle 11293 − cmin 11489 / cdiv 11917 2c2 12318 (,)cioo 13383 [,]cicc 13386 ↾_t crest 17466 topGenctg 17483 Topctop 22914 TopOnctopon 22931 Cn ccn 23247 IIcii 24914 ≃_phcphtpc 25014 *_𝑝cpco 25046

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-supp 8184 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-2o 8505 df-er 8743 df-map 8866 df-ixp 8936 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fsupp 9399 df-fi 9448 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-ioo 13387 df-icc 13390 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-hom 17321 df-cco 17322 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-mulg 19098 df-cntz 19347 df-cmn 19814 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-cn 23250 df-cnp 23251 df-tx 23585 df-hmeo 23778 df-xms 24345 df-ms 24346 df-tms 24347 df-ii 24916 df-htpy 25015 df-phtpy 25016 df-phtpc 25037 df-pco 25051

This theorem is referenced by: pcophtb 25075 pi1xfrcnvlem 25102

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