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Theorem pcopt2 22631

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 6104	. . . . . . . 8 ⊢ (𝑃‘((2 · 𝑥) − 1)) = (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1))
3		simpr 476	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹‘1) = 𝑌)
4		iiuni 22492	. . . . . . . . . . . . 13 ⊢ (0[,]1) = ∪ II
5		eqid 2610	. . . . . . . . . . . . 13 ⊢ ∪ 𝐽 = ∪ 𝐽
6	4, 5	cnf 20860	. . . . . . . . . . . 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 12162	. . . . . . . . . . 11 ⊢ 1 ∈ (0[,]1)
9		ffvelrn 6265	. . . . . . . . . . 11 ⊢ ((𝐹:(0[,]1)⟶∪ 𝐽 ∧ 1 ∈ (0[,]1)) → (𝐹‘1) ∈ ∪ 𝐽)
10	7, 8, 9	sylancl 693	. . . . . . . . . 10 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹‘1) ∈ ∪ 𝐽)
11	3, 10	eqeltrrd 2689	. . . . . . . . 9 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑌 ∈ ∪ 𝐽)
12		elii2 22543	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2)) → 𝑥 ∈ ((1 / 2)[,]1))
13		iihalf2 22540	. . . . . . . . . 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 6372	. . . . . . . . 9 ⊢ ((𝑌 ∈ ∪ 𝐽 ∧ ((2 · 𝑥) − 1) ∈ (0[,]1)) → (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1)) = 𝑌)
16	11, 14, 15	syl2an 493	. . . . . . . 8 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (((0[,]1) × {𝑌})‘((2 · 𝑥) − 1)) = 𝑌)
17	2, 16	syl5eq 2656	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝑃‘((2 · 𝑥) − 1)) = 𝑌)
18		simplr 788	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝐹‘1) = 𝑌)
19	17, 18	eqtr4d 2647	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑥 ∈ (0[,]1) ∧ ¬ 𝑥 ≤ (1 / 2))) → (𝑃‘((2 · 𝑥) − 1)) = (𝐹‘1))
20	19	anassrs 678	. . . . 5 ⊢ ((((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ 𝑥 ∈ (0[,]1)) ∧ ¬ 𝑥 ≤ (1 / 2)) → (𝑃‘((2 · 𝑥) − 1)) = (𝐹‘1))
21	20	ifeq2da 4067	. . . 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 4672	. . 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 472	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 ∈ (II Cn 𝐽))
24		cntop2 20855	. . . . . . . 8 ⊢ (𝐹 ∈ (II Cn 𝐽) → 𝐽 ∈ Top)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ Top)
26	5	toptopon 20548	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
27	25, 26	sylib 207	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐽 ∈ (TopOn‘∪ 𝐽))
28	1	pcoptcl 22629	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘∪ 𝐽) ∧ 𝑌 ∈ ∪ 𝐽) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
29	27, 11, 28	syl2anc 691	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑃 ∈ (II Cn 𝐽) ∧ (𝑃‘0) = 𝑌 ∧ (𝑃‘1) = 𝑌))
30	29	simp1d 1066	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝑃 ∈ (II Cn 𝐽))
31	23, 30	pcoval 22619	. . 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 4042	. . . . . . . . 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 22542	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↔ (𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)))
35		iihalf1 22538	. . . . . . . . 9 ⊢ (𝑥 ∈ (0[,](1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
36	34, 35	sylbir 224	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → (2 · 𝑥) ∈ (0[,]1))
37	33, 36	eqeltrd 2688	. . . . . . 7 ⊢ ((𝑥 ∈ (0[,]1) ∧ 𝑥 ≤ (1 / 2)) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
38	37	ex 449	. . . . . 6 ⊢ (𝑥 ∈ (0[,]1) → (𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1)))
39		iffalse 4045	. . . . . . 7 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
40	39, 8	syl6eqel 2696	. . . . . 6 ⊢ (¬ 𝑥 ≤ (1 / 2) → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) ∈ (0[,]1))
41	38, 40	pm2.61d1 170	. . . . 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 2611	. . . 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 6159	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 𝐹 = (𝑦 ∈ (0[,]1) ↦ (𝐹‘𝑦)))
45		fveq2 6103	. . . . 5 ⊢ (𝑦 = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) → (𝐹‘𝑦) = (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))
46		fvif 6114	. . . . 5 ⊢ (𝐹‘if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = if(𝑥 ≤ (1 / 2), (𝐹‘(2 · 𝑥)), (𝐹‘1))
47	45, 46	syl6eq 2660	. . . 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 6303	. . 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 2654	. 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 22490	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
51	50	a1i 11	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → II ∈ (TopOn‘(0[,]1)))
52	51	cnmptid 21274	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 𝑥) ∈ (II Cn II))
53		0elunit 12161	. . . . . 6 ⊢ 0 ∈ (0[,]1)
54	53	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ (0[,]1))
55	51, 51, 54	cnmptc 21275	. . . 4 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ 0) ∈ (II Cn II))
56		eqid 2610	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
57		eqid 2610	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
58		eqid 2610	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
59		dfii2 22493	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
60		0re 9919	. . . . . 6 ⊢ 0 ∈ ℝ
61	60	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 0 ∈ ℝ)
62		1re 9918	. . . . . 6 ⊢ 1 ∈ ℝ
63	62	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ ℝ)
64		halfre 11123	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
65		halfgt0 11125	. . . . . . . 8 ⊢ 0 < (1 / 2)
66	60, 64, 65	ltleii 10039	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
67		halflt1 11127	. . . . . . . 8 ⊢ (1 / 2) < 1
68	64, 62, 67	ltleii 10039	. . . . . . 7 ⊢ (1 / 2) ≤ 1
69	60, 62	elicc2i 12110	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
70	64, 66, 68, 69	mpbir3an 1237	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
71	70	a1i 11	. . . . 5 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (1 / 2) ∈ (0[,]1))
72		simprl 790	. . . . . . 7 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → 𝑦 = (1 / 2))
73	72	oveq2d 6565	. . . . . 6 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = (2 · (1 / 2)))
74		2cn 10968	. . . . . . 7 ⊢ 2 ∈ ℂ
75		2ne0 10990	. . . . . . 7 ⊢ 2 ≠ 0
76	74, 75	recidi 10635	. . . . . 6 ⊢ (2 · (1 / 2)) = 1
77	73, 76	syl6eq 2660	. . . . 5 ⊢ (((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) ∧ (𝑦 = (1 / 2) ∧ 𝑧 ∈ (0[,]1))) → (2 · 𝑦) = 1)
78		retopon 22377	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
79		iccssre 12126	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
80	60, 64, 79	mp2an 704	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
81		resttopon 20775	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
82	78, 80, 81	mp2an 704	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
83	82	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
84	83, 51	cnmpt1st 21281	. . . . . 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)))))
85	57	iihalf1cn 22539	. . . . . . 7 ⊢ (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II)
86	85	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,](1 / 2)) ↦ (2 · 𝑥)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II))
87		oveq2 6557	. . . . . 6 ⊢ (𝑥 = 𝑦 → (2 · 𝑥) = (2 · 𝑦))
88	83, 51, 84, 83, 86, 87	cnmpt21 21284	. . . . 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))
89		iccssre 12126	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
90	64, 62, 89	mp2an 704	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
91		resttopon 20775	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
92	78, 90, 91	mp2an 704	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
93	92	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
94	8	a1i 11	. . . . . 6 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → 1 ∈ (0[,]1))
95	93, 51, 51, 94	cnmpt2c 21283	. . . . 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))
96	56, 57, 58, 59, 61, 63, 71, 51, 77, 88, 95	cnmpt2pc 22535	. . . 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))
97		breq1 4586	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦 ≤ (1 / 2) ↔ 𝑥 ≤ (1 / 2)))
98		oveq2 6557	. . . . . 6 ⊢ (𝑦 = 𝑥 → (2 · 𝑦) = (2 · 𝑥))
99	97, 98	ifbieq1d 4059	. . . . 5 ⊢ (𝑦 = 𝑥 → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
100	99	adantr 480	. . . 4 ⊢ ((𝑦 = 𝑥 ∧ 𝑧 = 0) → if(𝑦 ≤ (1 / 2), (2 · 𝑦), 1) = if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
101	51, 52, 55, 51, 51, 96, 100	cnmpt12 21280	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) ∈ (II Cn II))
102		id 22	. . . . . . . 8 ⊢ (𝑥 = 0 → 𝑥 = 0)
103	102, 66	syl6eqbr 4622	. . . . . . 7 ⊢ (𝑥 = 0 → 𝑥 ≤ (1 / 2))
104	103, 32	syl 17	. . . . . 6 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = (2 · 𝑥))
105		oveq2 6557	. . . . . . 7 ⊢ (𝑥 = 0 → (2 · 𝑥) = (2 · 0))
106		2t0e0 11060	. . . . . . 7 ⊢ (2 · 0) = 0
107	105, 106	syl6eq 2660	. . . . . 6 ⊢ (𝑥 = 0 → (2 · 𝑥) = 0)
108	104, 107	eqtrd 2644	. . . . 5 ⊢ (𝑥 = 0 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 0)
109		eqid 2610	. . . . 5 ⊢ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)) = (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))
110		c0ex 9913	. . . . 5 ⊢ 0 ∈ V
111	108, 109, 110	fvmpt 6191	. . . 4 ⊢ (0 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
112	53, 111	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘0) = 0)
113	64, 62	ltnlei 10037	. . . . . . . 8 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
114	67, 113	mpbi 219	. . . . . . 7 ⊢ ¬ 1 ≤ (1 / 2)
115		breq1 4586	. . . . . . 7 ⊢ (𝑥 = 1 → (𝑥 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
116	114, 115	mtbiri 316	. . . . . 6 ⊢ (𝑥 = 1 → ¬ 𝑥 ≤ (1 / 2))
117	116, 39	syl 17	. . . . 5 ⊢ (𝑥 = 1 → if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1) = 1)
118		1ex 9914	. . . . 5 ⊢ 1 ∈ V
119	117, 109, 118	fvmpt 6191	. . . 4 ⊢ (1 ∈ (0[,]1) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
120	8, 119	mp1i 13	. . 3 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → ((𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1))‘1) = 1)
121	23, 101, 112, 120	reparpht 22606	. 2 ⊢ ((𝐹 ∈ (II Cn 𝐽) ∧ (𝐹‘1) = 𝑌) → (𝐹 ∘ (𝑥 ∈ (0[,]1) ↦ if(𝑥 ≤ (1 / 2), (2 · 𝑥), 1)))( ≃_ph‘𝐽)𝐹)
122	49, 121	eqbrtrd 4605	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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 ifcif 4036 {csn 4125 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 × cxp 5036 ran crn 5039 ∘ ccom 5042 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℝcr 9814 0cc0 9815 1c1 9816 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 2c2 10947 (,)cioo 12046 [,]cicc 12049 ↾_t crest 15904 topGenctg 15921 Topctop 20517 TopOnctopon 20518 Cn ccn 20838 IIcii 22486 ≃_phcphtpc 22576 *_𝑝cpco 22608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-icc 12053 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-mulg 17364 df-cntz 17573 df-cmn 18018 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-cn 20841 df-cnp 20842 df-tx 21175 df-hmeo 21368 df-xms 21935 df-ms 21936 df-tms 21937 df-ii 22488 df-htpy 22577 df-phtpy 22578 df-phtpc 22599 df-pco 22613

This theorem is referenced by: pcophtb 22637 pi1xfrcnvlem 22664

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