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Theorem htpycc 24141

Description: Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)

**Hypotheses**
Ref	Expression
htpycc.1	⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))))
htpycc.2	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
htpycc.4	⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
htpycc.5	⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
htpycc.6	⊢ (𝜑 → 𝐻 ∈ (𝐽 Cn 𝐾))
htpycc.7	⊢ (𝜑 → 𝐿 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
htpycc.8	⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻))

**Assertion**
Ref	Expression
htpycc	⊢ (𝜑 → 𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))

Distinct variable groups: 𝑥,𝑦,𝐽 𝑥,𝐾,𝑦 𝑥,𝐿,𝑦 𝑥,𝑀,𝑦 𝑥,𝑋,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝐻(𝑥,𝑦) 𝑁(𝑥,𝑦)

Proof of Theorem htpycc Dummy variables 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		htpycc.2	. 2 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
2		htpycc.4	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾))
3		htpycc.6	. 2 ⊢ (𝜑 → 𝐻 ∈ (𝐽 Cn 𝐾))
4		htpycc.1	. . 3 ⊢ 𝑁 = (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))))
5		iitopon 24040	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
7		eqid 2740	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
8		eqid 2740	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
9		eqid 2740	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
10		dfii2 24043	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
11		0red 10979	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
12		1red 10977	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
13		halfre 12187	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
14		halfge0 12190	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
15		1re 10976	. . . . . . . 8 ⊢ 1 ∈ ℝ
16		halflt1 12191	. . . . . . . 8 ⊢ (1 / 2) < 1
17	13, 15, 16	ltleii 11098	. . . . . . 7 ⊢ (1 / 2) ≤ 1
18		elicc01 13197	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
19	13, 14, 17, 18	mpbir3an 1340	. . . . . 6 ⊢ (1 / 2) ∈ (0[,]1)
20	19	a1i 11	. . . . 5 ⊢ (𝜑 → (1 / 2) ∈ (0[,]1))
21		htpycc.5	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾))
22		htpycc.7	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐿 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺))
23	1, 2, 21, 22	htpyi 24135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝐿0) = (𝐹‘𝑠) ∧ (𝑠𝐿1) = (𝐺‘𝑠)))
24	23	simprd 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝐺‘𝑠))
25		htpycc.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻))
26	1, 21, 3, 25	htpyi 24135	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝑀0) = (𝐺‘𝑠) ∧ (𝑠𝑀1) = (𝐻‘𝑠)))
27	26	simpld 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑀0) = (𝐺‘𝑠))
28	24, 27	eqtr4d 2783	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝑠𝑀0))
29	28	ralrimiva 3110	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0))
30		oveq1 7278	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝐿1) = (𝑥𝐿1))
31		oveq1 7278	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝑀0) = (𝑥𝑀0))
32	30, 31	eqeq12d 2756	. . . . . . . . 9 ⊢ (𝑠 = 𝑥 → ((𝑠𝐿1) = (𝑠𝑀0) ↔ (𝑥𝐿1) = (𝑥𝑀0)))
33	32	rspccva 3560	. . . . . . . 8 ⊢ ((∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐿1) = (𝑥𝑀0))
34	29, 33	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐿1) = (𝑥𝑀0))
35	34	adantrl 713	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿1) = (𝑥𝑀0))
36		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → 𝑦 = (1 / 2))
37	36	oveq2d 7287	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = (2 · (1 / 2)))
38		2cn 12048	. . . . . . . . 9 ⊢ 2 ∈ ℂ
39		2ne0 12077	. . . . . . . . 9 ⊢ 2 ≠ 0
40	38, 39	recidi 11706	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
41	37, 40	eqtrdi 2796	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = 1)
42	41	oveq2d 7287	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝐿1))
43	41	oveq1d 7286	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = (1 − 1))
44		1m1e0 12045	. . . . . . . 8 ⊢ (1 − 1) = 0
45	43, 44	eqtrdi 2796	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = 0)
46	45	oveq2d 7287	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑥𝑀0))
47	35, 42, 46	3eqtr4d 2790	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝑀((2 · 𝑦) − 1)))
48		retopon 23925	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
49		0re 10978	. . . . . . . . 9 ⊢ 0 ∈ ℝ
50		iccssre 13160	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
51	49, 13, 50	mp2an 689	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
52		resttopon 22310	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
53	48, 51, 52	mp2an 689	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
54	53	a1i 11	. . . . . 6 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
55	54, 1	cnmpt2nd 22818	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐽))
56	54, 1	cnmpt1st 22817	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))))
57	8	iihalf1cn 24093	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,](1 / 2)) ↦ (2 · 𝑧)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II)
58	57	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ (0[,](1 / 2)) ↦ (2 · 𝑧)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II))
59		oveq2 7279	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (2 · 𝑧) = (2 · 𝑦))
60	54, 1, 56, 54, 58, 59	cnmpt21 22820	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ (2 · 𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn II))
61	1, 2, 21	htpycn 24134	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×_t II) Cn 𝐾))
62	61, 22	sseldd 3927	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ((𝐽 ×_t II) Cn 𝐾))
63	54, 1, 55, 60, 62	cnmpt22f 22824	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ (𝑥𝐿(2 · 𝑦))) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐾))
64		iccssre 13160	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
65	13, 15, 64	mp2an 689	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
66		resttopon 22310	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
67	48, 65, 66	mp2an 689	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
68	67	a1i 11	. . . . . 6 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
69	68, 1	cnmpt2nd 22818	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn 𝐽))
70	68, 1	cnmpt1st 22817	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))))
71	9	iihalf2cn 24095	. . . . . . . 8 ⊢ (𝑧 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑧) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II)
72	71	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑧) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II))
73	59	oveq1d 7286	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((2 · 𝑧) − 1) = ((2 · 𝑦) − 1))
74	68, 1, 70, 68, 72, 73	cnmpt21 22820	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ ((2 · 𝑦) − 1)) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn II))
75	1, 21, 3	htpycn 24134	. . . . . . 7 ⊢ (𝜑 → (𝐺(𝐽 Htpy 𝐾)𝐻) ⊆ ((𝐽 ×_t II) Cn 𝐾))
76	75, 25	sseldd 3927	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×_t II) Cn 𝐾))
77	68, 1, 69, 74, 76	cnmpt22f 22824	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ (𝑥𝑀((2 · 𝑦) − 1))) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn 𝐾))
78	7, 8, 9, 10, 11, 12, 20, 1, 47, 63, 77	cnmpopc 24089	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (0[,]1), 𝑥 ∈ 𝑋 ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1)))) ∈ ((II ×_t 𝐽) Cn 𝐾))
79	6, 1, 78	cnmptcom 22827	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1)))) ∈ ((𝐽 ×_t II) Cn 𝐾))
80	4, 79	eqeltrid 2845	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×_t II) Cn 𝐾))
81		simpr 485	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
82		0elunit 13200	. . . 4 ⊢ 0 ∈ (0[,]1)
83		simpr 485	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
84	83, 14	eqbrtrdi 5118	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 ≤ (1 / 2))
85	84	iftrued 4473	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑥𝐿(2 · 𝑦)))
86		simpl 483	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
87	83	oveq2d 7287	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = (2 · 0))
88		2t0e0 12142	. . . . . . . 8 ⊢ (2 · 0) = 0
89	87, 88	eqtrdi 2796	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = 0)
90	86, 89	oveq12d 7289	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐿(2 · 𝑦)) = (𝑠𝐿0))
91	85, 90	eqtrd 2780	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝐿0))
92		ovex 7304	. . . . 5 ⊢ (𝑠𝐿0) ∈ V
93	91, 4, 92	ovmpoa 7422	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = (𝑠𝐿0))
94	81, 82, 93	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝑠𝐿0))
95	23	simpld 495	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿0) = (𝐹‘𝑠))
96	94, 95	eqtrd 2780	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝐹‘𝑠))
97		1elunit 13201	. . . 4 ⊢ 1 ∈ (0[,]1)
98	13, 15	ltnlei 11096	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
99	16, 98	mpbi 229	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
100		simpr 485	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
101	100	breq1d 5089	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑦 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
102	99, 101	mtbiri 327	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ¬ 𝑦 ≤ (1 / 2))
103	102	iffalsed 4476	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑥𝑀((2 · 𝑦) − 1)))
104		simpl 483	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
105	100	oveq2d 7287	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = (2 · 1))
106		2t1e2 12136	. . . . . . . . . 10 ⊢ (2 · 1) = 2
107	105, 106	eqtrdi 2796	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = 2)
108	107	oveq1d 7286	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = (2 − 1))
109		2m1e1 12099	. . . . . . . 8 ⊢ (2 − 1) = 1
110	108, 109	eqtrdi 2796	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = 1)
111	104, 110	oveq12d 7289	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑠𝑀1))
112	103, 111	eqtrd 2780	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝑀1))
113		ovex 7304	. . . . 5 ⊢ (𝑠𝑀1) ∈ V
114	112, 4, 113	ovmpoa 7422	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = (𝑠𝑀1))
115	81, 97, 114	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝑠𝑀1))
116	26	simprd 496	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑀1) = (𝐻‘𝑠))
117	115, 116	eqtrd 2780	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝐻‘𝑠))
118	1, 2, 3, 80, 96, 117	ishtpyd 24136	1 ⊢ (𝜑 → 𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ∀wral 3066 ⊆ wss 3892 ifcif 4465 class class class wbr 5079 ↦ cmpt 5162 ran crn 5591 ‘cfv 6432 (class class class)co 7271 ∈ cmpo 7273 ℝcr 10871 0cc0 10872 1c1 10873 · cmul 10877 < clt 11010 ≤ cle 11011 − cmin 11205 / cdiv 11632 2c2 12028 (,)cioo 13078 [,]cicc 13081 ↾_t crest 17129 topGenctg 17146 TopOnctopon 22057 Cn ccn 22373 ×_t ctx 22709 IIcii 24036 Htpy chtpy 24128

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 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-pre-sup 10950 ax-mulf 10952

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-isom 6441 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-supp 7969 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-2o 8289 df-er 8481 df-map 8600 df-ixp 8669 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-fsupp 9107 df-fi 9148 df-sup 9179 df-inf 9180 df-oi 9247 df-card 9698 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12437 df-uz 12582 df-q 12688 df-rp 12730 df-xneg 12847 df-xadd 12848 df-xmul 12849 df-ioo 13082 df-icc 13085 df-fz 13239 df-fzo 13382 df-seq 13720 df-exp 13781 df-hash 14043 df-cj 14808 df-re 14809 df-im 14810 df-sqrt 14944 df-abs 14945 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-starv 16975 df-sca 16976 df-vsca 16977 df-ip 16978 df-tset 16979 df-ple 16980 df-ds 16982 df-unif 16983 df-hom 16984 df-cco 16985 df-rest 17131 df-topn 17132 df-0g 17150 df-gsum 17151 df-topgen 17152 df-pt 17153 df-prds 17156 df-xrs 17211 df-qtop 17216 df-imas 17217 df-xps 17219 df-mre 17293 df-mrc 17294 df-acs 17296 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-mulg 18699 df-cntz 18921 df-cmn 19386 df-psmet 20587 df-xmet 20588 df-met 20589 df-bl 20590 df-mopn 20591 df-cnfld 20596 df-top 22041 df-topon 22058 df-topsp 22080 df-bases 22094 df-cld 22168 df-cn 22376 df-cnp 22377 df-tx 22711 df-hmeo 22904 df-xms 23471 df-ms 23472 df-tms 23473 df-ii 24038 df-htpy 24131

This theorem is referenced by: phtpycc 24152

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