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Theorem htpycc 23302

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 23205	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
7		eqid 2771	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
8		eqid 2771	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
9		eqid 2771	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
10		dfii2 23208	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
11		0red 10441	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
12		1red 10438	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
13		halfre 11659	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
14		halfge0 11662	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
15		1re 10437	. . . . . . . 8 ⊢ 1 ∈ ℝ
16		halflt1 11663	. . . . . . . 8 ⊢ (1 / 2) < 1
17	13, 15, 16	ltleii 10561	. . . . . . 7 ⊢ (1 / 2) ≤ 1
18		elicc01 12668	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
19	13, 14, 17, 18	mpbir3an 1322	. . . . . 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 23296	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝐿0) = (𝐹‘𝑠) ∧ (𝑠𝐿1) = (𝐺‘𝑠)))
24	23	simprd 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝐺‘𝑠))
25		htpycc.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻))
26	1, 21, 3, 25	htpyi 23296	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝑀0) = (𝐺‘𝑠) ∧ (𝑠𝑀1) = (𝐻‘𝑠)))
27	26	simpld 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑀0) = (𝐺‘𝑠))
28	24, 27	eqtr4d 2810	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝑠𝑀0))
29	28	ralrimiva 3125	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0))
30		oveq1 6981	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝐿1) = (𝑥𝐿1))
31		oveq1 6981	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝑀0) = (𝑥𝑀0))
32	30, 31	eqeq12d 2786	. . . . . . . . 9 ⊢ (𝑠 = 𝑥 → ((𝑠𝐿1) = (𝑠𝑀0) ↔ (𝑥𝐿1) = (𝑥𝑀0)))
33	32	rspccva 3527	. . . . . . . 8 ⊢ ((∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐿1) = (𝑥𝑀0))
34	29, 33	sylan 572	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐿1) = (𝑥𝑀0))
35	34	adantrl 704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿1) = (𝑥𝑀0))
36		simprl 759	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → 𝑦 = (1 / 2))
37	36	oveq2d 6990	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = (2 · (1 / 2)))
38		2cn 11513	. . . . . . . . 9 ⊢ 2 ∈ ℂ
39		2ne0 11549	. . . . . . . . 9 ⊢ 2 ≠ 0
40	38, 39	recidi 11170	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
41	37, 40	syl6eq 2823	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = 1)
42	41	oveq2d 6990	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝐿1))
43	41	oveq1d 6989	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = (1 − 1))
44		1m1e0 11510	. . . . . . . 8 ⊢ (1 − 1) = 0
45	43, 44	syl6eq 2823	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = 0)
46	45	oveq2d 6990	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑥𝑀0))
47	35, 42, 46	3eqtr4d 2817	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝑀((2 · 𝑦) − 1)))
48		retopon 23090	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
49		0re 10439	. . . . . . . . 9 ⊢ 0 ∈ ℝ
50		iccssre 12632	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
51	49, 13, 50	mp2an 680	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
52		resttopon 21488	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
53	48, 51, 52	mp2an 680	. . . . . . 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 21996	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐽))
56	54, 1	cnmpt1st 21995	. . . . . . 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 23254	. . . . . . . 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 6982	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (2 · 𝑧) = (2 · 𝑦))
60	54, 1, 56, 54, 58, 59	cnmpt21 21998	. . . . . 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 23295	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×_t II) Cn 𝐾))
62	61, 22	sseldd 3852	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ((𝐽 ×_t II) Cn 𝐾))
63	54, 1, 55, 60, 62	cnmpt22f 22002	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ (𝑥𝐿(2 · 𝑦))) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐾))
64		iccssre 12632	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
65	13, 15, 64	mp2an 680	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
66		resttopon 21488	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
67	48, 65, 66	mp2an 680	. . . . . . 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 21996	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn 𝐽))
70	68, 1	cnmpt1st 21995	. . . . . . 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 23256	. . . . . . . 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 6989	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((2 · 𝑧) − 1) = ((2 · 𝑦) − 1))
74	68, 1, 70, 68, 72, 73	cnmpt21 21998	. . . . . 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 23295	. . . . . . 7 ⊢ (𝜑 → (𝐺(𝐽 Htpy 𝐾)𝐻) ⊆ ((𝐽 ×_t II) Cn 𝐾))
76	75, 25	sseldd 3852	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×_t II) Cn 𝐾))
77	68, 1, 69, 74, 76	cnmpt22f 22002	. . . . 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 23250	. . . 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 22005	. . 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	syl5eqel 2863	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×_t II) Cn 𝐾))
81		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
82		0elunit 12669	. . . 4 ⊢ 0 ∈ (0[,]1)
83		simpr 477	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
84	83, 14	syl6eqbr 4964	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 ≤ (1 / 2))
85	84	iftrued 4352	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑥𝐿(2 · 𝑦)))
86		simpl 475	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
87	83	oveq2d 6990	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = (2 · 0))
88		2t0e0 11614	. . . . . . . 8 ⊢ (2 · 0) = 0
89	87, 88	syl6eq 2823	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = 0)
90	86, 89	oveq12d 6992	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐿(2 · 𝑦)) = (𝑠𝐿0))
91	85, 90	eqtrd 2807	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝐿0))
92		ovex 7006	. . . . 5 ⊢ (𝑠𝐿0) ∈ V
93	91, 4, 92	ovmpoa 7119	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = (𝑠𝐿0))
94	81, 82, 93	sylancl 578	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝑠𝐿0))
95	23	simpld 487	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿0) = (𝐹‘𝑠))
96	94, 95	eqtrd 2807	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝐹‘𝑠))
97		1elunit 12670	. . . 4 ⊢ 1 ∈ (0[,]1)
98	13, 15	ltnlei 10559	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
99	16, 98	mpbi 222	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
100		simpr 477	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
101	100	breq1d 4935	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑦 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
102	99, 101	mtbiri 319	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ¬ 𝑦 ≤ (1 / 2))
103	102	iffalsed 4355	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑥𝑀((2 · 𝑦) − 1)))
104		simpl 475	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
105	100	oveq2d 6990	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = (2 · 1))
106		2t1e2 11608	. . . . . . . . . 10 ⊢ (2 · 1) = 2
107	105, 106	syl6eq 2823	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = 2)
108	107	oveq1d 6989	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = (2 − 1))
109		2m1e1 11571	. . . . . . . 8 ⊢ (2 − 1) = 1
110	108, 109	syl6eq 2823	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = 1)
111	104, 110	oveq12d 6992	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑠𝑀1))
112	103, 111	eqtrd 2807	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝑀1))
113		ovex 7006	. . . . 5 ⊢ (𝑠𝑀1) ∈ V
114	112, 4, 113	ovmpoa 7119	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = (𝑠𝑀1))
115	81, 97, 114	sylancl 578	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝑠𝑀1))
116	26	simprd 488	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑀1) = (𝐻‘𝑠))
117	115, 116	eqtrd 2807	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝐻‘𝑠))
118	1, 2, 3, 80, 96, 117	ishtpyd 23297	1 ⊢ (𝜑 → 𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ⊆ wss 3822 ifcif 4344 class class class wbr 4925 ↦ cmpt 5004 ran crn 5404 ‘cfv 6185 (class class class)co 6974 ∈ cmpo 6976 ℝcr 10332 0cc0 10333 1c1 10334 · cmul 10338 < clt 10472 ≤ cle 10473 − cmin 10668 / cdiv 11096 2c2 11493 (,)cioo 12552 [,]cicc 12555 ↾_t crest 16548 topGenctg 16565 TopOnctopon 21237 Cn ccn 21551 ×_t ctx 21887 IIcii 23201 Htpy chtpy 23289

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 ax-mulf 10413

This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-of 7225 df-om 7395 df-1st 7499 df-2nd 7500 df-supp 7632 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-2o 7904 df-oadd 7907 df-er 8087 df-map 8206 df-ixp 8258 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fsupp 8627 df-fi 8668 df-sup 8699 df-inf 8700 df-oi 8767 df-card 9160 df-cda 9386 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-icc 12559 df-fz 12707 df-fzo 12848 df-seq 13183 df-exp 13243 df-hash 13504 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-sets 16344 df-ress 16345 df-plusg 16432 df-mulr 16433 df-starv 16434 df-sca 16435 df-vsca 16436 df-ip 16437 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-hom 16443 df-cco 16444 df-rest 16550 df-topn 16551 df-0g 16569 df-gsum 16570 df-topgen 16571 df-pt 16572 df-prds 16575 df-xrs 16629 df-qtop 16634 df-imas 16635 df-xps 16637 df-mre 16727 df-mrc 16728 df-acs 16730 df-mgm 17722 df-sgrp 17764 df-mnd 17775 df-submnd 17816 df-mulg 18024 df-cntz 18230 df-cmn 18680 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-cn 21554 df-cnp 21555 df-tx 21889 df-hmeo 22082 df-xms 22648 df-ms 22649 df-tms 22650 df-ii 23203 df-htpy 23292

This theorem is referenced by: phtpycc 23313

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