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Theorem htpycc 24907

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 24800	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
7		eqid 2733	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
8		eqid 2733	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
9		eqid 2733	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
10		dfii2 24803	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
11		0red 11122	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
12		1red 11120	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
13		halfre 12341	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
14		halfge0 12344	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
15		1re 11119	. . . . . . . 8 ⊢ 1 ∈ ℝ
16		halflt1 12345	. . . . . . . 8 ⊢ (1 / 2) < 1
17	13, 15, 16	ltleii 11243	. . . . . . 7 ⊢ (1 / 2) ≤ 1
18		elicc01 13368	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
19	13, 14, 17, 18	mpbir3an 1342	. . . . . 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 24901	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝐿0) = (𝐹‘𝑠) ∧ (𝑠𝐿1) = (𝐺‘𝑠)))
24	23	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝐺‘𝑠))
25		htpycc.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻))
26	1, 21, 3, 25	htpyi 24901	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝑀0) = (𝐺‘𝑠) ∧ (𝑠𝑀1) = (𝐻‘𝑠)))
27	26	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑀0) = (𝐺‘𝑠))
28	24, 27	eqtr4d 2771	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝑠𝑀0))
29	28	ralrimiva 3125	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0))
30		oveq1 7359	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝐿1) = (𝑥𝐿1))
31		oveq1 7359	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝑀0) = (𝑥𝑀0))
32	30, 31	eqeq12d 2749	. . . . . . . . 9 ⊢ (𝑠 = 𝑥 → ((𝑠𝐿1) = (𝑠𝑀0) ↔ (𝑥𝐿1) = (𝑥𝑀0)))
33	32	rspccva 3572	. . . . . . . 8 ⊢ ((∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐿1) = (𝑥𝑀0))
34	29, 33	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐿1) = (𝑥𝑀0))
35	34	adantrl 716	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿1) = (𝑥𝑀0))
36		simprl 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → 𝑦 = (1 / 2))
37	36	oveq2d 7368	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = (2 · (1 / 2)))
38		2cn 12207	. . . . . . . . 9 ⊢ 2 ∈ ℂ
39		2ne0 12236	. . . . . . . . 9 ⊢ 2 ≠ 0
40	38, 39	recidi 11859	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
41	37, 40	eqtrdi 2784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = 1)
42	41	oveq2d 7368	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝐿1))
43	41	oveq1d 7367	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = (1 − 1))
44		1m1e0 12204	. . . . . . . 8 ⊢ (1 − 1) = 0
45	43, 44	eqtrdi 2784	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = 0)
46	45	oveq2d 7368	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑥𝑀0))
47	35, 42, 46	3eqtr4d 2778	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝑀((2 · 𝑦) − 1)))
48		retopon 24679	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
49		0re 11121	. . . . . . . . 9 ⊢ 0 ∈ ℝ
50		iccssre 13331	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
51	49, 13, 50	mp2an 692	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
52		resttopon 23077	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
53	48, 51, 52	mp2an 692	. . . . . . 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 23585	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐽))
56	54, 1	cnmpt1st 23584	. . . . . . 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 24854	. . . . . . . 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 7360	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (2 · 𝑧) = (2 · 𝑦))
60	54, 1, 56, 54, 58, 59	cnmpt21 23587	. . . . . 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 24900	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×_t II) Cn 𝐾))
62	61, 22	sseldd 3931	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ((𝐽 ×_t II) Cn 𝐾))
63	54, 1, 55, 60, 62	cnmpt22f 23591	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ (𝑥𝐿(2 · 𝑦))) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐾))
64		iccssre 13331	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
65	13, 15, 64	mp2an 692	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
66		resttopon 23077	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
67	48, 65, 66	mp2an 692	. . . . . . 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 23585	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn 𝐽))
70	68, 1	cnmpt1st 23584	. . . . . . 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 24857	. . . . . . . 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 7367	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((2 · 𝑧) − 1) = ((2 · 𝑦) − 1))
74	68, 1, 70, 68, 72, 73	cnmpt21 23587	. . . . . 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 24900	. . . . . . 7 ⊢ (𝜑 → (𝐺(𝐽 Htpy 𝐾)𝐻) ⊆ ((𝐽 ×_t II) Cn 𝐾))
76	75, 25	sseldd 3931	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×_t II) Cn 𝐾))
77	68, 1, 69, 74, 76	cnmpt22f 23591	. . . . 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 24850	. . . 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 23594	. . 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 2837	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×_t II) Cn 𝐾))
81		simpr 484	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
82		0elunit 13371	. . . 4 ⊢ 0 ∈ (0[,]1)
83		simpr 484	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
84	83, 14	eqbrtrdi 5132	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 ≤ (1 / 2))
85	84	iftrued 4482	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑥𝐿(2 · 𝑦)))
86		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
87	83	oveq2d 7368	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = (2 · 0))
88		2t0e0 12296	. . . . . . . 8 ⊢ (2 · 0) = 0
89	87, 88	eqtrdi 2784	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = 0)
90	86, 89	oveq12d 7370	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐿(2 · 𝑦)) = (𝑠𝐿0))
91	85, 90	eqtrd 2768	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝐿0))
92		ovex 7385	. . . . 5 ⊢ (𝑠𝐿0) ∈ V
93	91, 4, 92	ovmpoa 7507	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = (𝑠𝐿0))
94	81, 82, 93	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝑠𝐿0))
95	23	simpld 494	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿0) = (𝐹‘𝑠))
96	94, 95	eqtrd 2768	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝐹‘𝑠))
97		1elunit 13372	. . . 4 ⊢ 1 ∈ (0[,]1)
98	13, 15	ltnlei 11241	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
99	16, 98	mpbi 230	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
100		simpr 484	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
101	100	breq1d 5103	. . . . . . . 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 4485	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑥𝑀((2 · 𝑦) − 1)))
104		simpl 482	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
105	100	oveq2d 7368	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = (2 · 1))
106		2t1e2 12290	. . . . . . . . . 10 ⊢ (2 · 1) = 2
107	105, 106	eqtrdi 2784	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = 2)
108	107	oveq1d 7367	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = (2 − 1))
109		2m1e1 12253	. . . . . . . 8 ⊢ (2 − 1) = 1
110	108, 109	eqtrdi 2784	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = 1)
111	104, 110	oveq12d 7370	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑠𝑀1))
112	103, 111	eqtrd 2768	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝑀1))
113		ovex 7385	. . . . 5 ⊢ (𝑠𝑀1) ∈ V
114	112, 4, 113	ovmpoa 7507	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = (𝑠𝑀1))
115	81, 97, 114	sylancl 586	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝑠𝑀1))
116	26	simprd 495	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑀1) = (𝐻‘𝑠))
117	115, 116	eqtrd 2768	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝐻‘𝑠))
118	1, 2, 3, 80, 96, 117	ishtpyd 24902	1 ⊢ (𝜑 → 𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3048 ⊆ wss 3898 ifcif 4474 class class class wbr 5093 ↦ cmpt 5174 ran crn 5620 ‘cfv 6486 (class class class)co 7352 ∈ cmpo 7354 ℝcr 11012 0cc0 11013 1c1 11014 · cmul 11018 < clt 11153 ≤ cle 11154 − cmin 11351 / cdiv 11781 2c2 12187 (,)cioo 13247 [,]cicc 13250 ↾_t crest 17326 topGenctg 17343 TopOnctopon 22826 Cn ccn 23140 ×_t ctx 23476 IIcii 24796 Htpy chtpy 24894

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-pre-sup 11091

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-supp 8097 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-2o 8392 df-er 8628 df-map 8758 df-ixp 8828 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-fsupp 9253 df-fi 9302 df-sup 9333 df-inf 9334 df-oi 9403 df-card 9839 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-div 11782 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-7 12200 df-8 12201 df-9 12202 df-n0 12389 df-z 12476 df-dec 12595 df-uz 12739 df-q 12849 df-rp 12893 df-xneg 13013 df-xadd 13014 df-xmul 13015 df-ioo 13251 df-icc 13254 df-fz 13410 df-fzo 13557 df-seq 13911 df-exp 13971 df-hash 14240 df-cj 15008 df-re 15009 df-im 15010 df-sqrt 15144 df-abs 15145 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-starv 17178 df-sca 17179 df-vsca 17180 df-ip 17181 df-tset 17182 df-ple 17183 df-ds 17185 df-unif 17186 df-hom 17187 df-cco 17188 df-rest 17328 df-topn 17329 df-0g 17347 df-gsum 17348 df-topgen 17349 df-pt 17350 df-prds 17353 df-xrs 17408 df-qtop 17413 df-imas 17414 df-xps 17416 df-mre 17490 df-mrc 17491 df-acs 17493 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-mulg 18983 df-cntz 19231 df-cmn 19696 df-psmet 21285 df-xmet 21286 df-met 21287 df-bl 21288 df-mopn 21289 df-cnfld 21294 df-top 22810 df-topon 22827 df-topsp 22849 df-bases 22862 df-cld 22935 df-cn 23143 df-cnp 23144 df-tx 23478 df-hmeo 23671 df-xms 24236 df-ms 24237 df-tms 24238 df-ii 24798 df-htpy 24897

This theorem is referenced by: phtpycc 24918

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