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Theorem htpycc 25120

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 25019	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
7		eqid 2763	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
8		eqid 2763	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
9		eqid 2763	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
10		dfii2 25022	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
11		0red 11212	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
12		1red 11210	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
13		halfre 12458	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
14		halfge0 12461	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
15		1re 11209	. . . . . . . 8 ⊢ 1 ∈ ℝ
16		halflt1 12462	. . . . . . . 8 ⊢ (1 / 2) < 1
17	13, 15, 16	ltleii 11334	. . . . . . 7 ⊢ (1 / 2) ≤ 1
18		elicc01 13494	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
19	13, 14, 17, 18	mpbir3an 1360	. . . . . 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 25114	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝐿0) = (𝐹‘𝑠) ∧ (𝑠𝐿1) = (𝐺‘𝑠)))
24	23	simprd 500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝐺‘𝑠))
25		htpycc.8	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (𝐺(𝐽 Htpy 𝐾)𝐻))
26	1, 21, 3, 25	htpyi 25114	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝑀0) = (𝐺‘𝑠) ∧ (𝑠𝑀1) = (𝐻‘𝑠)))
27	26	simpld 499	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑀0) = (𝐺‘𝑠))
28	24, 27	eqtr4d 2801	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝑠𝑀0))
29	28	ralrimiva 3157	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0))
30		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝐿1) = (𝑥𝐿1))
31		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝑀0) = (𝑥𝑀0))
32	30, 31	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑠 = 𝑥 → ((𝑠𝐿1) = (𝑠𝑀0) ↔ (𝑥𝐿1) = (𝑥𝑀0)))
33	32	rspccva 3581	. . . . . . . 8 ⊢ ((∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0) ∧ 𝑥 ∈ 𝑋) → (𝑥𝐿1) = (𝑥𝑀0))
34	29, 33	sylan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑥𝐿1) = (𝑥𝑀0))
35	34	adantrl 728	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿1) = (𝑥𝑀0))
36		simprl 782	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → 𝑦 = (1 / 2))
37	36	oveq2d 7428	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = (2 · (1 / 2)))
38		2thalfe1 12349	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
39	37, 38	eqtrdi 2814	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = 1)
40	39	oveq2d 7428	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝐿1))
41	39	oveq1d 7427	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = (1 − 1))
42		1m1e0 12314	. . . . . . . 8 ⊢ (1 − 1) = 0
43	41, 42	eqtrdi 2814	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = 0)
44	43	oveq2d 7428	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑥𝑀0))
45	35, 40, 44	3eqtr4d 2808	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝑀((2 · 𝑦) − 1)))
46		retopon 24901	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
47		0re 11211	. . . . . . . . 9 ⊢ 0 ∈ ℝ
48		iccssre 13457	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
49	47, 13, 48	mp2an 704	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
50		resttopon 23299	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ (0[,](1 / 2)) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
51	46, 49, 50	mp2an 704	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2)))
52	51	a1i 11	. . . . . 6 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ∈ (TopOn‘(0[,](1 / 2))))
53	52, 1	cnmpt2nd 23807	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐽))
54	52, 1	cnmpt1st 23806	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))))
55	8	iihalf1cn 25072	. . . . . . . 8 ⊢ (𝑧 ∈ (0[,](1 / 2)) ↦ (2 · 𝑧)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II)
56	55	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ (0[,](1 / 2)) ↦ (2 · 𝑧)) ∈ (((topGen‘ran (,)) ↾_t (0[,](1 / 2))) Cn II))
57		oveq2 7420	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (2 · 𝑧) = (2 · 𝑦))
58	52, 1, 54, 52, 56, 57	cnmpt21 23809	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ (2 · 𝑦)) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn II))
59	1, 2, 21	htpycn 25113	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×_t II) Cn 𝐾))
60	59, 22	sseldd 3939	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ((𝐽 ×_t II) Cn 𝐾))
61	52, 1, 53, 58, 60	cnmpt22f 23813	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ (𝑥𝐿(2 · 𝑦))) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐾))
62		iccssre 13457	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
63	13, 15, 62	mp2an 704	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
64		resttopon 23299	. . . . . . . 8 ⊢ (((topGen‘ran (,)) ∈ (TopOn‘ℝ) ∧ ((1 / 2)[,]1) ⊆ ℝ) → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
65	46, 63, 64	mp2an 704	. . . . . . 7 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1))
66	65	a1i 11	. . . . . 6 ⊢ (𝜑 → ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ∈ (TopOn‘((1 / 2)[,]1)))
67	66, 1	cnmpt2nd 23807	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn 𝐽))
68	66, 1	cnmpt1st 23806	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ 𝑦) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))))
69	9	iihalf2cn 25074	. . . . . . . 8 ⊢ (𝑧 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑧) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II)
70	69	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑧 ∈ ((1 / 2)[,]1) ↦ ((2 · 𝑧) − 1)) ∈ (((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) Cn II))
71	57	oveq1d 7427	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((2 · 𝑧) − 1) = ((2 · 𝑦) − 1))
72	66, 1, 68, 66, 70, 71	cnmpt21 23809	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ ((2 · 𝑦) − 1)) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn II))
73	1, 21, 3	htpycn 25113	. . . . . . 7 ⊢ (𝜑 → (𝐺(𝐽 Htpy 𝐾)𝐻) ⊆ ((𝐽 ×_t II) Cn 𝐾))
74	73, 25	sseldd 3939	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×_t II) Cn 𝐾))
75	66, 1, 67, 72, 74	cnmpt22f 23813	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ (𝑥𝑀((2 · 𝑦) − 1))) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn 𝐾))
76	7, 8, 9, 10, 11, 12, 20, 1, 45, 61, 75	cnmpopc 25068	. . . 4 ⊢ (𝜑 → (𝑦 ∈ (0[,]1), 𝑥 ∈ 𝑋 ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1)))) ∈ ((II ×_t 𝐽) Cn 𝐾))
77	6, 1, 76	cnmptcom 23816	. . 3 ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ (0[,]1) ↦ if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1)))) ∈ ((𝐽 ×_t II) Cn 𝐾))
78	4, 77	eqeltrid 2867	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×_t II) Cn 𝐾))
79		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
80		0elunit 13497	. . . 4 ⊢ 0 ∈ (0[,]1)
81		simpr 489	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
82	81, 14	eqbrtrdi 5151	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 ≤ (1 / 2))
83	82	iftrued 4496	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑥𝐿(2 · 𝑦)))
84		simpl 487	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
85	81	oveq2d 7428	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = (2 · 0))
86		2t0e0 12412	. . . . . . . 8 ⊢ (2 · 0) = 0
87	85, 86	eqtrdi 2814	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = 0)
88	84, 87	oveq12d 7430	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐿(2 · 𝑦)) = (𝑠𝐿0))
89	83, 88	eqtrd 2798	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝐿0))
90		ovex 7445	. . . . 5 ⊢ (𝑠𝐿0) ∈ V
91	89, 4, 90	ovmpoa 7567	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 0 ∈ (0[,]1)) → (𝑠𝑁0) = (𝑠𝐿0))
92	79, 80, 91	sylancl 597	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝑠𝐿0))
93	23	simpld 499	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿0) = (𝐹‘𝑠))
94	92, 93	eqtrd 2798	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝐹‘𝑠))
95		1elunit 13498	. . . 4 ⊢ 1 ∈ (0[,]1)
96	13, 15	ltnlei 11332	. . . . . . . . 9 ⊢ ((1 / 2) < 1 ↔ ¬ 1 ≤ (1 / 2))
97	16, 96	mpbi 233	. . . . . . . 8 ⊢ ¬ 1 ≤ (1 / 2)
98		simpr 489	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
99	98	breq1d 5120	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑦 ≤ (1 / 2) ↔ 1 ≤ (1 / 2)))
100	97, 99	mtbiri 330	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ¬ 𝑦 ≤ (1 / 2))
101	100	iffalsed 4499	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑥𝑀((2 · 𝑦) − 1)))
102		simpl 487	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
103	98	oveq2d 7428	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = (2 · 1))
104		2t1e2 12404	. . . . . . . . . 10 ⊢ (2 · 1) = 2
105	103, 104	eqtrdi 2814	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = 2)
106	105	oveq1d 7427	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = (2 − 1))
107		2m1e1 12366	. . . . . . . 8 ⊢ (2 − 1) = 1
108	106, 107	eqtrdi 2814	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = 1)
109	102, 108	oveq12d 7430	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑠𝑀1))
110	101, 109	eqtrd 2798	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝑀1))
111		ovex 7445	. . . . 5 ⊢ (𝑠𝑀1) ∈ V
112	110, 4, 111	ovmpoa 7567	. . . 4 ⊢ ((𝑠 ∈ 𝑋 ∧ 1 ∈ (0[,]1)) → (𝑠𝑁1) = (𝑠𝑀1))
113	79, 95, 112	sylancl 597	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝑠𝑀1))
114	26	simprd 500	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑀1) = (𝐻‘𝑠))
115	113, 114	eqtrd 2798	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝐻‘𝑠))
116	1, 2, 3, 78, 94, 115	ishtpyd 25115	1 ⊢ (𝜑 → 𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ⊆ wss 3906 ifcif 4488 class class class wbr 5110 ↦ cmpt 5193 ran crn 5664 ‘cfv 6538 (class class class)co 7412 ∈ cmpo 7414 ℝcr 11100 0cc0 11101 1c1 11102 · cmul 11106 < clt 11244 ≤ cle 11245 − cmin 11442 / cdiv 11872 2c2 12296 (,)cioo 13373 [,]cicc 13376 ↾_t crest 17474 topGenctg 17491 TopOnctopon 23048 Cn ccn 23362 ×_t ctx 23698 IIcii 25015 Htpy chtpy 25107

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-icc 13380 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-submnd 18843 df-mulg 19135 df-cntz 19388 df-cmn 19853 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-cn 23365 df-cnp 23366 df-tx 23700 df-hmeo 23893 df-xms 24458 df-ms 24459 df-tms 24460 df-ii 25017 df-htpy 25110

This theorem is referenced by: phtpycc 25131

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