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Theorem htpycc 25100

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 24999	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
6	5	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
7		eqid 2765	. . . . 5 ⊢ (topGen‘ran (,)) = (topGen‘ran (,))
8		eqid 2765	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t (0[,](1 / 2))) = ((topGen‘ran (,)) ↾_t (0[,](1 / 2)))
9		eqid 2765	. . . . 5 ⊢ ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) = ((topGen‘ran (,)) ↾_t ((1 / 2)[,]1))
10		dfii2 25002	. . . . 5 ⊢ II = ((topGen‘ran (,)) ↾_t (0[,]1))
11		0red 11199	. . . . 5 ⊢ (𝜑 → 0 ∈ ℝ)
12		1red 11197	. . . . 5 ⊢ (𝜑 → 1 ∈ ℝ)
13		halfre 12448	. . . . . . 7 ⊢ (1 / 2) ∈ ℝ
14		halfge0 12451	. . . . . . 7 ⊢ 0 ≤ (1 / 2)
15		1re 11196	. . . . . . . 8 ⊢ 1 ∈ ℝ
16		halflt1 12452	. . . . . . . 8 ⊢ (1 / 2) < 1
17	13, 15, 16	ltleii 11321	. . . . . . 7 ⊢ (1 / 2) ≤ 1
18		elicc01 13484	. . . . . . 7 ⊢ ((1 / 2) ∈ (0[,]1) ↔ ((1 / 2) ∈ ℝ ∧ 0 ≤ (1 / 2) ∧ (1 / 2) ≤ 1))
19	13, 14, 17, 18	mpbir3an 1358	. . . . . 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 25094	. . . . . . . . . . 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 25094	. . . . . . . . . . 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 2803	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐿1) = (𝑠𝑀0))
29	28	ralrimiva 3157	. . . . . . . 8 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 (𝑠𝐿1) = (𝑠𝑀0))
30		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝐿1) = (𝑥𝐿1))
31		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑠 = 𝑥 → (𝑠𝑀0) = (𝑥𝑀0))
32	30, 31	eqeq12d 2781	. . . . . . . . 9 ⊢ (𝑠 = 𝑥 → ((𝑠𝐿1) = (𝑠𝑀0) ↔ (𝑥𝐿1) = (𝑥𝑀0)))
33	32	rspccva 3583	. . . . . . . 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 7416	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = (2 · (1 / 2)))
38		2thalfe1 12339	. . . . . . . 8 ⊢ (2 · (1 / 2)) = 1
39	37, 38	eqtrdi 2816	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (2 · 𝑦) = 1)
40	39	oveq2d 7416	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝐿1))
41	39	oveq1d 7415	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = (1 − 1))
42		1m1e0 12304	. . . . . . . 8 ⊢ (1 − 1) = 0
43	41, 42	eqtrdi 2816	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → ((2 · 𝑦) − 1) = 0)
44	43	oveq2d 7416	. . . . . 6 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑥𝑀0))
45	35, 40, 44	3eqtr4d 2810	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 = (1 / 2) ∧ 𝑥 ∈ 𝑋)) → (𝑥𝐿(2 · 𝑦)) = (𝑥𝑀((2 · 𝑦) − 1)))
46		retopon 24881	. . . . . . . 8 ⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
47		0re 11198	. . . . . . . . 9 ⊢ 0 ∈ ℝ
48		iccssre 13447	. . . . . . . . 9 ⊢ ((0 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → (0[,](1 / 2)) ⊆ ℝ)
49	47, 13, 48	mp2an 704	. . . . . . . 8 ⊢ (0[,](1 / 2)) ⊆ ℝ
50		resttopon 23279	. . . . . . . 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 23787	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐽))
54	52, 1	cnmpt1st 23786	. . . . . . 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 25052	. . . . . . . 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 7408	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (2 · 𝑧) = (2 · 𝑦))
58	52, 1, 54, 52, 56, 57	cnmpt21 23789	. . . . . 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 25093	. . . . . . 7 ⊢ (𝜑 → (𝐹(𝐽 Htpy 𝐾)𝐺) ⊆ ((𝐽 ×_t II) Cn 𝐾))
60	59, 22	sseldd 3940	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ ((𝐽 ×_t II) Cn 𝐾))
61	52, 1, 53, 58, 60	cnmpt22f 23793	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ (0[,](1 / 2)), 𝑥 ∈ 𝑋 ↦ (𝑥𝐿(2 · 𝑦))) ∈ ((((topGen‘ran (,)) ↾_t (0[,](1 / 2))) ×_t 𝐽) Cn 𝐾))
62		iccssre 13447	. . . . . . . . 9 ⊢ (((1 / 2) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 2)[,]1) ⊆ ℝ)
63	13, 15, 62	mp2an 704	. . . . . . . 8 ⊢ ((1 / 2)[,]1) ⊆ ℝ
64		resttopon 23279	. . . . . . . 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 23787	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ ((1 / 2)[,]1), 𝑥 ∈ 𝑋 ↦ 𝑥) ∈ ((((topGen‘ran (,)) ↾_t ((1 / 2)[,]1)) ×_t 𝐽) Cn 𝐽))
68	66, 1	cnmpt1st 23786	. . . . . . 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 25054	. . . . . . . 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 7415	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((2 · 𝑧) − 1) = ((2 · 𝑦) − 1))
72	66, 1, 68, 66, 70, 71	cnmpt21 23789	. . . . . 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 25093	. . . . . . 7 ⊢ (𝜑 → (𝐺(𝐽 Htpy 𝐾)𝐻) ⊆ ((𝐽 ×_t II) Cn 𝐾))
74	73, 25	sseldd 3940	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ((𝐽 ×_t II) Cn 𝐾))
75	66, 1, 67, 72, 74	cnmpt22f 23793	. . . . 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 25048	. . . 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 23796	. . 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 2869	. 2 ⊢ (𝜑 → 𝑁 ∈ ((𝐽 ×_t II) Cn 𝐾))
79		simpr 489	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → 𝑠 ∈ 𝑋)
80		0elunit 13487	. . . 4 ⊢ 0 ∈ (0[,]1)
81		simpr 489	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
82	81, 14	eqbrtrdi 5144	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 ≤ (1 / 2))
83	82	iftrued 4491	. . . . . 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 7416	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = (2 · 0))
86		2t0e0 12402	. . . . . . . 8 ⊢ (2 · 0) = 0
87	85, 86	eqtrdi 2816	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (2 · 𝑦) = 0)
88	84, 87	oveq12d 7418	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑥𝐿(2 · 𝑦)) = (𝑠𝐿0))
89	83, 88	eqtrd 2800	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝐿0))
90		ovex 7433	. . . . 5 ⊢ (𝑠𝐿0) ∈ V
91	89, 4, 90	ovmpoa 7555	. . . 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 2800	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁0) = (𝐹‘𝑠))
95		1elunit 13488	. . . 4 ⊢ 1 ∈ (0[,]1)
96	13, 15	ltnlei 11319	. . . . . . . . 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 5115	. . . . . . . 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 4494	. . . . . 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 7416	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = (2 · 1))
104		2t1e2 12394	. . . . . . . . . 10 ⊢ (2 · 1) = 2
105	103, 104	eqtrdi 2816	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (2 · 𝑦) = 2)
106	105	oveq1d 7415	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = (2 − 1))
107		2m1e1 12356	. . . . . . . 8 ⊢ (2 − 1) = 1
108	106, 107	eqtrdi 2816	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((2 · 𝑦) − 1) = 1)
109	102, 108	oveq12d 7418	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑥𝑀((2 · 𝑦) − 1)) = (𝑠𝑀1))
110	101, 109	eqtrd 2800	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → if(𝑦 ≤ (1 / 2), (𝑥𝐿(2 · 𝑦)), (𝑥𝑀((2 · 𝑦) − 1))) = (𝑠𝑀1))
111		ovex 7433	. . . . 5 ⊢ (𝑠𝑀1) ∈ V
112	110, 4, 111	ovmpoa 7555	. . . 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 2800	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝑁1) = (𝐻‘𝑠))
116	1, 2, 3, 78, 94, 115	ishtpyd 25095	1 ⊢ (𝜑 → 𝑁 ∈ (𝐹(𝐽 Htpy 𝐾)𝐻))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1563 ∈ wcel 2145 ∀wral 3079 ⊆ wss 3907 ifcif 4483 class class class wbr 5105 ↦ cmpt 5186 ran crn 5653 ‘cfv 6525 (class class class)co 7400 ∈ cmpo 7402 ℝcr 11087 0cc0 11088 1c1 11089 · cmul 11093 < clt 11231 ≤ cle 11232 − cmin 11429 / cdiv 11859 2c2 12286 (,)cioo 13363 [,]cicc 13366 ↾_t crest 17463 topGenctg 17480 TopOnctopon 23028 Cn ccn 23342 ×_t ctx 23678 IIcii 24995 Htpy chtpy 25087

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2737 ax-rep 5232 ax-sep 5251 ax-nul 5261 ax-pow 5327 ax-pr 5395 ax-un 7722 ax-cnex 11144 ax-resscn 11145 ax-1cn 11146 ax-icn 11147 ax-addcl 11148 ax-addrcl 11149 ax-mulcl 11150 ax-mulrcl 11151 ax-mulcom 11152 ax-addass 11153 ax-mulass 11154 ax-distr 11155 ax-i2m1 11156 ax-1ne0 11157 ax-1rid 11158 ax-rnegex 11159 ax-rrecex 11160 ax-cnre 11161 ax-pre-lttri 11162 ax-pre-lttrn 11163 ax-pre-ltadd 11164 ax-pre-mulgt0 11165 ax-pre-sup 11166

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-nf 1807 df-sb 2094 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3370 df-reu 3371 df-rab 3418 df-v 3459 df-sbc 3748 df-csb 3856 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-pss 3927 df-nul 4289 df-if 4484 df-pw 4560 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4869 df-int 4909 df-iun 4954 df-iin 4955 df-br 5106 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5547 df-eprel 5552 df-po 5560 df-so 5561 df-fr 5605 df-se 5606 df-we 5607 df-xp 5658 df-rel 5659 df-cnv 5660 df-co 5661 df-dm 5662 df-rn 5663 df-res 5664 df-ima 5665 df-pred 6292 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6527 df-fn 6528 df-f 6529 df-f1 6530 df-fo 6531 df-f1o 6532 df-fv 6533 df-isom 6534 df-riota 7357 df-ov 7403 df-oprab 7404 df-mpo 7405 df-of 7664 df-om 7851 df-1st 7974 df-2nd 7975 df-supp 8145 df-frecs 8266 df-wrecs 8297 df-recs 8346 df-rdg 8385 df-1o 8441 df-2o 8442 df-er 8682 df-map 8814 df-ixp 8884 df-en 8932 df-dom 8933 df-sdom 8934 df-fin 8935 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9913 df-pnf 11233 df-mnf 11234 df-xr 11235 df-ltxr 11236 df-le 11237 df-sub 11431 df-neg 11432 df-div 11860 df-nn 12225 df-2 12294 df-3 12295 df-4 12296 df-5 12297 df-6 12298 df-7 12299 df-8 12300 df-9 12301 df-n0 12496 df-z 12583 df-dec 12703 df-uz 12854 df-q 12964 df-rp 13008 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13367 df-icc 13370 df-fz 13527 df-fzo 13674 df-seq 14029 df-exp 14089 df-hash 14358 df-cj 15140 df-re 15141 df-im 15142 df-sqrt 15276 df-abs 15277 df-struct 17197 df-sets 17214 df-slot 17232 df-ndx 17244 df-base 17260 df-ress 17281 df-plusg 17313 df-mulr 17314 df-starv 17315 df-sca 17316 df-vsca 17317 df-ip 17318 df-tset 17319 df-ple 17320 df-ds 17322 df-unif 17323 df-hom 17324 df-cco 17325 df-rest 17465 df-topn 17466 df-0g 17484 df-gsum 17485 df-topgen 17486 df-pt 17487 df-prds 17490 df-xrs 17546 df-qtop 17551 df-imas 17552 df-xps 17554 df-mre 17628 df-mrc 17629 df-acs 17631 df-mgm 18688 df-sgrp 18767 df-mnd 18783 df-submnd 18832 df-mulg 19125 df-cntz 19378 df-cmn 19843 df-psmet 21474 df-xmet 21475 df-met 21476 df-bl 21477 df-mopn 21478 df-cnfld 21483 df-top 23012 df-topon 23029 df-topsp 23051 df-bases 23064 df-cld 23137 df-cn 23345 df-cnp 23346 df-tx 23680 df-hmeo 23873 df-xms 24438 df-ms 24439 df-tms 24440 df-ii 24997 df-htpy 25090

This theorem is referenced by: phtpycc 25111

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