Metamath Proof Explorer


Theorem ntrclsk3

Description: The intersection of interiors of a every pair is a subset of the interior of the intersection of the pair if an only if the closure of the union of every pair is a subset of the union of closures of the pair. (Contributed by RP, 19-Jun-2021)

Ref Expression
Hypotheses ntrcls.o O = i V k 𝒫 i 𝒫 i j 𝒫 i i k i j
ntrcls.d D = O B
ntrcls.r φ I D K
Assertion ntrclsk3 φ s 𝒫 B t 𝒫 B I s I t I s t s 𝒫 B t 𝒫 B K s t K s K t

Proof

Step Hyp Ref Expression
1 ntrcls.o O = i V k 𝒫 i 𝒫 i j 𝒫 i i k i j
2 ntrcls.d D = O B
3 ntrcls.r φ I D K
4 fveq2 s = a I s = I a
5 4 ineq1d s = a I s I t = I a I t
6 ineq1 s = a s t = a t
7 6 fveq2d s = a I s t = I a t
8 5 7 sseq12d s = a I s I t I s t I a I t I a t
9 fveq2 t = b I t = I b
10 9 ineq2d t = b I a I t = I a I b
11 ineq2 t = b a t = a b
12 11 fveq2d t = b I a t = I a b
13 10 12 sseq12d t = b I a I t I a t I a I b I a b
14 8 13 cbvral2vw s 𝒫 B t 𝒫 B I s I t I s t a 𝒫 B b 𝒫 B I a I b I a b
15 2 3 ntrclsbex φ B V
16 difssd φ B s B
17 15 16 sselpwd φ B s 𝒫 B
18 17 adantr φ s 𝒫 B B s 𝒫 B
19 elpwi a 𝒫 B a B
20 simpl B V a B B V
21 difssd B V a B B a B
22 20 21 sselpwd B V a B B a 𝒫 B
23 simpr B V a B s = B a s = B a
24 23 difeq2d B V a B s = B a B s = B B a
25 24 eqeq2d B V a B s = B a a = B s a = B B a
26 eqcom a = B B a B B a = a
27 25 26 bitrdi B V a B s = B a a = B s B B a = a
28 dfss4 a B B B a = a
29 28 biimpi a B B B a = a
30 29 adantl B V a B B B a = a
31 22 27 30 rspcedvd B V a B s 𝒫 B a = B s
32 15 19 31 syl2an φ a 𝒫 B s 𝒫 B a = B s
33 simpl1 φ s 𝒫 B a = B s t 𝒫 B φ
34 difssd φ B t B
35 15 34 sselpwd φ B t 𝒫 B
36 33 35 syl φ s 𝒫 B a = B s t 𝒫 B B t 𝒫 B
37 elpwi b 𝒫 B b B
38 simpl B V b B B V
39 difssd B V b B B b B
40 38 39 sselpwd B V b B B b 𝒫 B
41 simpr B V b B t = B b t = B b
42 41 difeq2d B V b B t = B b B t = B B b
43 42 eqeq2d B V b B t = B b b = B t b = B B b
44 eqcom b = B B b B B b = b
45 43 44 bitrdi B V b B t = B b b = B t B B b = b
46 dfss4 b B B B b = b
47 46 biimpi b B B B b = b
48 47 adantl B V b B B B b = b
49 40 45 48 rspcedvd B V b B t 𝒫 B b = B t
50 15 37 49 syl2an φ b 𝒫 B t 𝒫 B b = B t
51 50 3ad2antl1 φ s 𝒫 B a = B s b 𝒫 B t 𝒫 B b = B t
52 simp13 φ s 𝒫 B a = B s t 𝒫 B b = B t a = B s
53 fveq2 a = B s I a = I B s
54 53 ineq1d a = B s I a I b = I B s I b
55 ineq1 a = B s a b = B s b
56 55 fveq2d a = B s I a b = I B s b
57 54 56 sseq12d a = B s I a I b I a b I B s I b I B s b
58 52 57 syl φ s 𝒫 B a = B s t 𝒫 B b = B t I a I b I a b I B s I b I B s b
59 fveq2 b = B t I b = I B t
60 59 ineq2d b = B t I B s I b = I B s I B t
61 ineq2 b = B t B s b = B s B t
62 difundi B s t = B s B t
63 61 62 eqtr4di b = B t B s b = B s t
64 63 fveq2d b = B t I B s b = I B s t
65 60 64 sseq12d b = B t I B s I b I B s b I B s I B t I B s t
66 65 3ad2ant3 φ s 𝒫 B a = B s t 𝒫 B b = B t I B s I b I B s b I B s I B t I B s t
67 simp11 φ s 𝒫 B a = B s t 𝒫 B b = B t φ
68 1 2 3 ntrclsiex φ I 𝒫 B 𝒫 B
69 68 15 jca φ I 𝒫 B 𝒫 B B V
70 67 69 syl φ s 𝒫 B a = B s t 𝒫 B b = B t I 𝒫 B 𝒫 B B V
71 elmapi I 𝒫 B 𝒫 B I : 𝒫 B 𝒫 B
72 71 adantr I 𝒫 B 𝒫 B B V I : 𝒫 B 𝒫 B
73 simpr I 𝒫 B 𝒫 B B V B V
74 difssd I 𝒫 B 𝒫 B B V B s B
75 73 74 sselpwd I 𝒫 B 𝒫 B B V B s 𝒫 B
76 72 75 ffvelrnd I 𝒫 B 𝒫 B B V I B s 𝒫 B
77 76 elpwid I 𝒫 B 𝒫 B B V I B s B
78 orc I B s B I B s B I B t B
79 inss I B s B I B t B I B s I B t B
80 77 78 79 3syl I 𝒫 B 𝒫 B B V I B s I B t B
81 difssd I 𝒫 B 𝒫 B B V B s t B
82 73 81 sselpwd I 𝒫 B 𝒫 B B V B s t 𝒫 B
83 72 82 ffvelrnd I 𝒫 B 𝒫 B B V I B s t 𝒫 B
84 83 elpwid I 𝒫 B 𝒫 B B V I B s t B
85 80 84 jca I 𝒫 B 𝒫 B B V I B s I B t B I B s t B
86 sscon34b I B s I B t B I B s t B I B s I B t I B s t B I B s t B I B s I B t
87 70 85 86 3syl φ s 𝒫 B a = B s t 𝒫 B b = B t I B s I B t I B s t B I B s t B I B s I B t
88 difindi B I B s I B t = B I B s B I B t
89 88 sseq2i B I B s t B I B s I B t B I B s t B I B s B I B t
90 89 a1i φ s 𝒫 B a = B s t 𝒫 B b = B t B I B s t B I B s I B t B I B s t B I B s B I B t
91 67 15 syl φ s 𝒫 B a = B s t 𝒫 B b = B t B V
92 67 68 syl φ s 𝒫 B a = B s t 𝒫 B b = B t I 𝒫 B 𝒫 B
93 simp12 φ s 𝒫 B a = B s t 𝒫 B b = B t s 𝒫 B
94 rp-simp2 φ s 𝒫 B a = B s t 𝒫 B b = B t t 𝒫 B
95 simpl2 φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B B V
96 simpl3 φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B I 𝒫 B 𝒫 B
97 eqid D I = D I
98 simpl B V s 𝒫 B t 𝒫 B B V
99 simprl B V s 𝒫 B t 𝒫 B s 𝒫 B
100 99 elpwid B V s 𝒫 B t 𝒫 B s B
101 simprr B V s 𝒫 B t 𝒫 B t 𝒫 B
102 101 elpwid B V s 𝒫 B t 𝒫 B t B
103 100 102 unssd B V s 𝒫 B t 𝒫 B s t B
104 98 103 sselpwd B V s 𝒫 B t 𝒫 B s t 𝒫 B
105 104 3ad2antl2 φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B s t 𝒫 B
106 eqid D I s t = D I s t
107 1 2 95 96 97 105 106 dssmapfv3d φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B D I s t = B I B s t
108 simpl1 φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B φ
109 1 2 3 ntrclsfv1 φ D I = K
110 109 fveq1d φ D I s t = K s t
111 108 110 syl φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B D I s t = K s t
112 107 111 eqtr3d φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B B I B s t = K s t
113 simprl φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B s 𝒫 B
114 eqid D I s = D I s
115 1 2 95 96 97 113 114 dssmapfv3d φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B D I s = B I B s
116 109 fveq1d φ D I s = K s
117 108 116 syl φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B D I s = K s
118 115 117 eqtr3d φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B B I B s = K s
119 simprr φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B t 𝒫 B
120 eqid D I t = D I t
121 1 2 95 96 97 119 120 dssmapfv3d φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B D I t = B I B t
122 109 fveq1d φ D I t = K t
123 108 122 syl φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B D I t = K t
124 121 123 eqtr3d φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B B I B t = K t
125 118 124 uneq12d φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B B I B s B I B t = K s K t
126 112 125 sseq12d φ B V I 𝒫 B 𝒫 B s 𝒫 B t 𝒫 B B I B s t B I B s B I B t K s t K s K t
127 67 91 92 93 94 126 syl32anc φ s 𝒫 B a = B s t 𝒫 B b = B t B I B s t B I B s B I B t K s t K s K t
128 87 90 127 3bitrd φ s 𝒫 B a = B s t 𝒫 B b = B t I B s I B t I B s t K s t K s K t
129 58 66 128 3bitrd φ s 𝒫 B a = B s t 𝒫 B b = B t I a I b I a b K s t K s K t
130 36 51 129 ralxfrd2 φ s 𝒫 B a = B s b 𝒫 B I a I b I a b t 𝒫 B K s t K s K t
131 18 32 130 ralxfrd2 φ a 𝒫 B b 𝒫 B I a I b I a b s 𝒫 B t 𝒫 B K s t K s K t
132 14 131 syl5bb φ s 𝒫 B t 𝒫 B I s I t I s t s 𝒫 B t 𝒫 B K s t K s K t