Theorem stb3xpl 16743

Description: The explicit equivalent of a constructed 3-member structure. This theorem lets us go back and forth between constructed and explicit representations.

Hypotheses

Ref	Expression
stb3xpl.s	|- S = (g e. _V |-> (g` 1))
stb3xpl.t	|- T = (g e. _V |-> (g` 2))
stb3xpl.u	|- U = (g e. _V |-> (g` 3))
stb3xpl.3	|- X e. _V
stb3xpl.4	|- Y e. _V
stb3xpl.z	|- Z e. _V
stb3xpl.5	|- F = StrBldr(3, f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z))

Assertion

Ref	Expression
stb3xpl	|- F = {<.1, X>., <.2, Y>., <.3, Z>.}

Distinct variable groups:   f,X   f,Y   f,Z

Proof of Theorem stb3xpl

Step	Hyp	Ref	Expression
1		visset 2295	. . . . . . . . 9 |- m e. _V
2	1	eltp 3074	. . . . . . . 8 |- (m e. {1, (1 + 1), (1 + 2)} <-> (m = 1 \/ m = (1 + 1) \/ m = (1 + 2)))
3		df-3 7155	. . . . . . . . . . . 12 |- 3 = (2 + 1)
4		2cn 7164	. . . . . . . . . . . . 13 |- 2 e. CC
5		ax1cn 6422	. . . . . . . . . . . . 13 |- 1 e. CC
6	4, 5	addcomi 6475	. . . . . . . . . . . 12 |- (2 + 1) = (1 + 2)
7	3, 6	eqtri 1908	. . . . . . . . . . 11 |- 3 = (1 + 2)
8	7	opreq2i 4893	. . . . . . . . . 10 |- (1...3) = (1...(1 + 2))
9		1z 7368	. . . . . . . . . . 11 |- 1 e. ZZ
10		fztp 7686	. . . . . . . . . . 11 |- (1 e. ZZ -> (1...(1 + 2)) = {1, (1 + 1), (1 + 2)})
11	9, 10	ax-mp 7	. . . . . . . . . 10 |- (1...(1 + 2)) = {1, (1 + 1), (1 + 2)}
12	8, 11	eqtri 1908	. . . . . . . . 9 |- (1...3) = {1, (1 + 1), (1 + 2)}
13	12	eleq2i 1961	. . . . . . . 8 |- (m e. (1...3) <-> m e. {1, (1 + 1), (1 + 2)})
14		biid 187	. . . . . . . . 9 |- (m = 1 <-> m = 1)
15		df-2 7154	. . . . . . . . . 10 |- 2 = (1 + 1)
16	15	eqeq2i 1894	. . . . . . . . 9 |- (m = 2 <-> m = (1 + 1))
17	7	eqeq2i 1894	. . . . . . . . 9 |- (m = 3 <-> m = (1 + 2))
18	14, 16, 17	3orbi123i 1057	. . . . . . . 8 |- ((m = 1 \/ m = 2 \/ m = 3) <-> (m = 1 \/ m = (1 + 1) \/ m = (1 + 2)))
19	2, 13, 18	3bitr4i 200	. . . . . . 7 |- (m e. (1...3) <-> (m = 1 \/ m = 2 \/ m = 3))
20		1lt2 7212	. . . . . . . . . . . 12 |- 1 < 2
21		1re 6598	. . . . . . . . . . . . 13 |- 1 e. RR
22		2re 7163	. . . . . . . . . . . . 13 |- 2 e. RR
23	21, 22	ltnei 6758	. . . . . . . . . . . 12 |- (1 < 2 -> 2 =/= 1)
24	20, 23	ax-mp 7	. . . . . . . . . . 11 |- 2 =/= 1
25		necom 2094	. . . . . . . . . . 11 |- (2 =/= 1 <-> 1 =/= 2)
26	24, 25	mpbi 206	. . . . . . . . . 10 |- 1 =/= 2
27		1lt3 7214	. . . . . . . . . . . 12 |- 1 < 3
28		3re 7165	. . . . . . . . . . . . 13 |- 3 e. RR
29	21, 28	ltnei 6758	. . . . . . . . . . . 12 |- (1 < 3 -> 3 =/= 1)
30	27, 29	ax-mp 7	. . . . . . . . . . 11 |- 3 =/= 1
31		necom 2094	. . . . . . . . . . 11 |- (3 =/= 1 <-> 1 =/= 3)
32	30, 31	mpbi 206	. . . . . . . . . 10 |- 1 =/= 3
33		2lt3 7213	. . . . . . . . . . . 12 |- 2 < 3
34	22, 28	ltnei 6758	. . . . . . . . . . . 12 |- (2 < 3 -> 3 =/= 2)
35	33, 34	ax-mp 7	. . . . . . . . . . 11 |- 3 =/= 2
36		necom 2094	. . . . . . . . . . 11 |- (3 =/= 2 <-> 2 =/= 3)
37	35, 36	mpbi 206	. . . . . . . . . 10 |- 2 =/= 3
38	21	elisseti 2301	. . . . . . . . . . 11 |- 1 e. _V
39	22	elisseti 2301	. . . . . . . . . . 11 |- 2 e. _V
40	28	elisseti 2301	. . . . . . . . . . 11 |- 3 e. _V
41		stb3xpl.3	. . . . . . . . . . 11 |- X e. _V
42		stb3xpl.4	. . . . . . . . . . 11 |- Y e. _V
43		stb3xpl.z	. . . . . . . . . . 11 |- Z e. _V
44	38, 39, 40, 41, 42, 43	fvtp1 4761	. . . . . . . . . 10 |- ((1 =/= 2 /\ 1 =/= 3 /\ 2 =/= 3) -> ({<.1, X>., <.2, Y>., <.3, Z>.}` 1) = X)
45	26, 32, 37, 44	mp3an 1191	. . . . . . . . 9 |- ({<.1, X>., <.2, Y>., <.3, Z>.}` 1) = X
46		fveq2 4681	. . . . . . . . 9 |- (m = 1 -> ({<.1, X>., <.2, Y>., <.3, Z>.}` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` 1))
47		iftrue 2989	. . . . . . . . 9 |- (m = 1 -> if(m = 1, X, if(m = 2, Y, Z)) = X)
48	45, 46, 47	3eqtr4a 1954	. . . . . . . 8 |- (m = 1 -> ({<.1, X>., <.2, Y>., <.3, Z>.}` m) = if(m = 1, X, if(m = 2, Y, Z)))
49	38, 39, 40, 41, 42, 43	fvtp2 4762	. . . . . . . . . 10 |- ((1 =/= 2 /\ 1 =/= 3 /\ 2 =/= 3) -> ({<.1, X>., <.2, Y>., <.3, Z>.}` 2) = Y)
50	26, 32, 37, 49	mp3an 1191	. . . . . . . . 9 |- ({<.1, X>., <.2, Y>., <.3, Z>.}` 2) = Y
51		fveq2 4681	. . . . . . . . 9 |- (m = 2 -> ({<.1, X>., <.2, Y>., <.3, Z>.}` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` 2))
52		pm13.181 2086	. . . . . . . . . . . . 13 |- ((m = 2 /\ 2 =/= 1) -> m =/= 1)
53	24, 52	mpan2 760	. . . . . . . . . . . 12 |- (m = 2 -> m =/= 1)
54		df-ne 2019	. . . . . . . . . . . 12 |- (m =/= 1 <-> -. m = 1)
55	53, 54	sylib 215	. . . . . . . . . . 11 |- (m = 2 -> -. m = 1)
56		iffalse 2991	. . . . . . . . . . 11 |- (-. m = 1 -> if(m = 1, X, if(m = 2, Y, Z)) = if(m = 2, Y, Z))
57	55, 56	syl 12	. . . . . . . . . 10 |- (m = 2 -> if(m = 1, X, if(m = 2, Y, Z)) = if(m = 2, Y, Z))
58		iftrue 2989	. . . . . . . . . 10 |- (m = 2 -> if(m = 2, Y, Z) = Y)
59	57, 58	eqtrd 1925	. . . . . . . . 9 |- (m = 2 -> if(m = 1, X, if(m = 2, Y, Z)) = Y)
60	50, 51, 59	3eqtr4a 1954	. . . . . . . 8 |- (m = 2 -> ({<.1, X>., <.2, Y>., <.3, Z>.}` m) = if(m = 1, X, if(m = 2, Y, Z)))
61	38, 39, 40, 41, 42, 43	fvtp3 4763	. . . . . . . . . 10 |- ((1 =/= 2 /\ 1 =/= 3 /\ 2 =/= 3) -> ({<.1, X>., <.2, Y>., <.3, Z>.}` 3) = Z)
62	26, 32, 37, 61	mp3an 1191	. . . . . . . . 9 |- ({<.1, X>., <.2, Y>., <.3, Z>.}` 3) = Z
63		fveq2 4681	. . . . . . . . 9 |- (m = 3 -> ({<.1, X>., <.2, Y>., <.3, Z>.}` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` 3))
64		pm13.181 2086	. . . . . . . . . . . . 13 |- ((m = 3 /\ 3 =/= 1) -> m =/= 1)
65	30, 64	mpan2 760	. . . . . . . . . . . 12 |- (m = 3 -> m =/= 1)
66	65, 54	sylib 215	. . . . . . . . . . 11 |- (m = 3 -> -. m = 1)
67	66, 56	syl 12	. . . . . . . . . 10 |- (m = 3 -> if(m = 1, X, if(m = 2, Y, Z)) = if(m = 2, Y, Z))
68		pm13.181 2086	. . . . . . . . . . . . 13 |- ((m = 3 /\ 3 =/= 2) -> m =/= 2)
69	35, 68	mpan2 760	. . . . . . . . . . . 12 |- (m = 3 -> m =/= 2)
70		df-ne 2019	. . . . . . . . . . . 12 |- (m =/= 2 <-> -. m = 2)
71	69, 70	sylib 215	. . . . . . . . . . 11 |- (m = 3 -> -. m = 2)
72		iffalse 2991	. . . . . . . . . . 11 |- (-. m = 2 -> if(m = 2, Y, Z) = Z)
73	71, 72	syl 12	. . . . . . . . . 10 |- (m = 3 -> if(m = 2, Y, Z) = Z)
74	67, 73	eqtrd 1925	. . . . . . . . 9 |- (m = 3 -> if(m = 1, X, if(m = 2, Y, Z)) = Z)
75	62, 63, 74	3eqtr4a 1954	. . . . . . . 8 |- (m = 3 -> ({<.1, X>., <.2, Y>., <.3, Z>.}` m) = if(m = 1, X, if(m = 2, Y, Z)))
76	48, 60, 75	3jaoi 1160	. . . . . . 7 |- ((m = 1 \/ m = 2 \/ m = 3) -> ({<.1, X>., <.2, Y>., <.3, Z>.}` m) = if(m = 1, X, if(m = 2, Y, Z)))
77	19, 76	sylbi 216	. . . . . 6 |- (m e. (1...3) -> ({<.1, X>., <.2, Y>., <.3, Z>.}` m) = if(m = 1, X, if(m = 2, Y, Z)))
78	77	eqeq2d 1895	. . . . 5 |- (m e. (1...3) -> ((F` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` m) <-> (F` m) = if(m = 1, X, if(m = 2, Y, Z))))
79	78	ralbiia 2133	. . . 4 |- (A.m e. (1...3)(F` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` m) <-> A.m e. (1...3)(F` m) = if(m = 1, X, if(m = 2, Y, Z)))
80		fztpval 7688	. . . 4 |- (A.m e. (1...3)(F` m) = if(m = 1, X, if(m = 2, Y, Z)) <-> ((F` 1) = X /\ (F` 2) = Y /\ (F` 3) = Z))
81	79, 80	bitri 190	. . 3 |- (A.m e. (1...3)(F` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` m) <-> ((F` 1) = X /\ (F` 2) = Y /\ (F` 3) = Z))
82		visset 2295	. . . . . . 7 |- f e. _V
83		stb3xpl.s	. . . . . . 7 |- S = (g e. _V |-> (g` 1))
84	82, 83	strcval 16732	. . . . . 6 |- (S` f) = (f` 1)
85	84	eqeq1i 1891	. . . . 5 |- ((S` f) = X <-> (f` 1) = X)
86		stb3xpl.t	. . . . . . 7 |- T = (g e. _V |-> (g` 2))
87	82, 86	strcval 16732	. . . . . 6 |- (T` f) = (f` 2)
88	87	eqeq1i 1891	. . . . 5 |- ((T` f) = Y <-> (f` 2) = Y)
89		stb3xpl.u	. . . . . . 7 |- U = (g e. _V |-> (g` 3))
90	82, 89	strcval 16732	. . . . . 6 |- (U` f) = (f` 3)
91	90	eqeq1i 1891	. . . . 5 |- ((U` f) = Z <-> (f` 3) = Z)
92	85, 88, 91	3anbi123i 1056	. . . 4 |- (((S` f) = X /\ (T` f) = Y /\ (U` f) = Z) <-> ((f` 1) = X /\ (f` 2) = Y /\ (f` 3) = Z))
93		stb3xpl.5	. . . 4 |- F = StrBldr(3, f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z))
94	41, 42, 43, 92, 93	stb3val1 16739	. . 3 |- (F` 1) = X
95	41, 42, 43, 92, 93	stb3val2 16740	. . 3 |- (F` 2) = Y
96	41, 42, 43, 92, 93	stb3val3 16741	. . 3 |- (F` 3) = Z
97	81, 94, 95, 96	mpbir3an 1052	. 2 |- A.m e. (1...3)(F` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` m)
98	41, 42, 43, 92, 93	stb3el 16737	. . . . 5 |- F e. (Struct(3, f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z)) \ Struct((3 + 1), f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z)))
99		3nn 7184	. . . . . 6 |- 3 e. NN
100		hbstb1 16727	. . . . . . . 8 |- (m e. StrBldr(3, f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z)) -> A.f m e. StrBldr(3, f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z)))
101	93, 100	hbxfr 1992	. . . . . . 7 |- (m e. F -> A.f m e. F)
102		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (g` 1) -> A.h m e. (g` 1))
103		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (h` 1) -> A.g m e. (h` 1))
104		fveq1 4680	. . . . . . . . . . . . 13 |- (g = h -> (g` 1) = (h` 1))
105	102, 103, 104	cbvmpt 5011	. . . . . . . . . . . 12 |- (g e. _V |-> (g` 1)) = (h e. _V |-> (h` 1))
106		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (h` 1) -> A.f m e. (h` 1))
107		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (f` 1) -> A.h m e. (f` 1))
108		fveq1 4680	. . . . . . . . . . . . 13 |- (h = f -> (h` 1) = (f` 1))
109	106, 107, 108	cbvmpt 5011	. . . . . . . . . . . 12 |- (h e. _V |-> (h` 1)) = (f e. _V |-> (f` 1))
110	83, 105, 109	3eqtri 1912	. . . . . . . . . . 11 |- S = (f e. _V |-> (f` 1))
111		hbmpt1 5010	. . . . . . . . . . 11 |- (m e. (f e. _V |-> (f` 1)) -> A.f m e. (f e. _V |-> (f` 1)))
112	110, 111	hbxfr 1992	. . . . . . . . . 10 |- (m e. S -> A.f m e. S)
113	112, 101	hbfv 4686	. . . . . . . . 9 |- (m e. (S` F) -> A.f m e. (S` F))
114		ax-17 1317	. . . . . . . . 9 |- (m e. X -> A.f m e. X)
115	113, 114	hbeq 1995	. . . . . . . 8 |- ((S` F) = X -> A.f(S` F) = X)
116		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (g` 2) -> A.h m e. (g` 2))
117		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (h` 2) -> A.g m e. (h` 2))
118		fveq1 4680	. . . . . . . . . . . . 13 |- (g = h -> (g` 2) = (h` 2))
119	116, 117, 118	cbvmpt 5011	. . . . . . . . . . . 12 |- (g e. _V |-> (g` 2)) = (h e. _V |-> (h` 2))
120		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (h` 2) -> A.f m e. (h` 2))
121		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (f` 2) -> A.h m e. (f` 2))
122		fveq1 4680	. . . . . . . . . . . . 13 |- (h = f -> (h` 2) = (f` 2))
123	120, 121, 122	cbvmpt 5011	. . . . . . . . . . . 12 |- (h e. _V |-> (h` 2)) = (f e. _V |-> (f` 2))
124	86, 119, 123	3eqtri 1912	. . . . . . . . . . 11 |- T = (f e. _V |-> (f` 2))
125		hbmpt1 5010	. . . . . . . . . . 11 |- (m e. (f e. _V |-> (f` 2)) -> A.f m e. (f e. _V |-> (f` 2)))
126	124, 125	hbxfr 1992	. . . . . . . . . 10 |- (m e. T -> A.f m e. T)
127	126, 101	hbfv 4686	. . . . . . . . 9 |- (m e. (T` F) -> A.f m e. (T` F))
128		ax-17 1317	. . . . . . . . 9 |- (m e. Y -> A.f m e. Y)
129	127, 128	hbeq 1995	. . . . . . . 8 |- ((T` F) = Y -> A.f(T` F) = Y)
130		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (g` 3) -> A.h m e. (g` 3))
131		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (h` 3) -> A.g m e. (h` 3))
132		fveq1 4680	. . . . . . . . . . . . 13 |- (g = h -> (g` 3) = (h` 3))
133	130, 131, 132	cbvmpt 5011	. . . . . . . . . . . 12 |- (g e. _V |-> (g` 3)) = (h e. _V |-> (h` 3))
134		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (h` 3) -> A.f m e. (h` 3))
135		ax-17 1317	. . . . . . . . . . . . 13 |- (m e. (f` 3) -> A.h m e. (f` 3))
136		fveq1 4680	. . . . . . . . . . . . 13 |- (h = f -> (h` 3) = (f` 3))
137	134, 135, 136	cbvmpt 5011	. . . . . . . . . . . 12 |- (h e. _V |-> (h` 3)) = (f e. _V |-> (f` 3))
138	89, 133, 137	3eqtri 1912	. . . . . . . . . . 11 |- U = (f e. _V |-> (f` 3))
139		hbmpt1 5010	. . . . . . . . . . 11 |- (m e. (f e. _V |-> (f` 3)) -> A.f m e. (f e. _V |-> (f` 3)))
140	138, 139	hbxfr 1992	. . . . . . . . . 10 |- (m e. U -> A.f m e. U)
141	140, 101	hbfv 4686	. . . . . . . . 9 |- (m e. (U` F) -> A.f m e. (U` F))
142		ax-17 1317	. . . . . . . . 9 |- (m e. Z -> A.f m e. Z)
143	141, 142	hbeq 1995	. . . . . . . 8 |- ((U` F) = Z -> A.f(U` F) = Z)
144	115, 129, 143	hb3an 1359	. . . . . . 7 |- (((S` F) = X /\ (T` F) = Y /\ (U` F) = Z) -> A.f((S` F) = X /\ (T` F) = Y /\ (U` F) = Z))
145		fveq2 4681	. . . . . . . . 9 |- (f = F -> (S` f) = (S` F))
146	145	eqeq1d 1892	. . . . . . . 8 |- (f = F -> ((S` f) = X <-> (S` F) = X))
147		fveq2 4681	. . . . . . . . 9 |- (f = F -> (T` f) = (T` F))
148	147	eqeq1d 1892	. . . . . . . 8 |- (f = F -> ((T` f) = Y <-> (T` F) = Y))
149		fveq2 4681	. . . . . . . . 9 |- (f = F -> (U` f) = (U` F))
150	149	eqeq1d 1892	. . . . . . . 8 |- (f = F -> ((U` f) = Z <-> (U` F) = Z))
151	146, 148, 150	3anbi123d 1168	. . . . . . 7 |- (f = F -> (((S` f) = X /\ (T` f) = Y /\ (U` f) = Z) <-> ((S` F) = X /\ (T` F) = Y /\ (U` F) = Z)))
152	101, 144, 151	elstrdiff 16720	. . . . . 6 |- (3 e. NN -> (F e. (Struct(3, f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z)) \ Struct((3 + 1), f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z))) <-> (F Fn (1...3) /\ ((S` F) = X /\ (T` F) = Y /\ (U` F) = Z))))
153	99, 152	ax-mp 7	. . . . 5 |- (F e. (Struct(3, f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z)) \ Struct((3 + 1), f, ((S` f) = X /\ (T` f) = Y /\ (U` f) = Z))) <-> (F Fn (1...3) /\ ((S` F) = X /\ (T` F) = Y /\ (U` F) = Z)))
154	98, 153	mpbi 206	. . . 4 |- (F Fn (1...3) /\ ((S` F) = X /\ (T` F) = Y /\ (U` F) = Z))
155	154	simpli 347	. . 3 |- F Fn (1...3)
156	38, 39, 40, 41, 42, 43	fntp 4471	. . . . 5 |- ((1 =/= 2 /\ 1 =/= 3 /\ 2 =/= 3) -> {<.1, X>., <.2, Y>., <.3, Z>.} Fn {1, 2, 3})
157	26, 32, 37, 156	mp3an 1191	. . . 4 |- {<.1, X>., <.2, Y>., <.3, Z>.} Fn {1, 2, 3}
158		tpeq3 3102	. . . . . . . 8 |- (3 = (1 + 2) -> {1, 2, 3} = {1, 2, (1 + 2)})
159	7, 158	ax-mp 7	. . . . . . 7 |- {1, 2, 3} = {1, 2, (1 + 2)}
160		tpeq2 3101	. . . . . . . 8 |- (2 = (1 + 1) -> {1, 2, (1 + 2)} = {1, (1 + 1), (1 + 2)})
161	15, 160	ax-mp 7	. . . . . . 7 |- {1, 2, (1 + 2)} = {1, (1 + 1), (1 + 2)}
162	159, 161	eqtri 1908	. . . . . 6 |- {1, 2, 3} = {1, (1 + 1), (1 + 2)}
163	11, 8, 162	3eqtr4i 1921	. . . . 5 |- (1...3) = {1, 2, 3}
164	163	fneq2i 4508	. . . 4 |- ({<.1, X>., <.2, Y>., <.3, Z>.} Fn (1...3) <-> {<.1, X>., <.2, Y>., <.3, Z>.} Fn {1, 2, 3})
165	157, 164	mpbir 207	. . 3 |- {<.1, X>., <.2, Y>., <.3, Z>.} Fn (1...3)
166		eqfnfv2 4767	. . 3 |- ((F Fn (1...3) /\ {<.1, X>., <.2, Y>., <.3, Z>.} Fn (1...3)) -> (F = {<.1, X>., <.2, Y>., <.3, Z>.} <-> A.m e. (1...3)(F` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` m)))
167	155, 165, 166	mp2an 761	. 2 |- (F = {<.1, X>., <.2, Y>., <.3, Z>.} <-> A.m e. (1...3)(F` m) = ({<.1, X>., <.2, Y>., <.3, Z>.}` m))
168	97, 167	mpbir 207	1 |- F = {<.1, X>., <.2, Y>., <.3, Z>.}

Syntax hints:  -. wn 2   <-> wb 163   /\ wa 240   \/ w3o 857   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  _Vcvv 2292   \ cdif 2590  ifcif 2982  <.cop 3046  {ctp 3051   class class class wbr 3338   Fn wfn 3993  ` cfv 3998  (class class class)co 4884   e. cmpt 5004  RRcr 6385  1c1 6387   + caddc 6389  NNcn 6449  ZZcz 6451   < clt 6653  2c2 7145  3c3 7146  ...cfz 7637  Structcstru 16707  StrBldrccstr 16724

This theorem is referenced by:  stb3strx 16754  stb3v1x 16755  stb3v2x 16756  stb3v3x 16757

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-n 7108  df-2 7154  df-3 7155  df-n0 7309  df-z 7345  df-uz 7587  df-fz 7638  df-struct 16708  df-strbldr 16725

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