Theorem spacind 6288

Description: Inductive law for the special set generator. (Contributed by SF, 13-Mar-2015.)

Assertion

Ref	Expression
spacind	|- (((M e. NC /\ S e. V) /\ (M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S))) -> ( Spac ` M) C_ S)

Distinct variable groups:   x,M   x,S

Allowed substitution hint:   V(x)

Proof of Theorem spacind

Dummy variables z q p are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 2868	. 2 |- (S e. V -> S e. _V)
2		spacval 6283	. . . . 5 |- (M e. NC -> ( Spac ` M) = Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))
3	2	adantr 451	. . . 4 |- ((M e. NC /\ S e. _V) -> ( Spac ` M) = Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))
4	3	adantr 451	. . 3 |- (((M e. NC /\ S e. _V) /\ (M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S))) -> ( Spac ` M) = Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}))
5		simplr 731	. . . 4 |- (((M e. NC /\ S e. _V) /\ (M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S))) -> S e. _V)
6		snssi 3853	. . . . . 6 |- (M e. S -> {M} C_ S)
7	6	adantr 451	. . . . 5 |- ((M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S)) -> {M} C_ S)
8	7	adantl 452	. . . 4 |- (((M e. NC /\ S e. _V) /\ (M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S))) -> {M} C_ S)
9		spacssnc 6285	. . . . . . . . . . 11 |- (M e. NC -> ( Spac ` M) C_ NC )
10	9	sseld 3273	. . . . . . . . . 10 |- (M e. NC -> (x e. ( Spac ` M) -> x e. NC ))
11		2nc 6169	. . . . . . . . . . . . . . . . . . . 20 |- 2c e. NC
12		ceclr 6188	. . . . . . . . . . . . . . . . . . . . 21 |- ((2c e. NC /\ x e. NC /\ (2c ↑c x) e. NC ) -> ((2c ↑c 0c) e. NC /\ (x ↑c 0c) e. NC ))
13	12	simprd 449	. . . . . . . . . . . . . . . . . . . 20 |- ((2c e. NC /\ x e. NC /\ (2c ↑c x) e. NC ) -> (x ↑c 0c) e. NC )
14	11, 13	mp3an1 1264	. . . . . . . . . . . . . . . . . . 19 |- ((x e. NC /\ (2c ↑c x) e. NC ) -> (x ↑c 0c) e. NC )
15	14	ex 423	. . . . . . . . . . . . . . . . . 18 |- (x e. NC -> ((2c ↑c x) e. NC -> (x ↑c 0c) e. NC ))
16	15	imim1d 69	. . . . . . . . . . . . . . . . 17 |- (x e. NC -> (((x ↑c 0c) e. NC -> (2c ↑c x) e. S) -> ((2c ↑c x) e. NC -> (2c ↑c x) e. S)))
17	16	a1dd 42	. . . . . . . . . . . . . . . 16 |- (x e. NC -> (((x ↑c 0c) e. NC -> (2c ↑c x) e. S) -> (x e. NC -> ((2c ↑c x) e. NC -> (2c ↑c x) e. S))))
18	17	adantl 452	. . . . . . . . . . . . . . 15 |- ((M e. NC /\ x e. NC ) -> (((x ↑c 0c) e. NC -> (2c ↑c x) e. S) -> (x e. NC -> ((2c ↑c x) e. NC -> (2c ↑c x) e. S))))
19		3anass 938	. . . . . . . . . . . . . . . . . . 19 |- ((x e. NC /\ z e. NC /\ z = (2c ↑c x)) <-> (x e. NC /\ (z e. NC /\ z = (2c ↑c x))))
20	19	imbi1i 315	. . . . . . . . . . . . . . . . . 18 |- (((x e. NC /\ z e. NC /\ z = (2c ↑c x)) -> z e. S) <-> ((x e. NC /\ (z e. NC /\ z = (2c ↑c x))) -> z e. S))
21		impexp 433	. . . . . . . . . . . . . . . . . 18 |- (((x e. NC /\ (z e. NC /\ z = (2c ↑c x))) -> z e. S) <-> (x e. NC -> ((z e. NC /\ z = (2c ↑c x)) -> z e. S)))
22	20, 21	bitri 240	. . . . . . . . . . . . . . . . 17 |- (((x e. NC /\ z e. NC /\ z = (2c ↑c x)) -> z e. S) <-> (x e. NC -> ((z e. NC /\ z = (2c ↑c x)) -> z e. S)))
23	22	albii 1566	. . . . . . . . . . . . . . . 16 |- (A.z((x e. NC /\ z e. NC /\ z = (2c ↑c x)) -> z e. S) <-> A.z(x e. NC -> ((z e. NC /\ z = (2c ↑c x)) -> z e. S)))
24		19.21v 1890	. . . . . . . . . . . . . . . . 17 |- (A.z(x e. NC -> ((z e. NC /\ z = (2c ↑c x)) -> z e. S)) <-> (x e. NC -> A.z((z e. NC /\ z = (2c ↑c x)) -> z e. S)))
25		impexp 433	. . . . . . . . . . . . . . . . . . . . 21 |- (((z e. NC /\ z = (2c ↑c x)) -> z e. S) <-> (z e. NC -> (z = (2c ↑c x) -> z e. S)))
26		bi2.04 350	. . . . . . . . . . . . . . . . . . . . 21 |- ((z e. NC -> (z = (2c ↑c x) -> z e. S)) <-> (z = (2c ↑c x) -> (z e. NC -> z e. S)))
27	25, 26	bitri 240	. . . . . . . . . . . . . . . . . . . 20 |- (((z e. NC /\ z = (2c ↑c x)) -> z e. S) <-> (z = (2c ↑c x) -> (z e. NC -> z e. S)))
28	27	albii 1566	. . . . . . . . . . . . . . . . . . 19 |- (A.z((z e. NC /\ z = (2c ↑c x)) -> z e. S) <-> A.z(z = (2c ↑c x) -> (z e. NC -> z e. S)))
29		ovex 5552	. . . . . . . . . . . . . . . . . . . 20 |- (2c ↑c x) e. _V
30		eleq1 2413	. . . . . . . . . . . . . . . . . . . . 21 |- (z = (2c ↑c x) -> (z e. NC <-> (2c ↑c x) e. NC ))
31		eleq1 2413	. . . . . . . . . . . . . . . . . . . . 21 |- (z = (2c ↑c x) -> (z e. S <-> (2c ↑c x) e. S))
32	30, 31	imbi12d 311	. . . . . . . . . . . . . . . . . . . 20 |- (z = (2c ↑c x) -> ((z e. NC -> z e. S) <-> ((2c ↑c x) e. NC -> (2c ↑c x) e. S)))
33	29, 32	ceqsalv 2886	. . . . . . . . . . . . . . . . . . 19 |- (A.z(z = (2c ↑c x) -> (z e. NC -> z e. S)) <-> ((2c ↑c x) e. NC -> (2c ↑c x) e. S))
34	28, 33	bitri 240	. . . . . . . . . . . . . . . . . 18 |- (A.z((z e. NC /\ z = (2c ↑c x)) -> z e. S) <-> ((2c ↑c x) e. NC -> (2c ↑c x) e. S))
35	34	imbi2i 303	. . . . . . . . . . . . . . . . 17 |- ((x e. NC -> A.z((z e. NC /\ z = (2c ↑c x)) -> z e. S)) <-> (x e. NC -> ((2c ↑c x) e. NC -> (2c ↑c x) e. S)))
36	24, 35	bitri 240	. . . . . . . . . . . . . . . 16 |- (A.z(x e. NC -> ((z e. NC /\ z = (2c ↑c x)) -> z e. S)) <-> (x e. NC -> ((2c ↑c x) e. NC -> (2c ↑c x) e. S)))
37	23, 36	bitri 240	. . . . . . . . . . . . . . 15 |- (A.z((x e. NC /\ z e. NC /\ z = (2c ↑c x)) -> z e. S) <-> (x e. NC -> ((2c ↑c x) e. NC -> (2c ↑c x) e. S)))
38	18, 37	syl6ibr 218	. . . . . . . . . . . . . 14 |- ((M e. NC /\ x e. NC ) -> (((x ↑c 0c) e. NC -> (2c ↑c x) e. S) -> A.z((x e. NC /\ z e. NC /\ z = (2c ↑c x)) -> z e. S)))
39		vex 2863	. . . . . . . . . . . . . . . . 17 |- x e. _V
40		vex 2863	. . . . . . . . . . . . . . . . 17 |- z e. _V
41		eleq1 2413	. . . . . . . . . . . . . . . . . 18 |- (p = x -> (p e. NC <-> x e. NC ))
42		oveq2 5532	. . . . . . . . . . . . . . . . . . 19 |- (p = x -> (2c ↑c p) = (2c ↑c x))
43	42	eqeq2d 2364	. . . . . . . . . . . . . . . . . 18 |- (p = x -> (q = (2c ↑c p) <-> q = (2c ↑c x)))
44	41, 43	3anbi13d 1254	. . . . . . . . . . . . . . . . 17 |- (p = x -> ((p e. NC /\ q e. NC /\ q = (2c ↑c p)) <-> (x e. NC /\ q e. NC /\ q = (2c ↑c x))))
45		eleq1 2413	. . . . . . . . . . . . . . . . . 18 |- (q = z -> (q e. NC <-> z e. NC ))
46		eqeq1 2359	. . . . . . . . . . . . . . . . . 18 |- (q = z -> (q = (2c ↑c x) <-> z = (2c ↑c x)))
47	45, 46	3anbi23d 1255	. . . . . . . . . . . . . . . . 17 |- (q = z -> ((x e. NC /\ q e. NC /\ q = (2c ↑c x)) <-> (x e. NC /\ z e. NC /\ z = (2c ↑c x))))
48		eqid 2353	. . . . . . . . . . . . . . . . 17 |- {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))} = {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}
49	39, 40, 44, 47, 48	brab 4710	. . . . . . . . . . . . . . . 16 |- (x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z <-> (x e. NC /\ z e. NC /\ z = (2c ↑c x)))
50	49	imbi1i 315	. . . . . . . . . . . . . . 15 |- ((x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S) <-> ((x e. NC /\ z e. NC /\ z = (2c ↑c x)) -> z e. S))
51	50	albii 1566	. . . . . . . . . . . . . 14 |- (A.z(x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S) <-> A.z((x e. NC /\ z e. NC /\ z = (2c ↑c x)) -> z e. S))
52	38, 51	syl6ibr 218	. . . . . . . . . . . . 13 |- ((M e. NC /\ x e. NC ) -> (((x ↑c 0c) e. NC -> (2c ↑c x) e. S) -> A.z(x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S)))
53	52	imim2d 48	. . . . . . . . . . . 12 |- ((M e. NC /\ x e. NC ) -> ((x e. S -> ((x ↑c 0c) e. NC -> (2c ↑c x) e. S)) -> (x e. S -> A.z(x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S))))
54		impexp 433	. . . . . . . . . . . 12 |- (((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S) <-> (x e. S -> ((x ↑c 0c) e. NC -> (2c ↑c x) e. S)))
55		impexp 433	. . . . . . . . . . . . . 14 |- (((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S) <-> (x e. S -> (x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S)))
56	55	albii 1566	. . . . . . . . . . . . 13 |- (A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S) <-> A.z(x e. S -> (x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S)))
57		19.21v 1890	. . . . . . . . . . . . 13 |- (A.z(x e. S -> (x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S)) <-> (x e. S -> A.z(x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S)))
58	56, 57	bitri 240	. . . . . . . . . . . 12 |- (A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S) <-> (x e. S -> A.z(x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z -> z e. S)))
59	53, 54, 58	3imtr4g 261	. . . . . . . . . . 11 |- ((M e. NC /\ x e. NC ) -> (((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S) -> A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S)))
60	59	ex 423	. . . . . . . . . 10 |- (M e. NC -> (x e. NC -> (((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S) -> A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S))))
61	10, 60	syld 40	. . . . . . . . 9 |- (M e. NC -> (x e. ( Spac ` M) -> (((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S) -> A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S))))
62	61	imp 418	. . . . . . . 8 |- ((M e. NC /\ x e. ( Spac ` M)) -> (((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S) -> A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S)))
63	62	ralimdva 2693	. . . . . . 7 |- (M e. NC -> (A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S) -> A.x e. ( Spac ` M)A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S)))
64		raleq 2808	. . . . . . . 8 |- (( Spac ` M) = Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) -> (A.x e. ( Spac ` M)A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S) <-> A.x e. Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S)))
65	2, 64	syl 15	. . . . . . 7 |- (M e. NC -> (A.x e. ( Spac ` M)A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S) <-> A.x e. Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S)))
66	63, 65	sylibd 205	. . . . . 6 |- (M e. NC -> (A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S) -> A.x e. Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S)))
67	66	imp 418	. . . . 5 |- ((M e. NC /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S)) -> A.x e. Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S))
68	67	ad2ant2rl 729	. . . 4 |- (((M e. NC /\ S e. _V) /\ (M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S))) -> A.x e. Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S))
69		snex 4112	. . . . 5 |- {M} e. _V
70		spacvallem1 6282	. . . . 5 |- {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))} e. _V
71		eqid 2353	. . . . 5 |- Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) = Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})
72	69, 70, 71	clos1induct 5881	. . . 4 |- ((S e. _V /\ {M} C_ S /\ A.x e. Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))})A.z((x e. S /\ x{<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}z) -> z e. S)) -> Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) C_ S)
73	5, 8, 68, 72	syl3anc 1182	. . 3 |- (((M e. NC /\ S e. _V) /\ (M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S))) -> Clos1 ({M}, {<.p, q>. | (p e. NC /\ q e. NC /\ q = (2c ↑c p))}) C_ S)
74	4, 73	eqsstrd 3306	. 2 |- (((M e. NC /\ S e. _V) /\ (M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S))) -> ( Spac ` M) C_ S)
75	1, 74	sylanl2 632	1 |- (((M e. NC /\ S e. V) /\ (M e. S /\ A.x e. ( Spac ` M)((x e. S /\ (x ↑c 0c) e. NC ) -> (2c ↑c x) e. S))) -> ( Spac ` M) C_ S)

This proof depends on syntax axioms:   -> wi 4   <-> wb 176   /\ wa 358   /\ w3a 934  A.wal 1540   = wceq 1642   e. wcel 1710  A.wral 2615  _Vcvv 2860   C_ wss 3258  {csn 3738  0_cc0c 4375  {copab 4623   class class class wbr 4640  ` cfv 4782  (class class class)co 5526   Clos1 cclos1 5873   NC cncs 6089  2_cc2c 6095   ↑_c cce 6097   Sp_ac cspac 6274

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-si 4729  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-ov 5527  df-oprab 5529  df-mpt 5653  df-mpt2 5655  df-txp 5737  df-fix 5741  df-compose 5749  df-ins2 5751  df-ins3 5753  df-image 5755  df-ins4 5757  df-si3 5759  df-funs 5761  df-fns 5763  df-pw1fn 5767  df-fullfun 5769  df-clos1 5874  df-trans 5900  df-sym 5909  df-er 5910  df-ec 5948  df-qs 5952  df-map 6002  df-en 6030  df-ncs 6099  df-nc 6102  df-2c 6105  df-ce 6107  df-spac 6275

This theorem is used by:  spacis  6289  nchoicelem6  6295