Theorem spaccl 6287

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

Assertion

Ref	Expression
spaccl	|- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> (2c ↑c N) e. ( Spac ` M))

Proof of Theorem spaccl

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 956	. . . 4 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> N e. ( Spac ` M))
2		spacval 6283	. . . . 5 |- (M e. NC -> ( Spac ` M) = Clos1 ({M}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}))
3	2	3ad2ant1 976	. . . 4 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> ( Spac ` M) = Clos1 ({M}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}))
4	1, 3	eleqtrd 2429	. . 3 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> N e. Clos1 ({M}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}))
5		spacssnc 6285	. . . . . 6 |- (M e. NC -> ( Spac ` M) C_ NC )
6	5	sselda 3274	. . . . 5 |- ((M e. NC /\ N e. ( Spac ` M)) -> N e. NC )
7	6	3adant3 975	. . . 4 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> N e. NC )
8		simp3 957	. . . . 5 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> (N ↑c 0c) e. NC )
9		2nnc 6168	. . . . . 6 |- 2c e. Nn
10		ceclnn1 6190	. . . . . 6 |- ((2c e. Nn /\ N e. NC /\ (N ↑c 0c) e. NC ) -> (2c ↑c N) e. NC )
11	9, 10	mp3an1 1264	. . . . 5 |- ((N e. NC /\ (N ↑c 0c) e. NC ) -> (2c ↑c N) e. NC )
12	7, 8, 11	syl2anc 642	. . . 4 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> (2c ↑c N) e. NC )
13		eqidd 2354	. . . 4 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> (2c ↑c N) = (2c ↑c N))
14		ovex 5552	. . . . . 6 |- (2c ↑c N) e. _V
15		eleq1 2413	. . . . . . . 8 |- (x = N -> (x e. NC <-> N e. NC ))
16		oveq2 5532	. . . . . . . . 9 |- (x = N -> (2c ↑c x) = (2c ↑c N))
17	16	eqeq2d 2364	. . . . . . . 8 |- (x = N -> (y = (2c ↑c x) <-> y = (2c ↑c N)))
18	15, 17	3anbi13d 1254	. . . . . . 7 |- (x = N -> ((x e. NC /\ y e. NC /\ y = (2c ↑c x)) <-> (N e. NC /\ y e. NC /\ y = (2c ↑c N))))
19		eleq1 2413	. . . . . . . 8 |- (y = (2c ↑c N) -> (y e. NC <-> (2c ↑c N) e. NC ))
20		eqeq1 2359	. . . . . . . 8 |- (y = (2c ↑c N) -> (y = (2c ↑c N) <-> (2c ↑c N) = (2c ↑c N)))
21	19, 20	3anbi23d 1255	. . . . . . 7 |- (y = (2c ↑c N) -> ((N e. NC /\ y e. NC /\ y = (2c ↑c N)) <-> (N e. NC /\ (2c ↑c N) e. NC /\ (2c ↑c N) = (2c ↑c N))))
22		eqid 2353	. . . . . . 7 |- {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))} = {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}
23	18, 21, 22	brabg 4707	. . . . . 6 |- ((N e. ( Spac ` M) /\ (2c ↑c N) e. _V) -> (N{<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))} (2c ↑c N) <-> (N e. NC /\ (2c ↑c N) e. NC /\ (2c ↑c N) = (2c ↑c N))))
24	14, 23	mpan2 652	. . . . 5 |- (N e. ( Spac ` M) -> (N{<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))} (2c ↑c N) <-> (N e. NC /\ (2c ↑c N) e. NC /\ (2c ↑c N) = (2c ↑c N))))
25	24	3ad2ant2 977	. . . 4 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> (N{<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))} (2c ↑c N) <-> (N e. NC /\ (2c ↑c N) e. NC /\ (2c ↑c N) = (2c ↑c N))))
26	7, 12, 13, 25	mpbir3and 1135	. . 3 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> N{<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))} (2c ↑c N))
27		eqid 2353	. . . 4 |- Clos1 ({M}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}) = Clos1 ({M}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))})
28	27	clos1conn 5880	. . 3 |- ((N e. Clos1 ({M}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}) /\ N{<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))} (2c ↑c N)) -> (2c ↑c N) e. Clos1 ({M}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}))
29	4, 26, 28	syl2anc 642	. 2 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> (2c ↑c N) e. Clos1 ({M}, {<.x, y>. | (x e. NC /\ y e. NC /\ y = (2c ↑c x))}))
30	29, 3	eleqtrrd 2430	1 |- ((M e. NC /\ N e. ( Spac ` M) /\ (N ↑c 0c) e. NC ) -> (2c ↑c N) e. ( Spac ` M))

Syntax hints:   -> wi 4   <-> wb 176   /\ w3a 934   = wceq 1642   e. wcel 1710  _Vcvv 2860  {csn 3738   Nn cnnc 4374  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 theorem was proved from 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 theorem 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 referenced by:  nchoicelem6  6295