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Theorem spaccl 6286
 Description: Closure law for the special set generator. (Contributed by SF, 13-Mar-2015.)
Assertion
Ref Expression
spaccl ((M NC N ( SpacM) (Nc 0c) NC ) → (2cc N) ( SpacM))

Proof of Theorem spaccl
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 956 . . . 4 ((M NC N ( SpacM) (Nc 0c) NC ) → N ( SpacM))
2 spacval 6282 . . . . 5 (M NC → ( SpacM) = Clos1 ({M}, {x, y (x NC y NC y = (2cc x))}))
323ad2ant1 976 . . . 4 ((M NC N ( SpacM) (Nc 0c) NC ) → ( SpacM) = Clos1 ({M}, {x, y (x NC y NC y = (2cc x))}))
41, 3eleqtrd 2429 . . 3 ((M NC N ( SpacM) (Nc 0c) NC ) → N Clos1 ({M}, {x, y (x NC y NC y = (2cc x))}))
5 spacssnc 6284 . . . . . 6 (M NC → ( SpacM) NC )
65sselda 3273 . . . . 5 ((M NC N ( SpacM)) → N NC )
763adant3 975 . . . 4 ((M NC N ( SpacM) (Nc 0c) NC ) → N NC )
8 simp3 957 . . . . 5 ((M NC N ( SpacM) (Nc 0c) NC ) → (Nc 0c) NC )
9 2nnc 6167 . . . . . 6 2c Nn
10 ceclnn1 6189 . . . . . 6 ((2c Nn N NC (Nc 0c) NC ) → (2cc N) NC )
119, 10mp3an1 1264 . . . . 5 ((N NC (Nc 0c) NC ) → (2cc N) NC )
127, 8, 11syl2anc 642 . . . 4 ((M NC N ( SpacM) (Nc 0c) NC ) → (2cc N) NC )
13 eqidd 2354 . . . 4 ((M NC N ( SpacM) (Nc 0c) NC ) → (2cc N) = (2cc N))
14 ovex 5551 . . . . . 6 (2cc N) V
15 eleq1 2413 . . . . . . . 8 (x = N → (x NCN NC ))
16 oveq2 5531 . . . . . . . . 9 (x = N → (2cc x) = (2cc N))
1716eqeq2d 2364 . . . . . . . 8 (x = N → (y = (2cc x) ↔ y = (2cc N)))
1815, 173anbi13d 1254 . . . . . . 7 (x = N → ((x NC y NC y = (2cc x)) ↔ (N NC y NC y = (2cc N))))
19 eleq1 2413 . . . . . . . 8 (y = (2cc N) → (y NC ↔ (2cc N) NC ))
20 eqeq1 2359 . . . . . . . 8 (y = (2cc N) → (y = (2cc N) ↔ (2cc N) = (2cc N)))
2119, 203anbi23d 1255 . . . . . . 7 (y = (2cc N) → ((N NC y NC y = (2cc N)) ↔ (N NC (2cc N) NC (2cc N) = (2cc N))))
22 eqid 2353 . . . . . . 7 {x, y (x NC y NC y = (2cc x))} = {x, y (x NC y NC y = (2cc x))}
2318, 21, 22brabg 4706 . . . . . 6 ((N ( SpacM) (2cc N) V) → (N{x, y (x NC y NC y = (2cc x))} (2cc N) ↔ (N NC (2cc N) NC (2cc N) = (2cc N))))
2414, 23mpan2 652 . . . . 5 (N ( SpacM) → (N{x, y (x NC y NC y = (2cc x))} (2cc N) ↔ (N NC (2cc N) NC (2cc N) = (2cc N))))
25243ad2ant2 977 . . . 4 ((M NC N ( SpacM) (Nc 0c) NC ) → (N{x, y (x NC y NC y = (2cc x))} (2cc N) ↔ (N NC (2cc N) NC (2cc N) = (2cc N))))
267, 12, 13, 25mpbir3and 1135 . . 3 ((M NC N ( SpacM) (Nc 0c) NC ) → N{x, y (x NC y NC y = (2cc x))} (2cc N))
27 eqid 2353 . . . 4 Clos1 ({M}, {x, y (x NC y NC y = (2cc x))}) = Clos1 ({M}, {x, y (x NC y NC y = (2cc x))})
2827clos1conn 5879 . . 3 ((N Clos1 ({M}, {x, y (x NC y NC y = (2cc x))}) N{x, y (x NC y NC y = (2cc x))} (2cc N)) → (2cc N) Clos1 ({M}, {x, y (x NC y NC y = (2cc x))}))
294, 26, 28syl2anc 642 . 2 ((M NC N ( SpacM) (Nc 0c) NC ) → (2cc N) Clos1 ({M}, {x, y (x NC y NC y = (2cc x))}))
3029, 3eleqtrrd 2430 1 ((M NC N ( SpacM) (Nc 0c) NC ) → (2cc N) ( SpacM))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2859  {csn 3737   Nn cnnc 4373  0cc0c 4374  {copab 4622   class class class wbr 4639   ‘cfv 4781  (class class class)co 5525   Clos1 cclos1 5872   NC cncs 6088  2cc2c 6094   ↑c cce 6096   Spac cspac 6273 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-1st 4723  df-swap 4724  df-sset 4725  df-co 4726  df-ima 4727  df-si 4728  df-id 4767  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-res 4788  df-fun 4789  df-fn 4790  df-f 4791  df-f1 4792  df-fo 4793  df-f1o 4794  df-fv 4795  df-2nd 4797  df-ov 5526  df-oprab 5528  df-mpt 5652  df-mpt2 5654  df-txp 5736  df-fix 5740  df-compose 5748  df-ins2 5750  df-ins3 5752  df-image 5754  df-ins4 5756  df-si3 5758  df-funs 5760  df-fns 5762  df-pw1fn 5766  df-fullfun 5768  df-clos1 5873  df-trans 5899  df-sym 5908  df-er 5909  df-ec 5947  df-qs 5951  df-map 6001  df-en 6029  df-ncs 6098  df-nc 6101  df-2c 6104  df-ce 6106  df-spac 6274 This theorem is referenced by:  nchoicelem6  6294
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