New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  ndmovdistr GIF version

Theorem ndmovdistr 5619
 Description: Any operation is distributive outside its domain, if the domain doesn't contain the empty set. (Contributed by set.mm contributors, 24-Aug-1995.)
Hypotheses
Ref Expression
ndmov.1 B V
ndmov.2 dom F = (S × S)
ndmov.4 C V
ndmov.5 ¬ S
ndmov.6 dom G = (S × S)
Assertion
Ref Expression
ndmovdistr (¬ (A S B S C S) → (AG(BFC)) = ((AGB)F(AGC)))

Proof of Theorem ndmovdistr
StepHypRef Expression
1 ndmov.4 . . . . . . 7 C V
2 ndmov.2 . . . . . . 7 dom F = (S × S)
3 ndmov.5 . . . . . . 7 ¬ S
41, 2, 3ndmovrcl 5616 . . . . . 6 ((BFC) S → (B S C S))
54anim2i 552 . . . . 5 ((A S (BFC) S) → (A S (B S C S)))
6 3anass 938 . . . . 5 ((A S B S C S) ↔ (A S (B S C S)))
75, 6sylibr 203 . . . 4 ((A S (BFC) S) → (A S B S C S))
87con3i 127 . . 3 (¬ (A S B S C S) → ¬ (A S (BFC) S))
9 ovex 5551 . . . 4 (BFC) V
10 ndmov.6 . . . 4 dom G = (S × S)
119, 10ndmov 5615 . . 3 (¬ (A S (BFC) S) → (AG(BFC)) = )
128, 11syl 15 . 2 (¬ (A S B S C S) → (AG(BFC)) = )
13 ndmov.1 . . . . . . 7 B V
1413, 10, 3ndmovrcl 5616 . . . . . 6 ((AGB) S → (A S B S))
151, 10, 3ndmovrcl 5616 . . . . . 6 ((AGC) S → (A S C S))
1614, 15anim12i 549 . . . . 5 (((AGB) S (AGC) S) → ((A S B S) (A S C S)))
17 anandi 801 . . . . . 6 ((A S (B S C S)) ↔ ((A S B S) (A S C S)))
186, 17bitri 240 . . . . 5 ((A S B S C S) ↔ ((A S B S) (A S C S)))
1916, 18sylibr 203 . . . 4 (((AGB) S (AGC) S) → (A S B S C S))
2019con3i 127 . . 3 (¬ (A S B S C S) → ¬ ((AGB) S (AGC) S))
21 ovex 5551 . . . 4 (AGC) V
2221, 2ndmov 5615 . . 3 (¬ ((AGB) S (AGC) S) → ((AGB)F(AGC)) = )
2320, 22syl 15 . 2 (¬ (A S B S C S) → ((AGB)F(AGC)) = )
2412, 23eqtr4d 2388 1 (¬ (A S B S C S) → (AG(BFC)) = ((AGB)F(AGC)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2859  ∅c0 3550   × cxp 4770  dom cdm 4772  (class class class)co 5525 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 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-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787  df-fv 4795  df-ov 5526 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator