Theorem ndmovdistr 5620

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

Step	Hyp	Ref	Expression
1		ndmov.4	. . . . . . 7 ⊢ C ∈ V
2		ndmov.2	. . . . . . 7 ⊢ dom F = (S × S)
3		ndmov.5	. . . . . . 7 ⊢ ¬ ∅ ∈ S
4	1, 2, 3	ndmovrcl 5617	. . . . . 6 ⊢ ((BFC) ∈ S → (B ∈ S ∧ C ∈ S))
5	4	anim2i 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)))
7	5, 6	sylibr 203	. . . 4 ⊢ ((A ∈ S ∧ (BFC) ∈ S) → (A ∈ S ∧ B ∈ S ∧ C ∈ S))
8	7	con3i 127	. . 3 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ¬ (A ∈ S ∧ (BFC) ∈ S))
9		ovex 5552	. . . 4 ⊢ (BFC) ∈ V
10		ndmov.6	. . . 4 ⊢ dom G = (S × S)
11	9, 10	ndmov 5616	. . 3 ⊢ (¬ (A ∈ S ∧ (BFC) ∈ S) → (AG(BFC)) = ∅)
12	8, 11	syl 15	. 2 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → (AG(BFC)) = ∅)
13		ndmov.1	. . . . . . 7 ⊢ B ∈ V
14	13, 10, 3	ndmovrcl 5617	. . . . . 6 ⊢ ((AGB) ∈ S → (A ∈ S ∧ B ∈ S))
15	1, 10, 3	ndmovrcl 5617	. . . . . 6 ⊢ ((AGC) ∈ S → (A ∈ S ∧ C ∈ S))
16	14, 15	anim12i 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)))
18	6, 17	bitri 240	. . . . 5 ⊢ ((A ∈ S ∧ B ∈ S ∧ C ∈ S) ↔ ((A ∈ S ∧ B ∈ S) ∧ (A ∈ S ∧ C ∈ S)))
19	16, 18	sylibr 203	. . . 4 ⊢ (((AGB) ∈ S ∧ (AGC) ∈ S) → (A ∈ S ∧ B ∈ S ∧ C ∈ S))
20	19	con3i 127	. . 3 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ¬ ((AGB) ∈ S ∧ (AGC) ∈ S))
21		ovex 5552	. . . 4 ⊢ (AGC) ∈ V
22	21, 2	ndmov 5616	. . 3 ⊢ (¬ ((AGB) ∈ S ∧ (AGC) ∈ S) → ((AGB)F(AGC)) = ∅)
23	20, 22	syl 15	. 2 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ((AGB)F(AGC)) = ∅)
24	12, 23	eqtr4d 2388	1 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → (AG(BFC)) = ((AGB)F(AGC)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358   ∧ w3a 934   = wceq 1642   ∈ wcel 1710  Vcvv 2860  ∅c0 3551   × cxp 4771  dom cdm 4773  (class class class)co 5526

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-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-fv 4796  df-ov 5527

This theorem is referenced by: (None)