Theorem ndmovass 5618

Description: Any operation is associative 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

Assertion

Ref	Expression
ndmovass	⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ((AFB)FC) = (AF(BFC)))

Proof of Theorem ndmovass

Step	Hyp	Ref	Expression
1		ndmov.1	. . . . . . 7 ⊢ B ∈ V
2		ndmov.2	. . . . . . 7 ⊢ dom F = (S × S)
3		ndmov.5	. . . . . . 7 ⊢ ¬ ∅ ∈ S
4	1, 2, 3	ndmovrcl 5616	. . . . . 6 ⊢ ((AFB) ∈ S → (A ∈ S ∧ B ∈ S))
5	4	anim1i 551	. . . . 5 ⊢ (((AFB) ∈ S ∧ C ∈ S) → ((A ∈ S ∧ B ∈ S) ∧ C ∈ S))
6		df-3an 936	. . . . 5 ⊢ ((A ∈ S ∧ B ∈ S ∧ C ∈ S) ↔ ((A ∈ S ∧ B ∈ S) ∧ C ∈ S))
7	5, 6	sylibr 203	. . . 4 ⊢ (((AFB) ∈ S ∧ C ∈ S) → (A ∈ S ∧ B ∈ S ∧ C ∈ S))
8	7	con3i 127	. . 3 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ¬ ((AFB) ∈ S ∧ C ∈ S))
9		ndmov.4	. . . 4 ⊢ C ∈ V
10	9, 2	ndmov 5615	. . 3 ⊢ (¬ ((AFB) ∈ S ∧ C ∈ S) → ((AFB)FC) = ∅)
11	8, 10	syl 15	. 2 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ((AFB)FC) = ∅)
12	9, 2, 3	ndmovrcl 5616	. . . . . 6 ⊢ ((BFC) ∈ S → (B ∈ S ∧ C ∈ S))
13	12	anim2i 552	. . . . 5 ⊢ ((A ∈ S ∧ (BFC) ∈ S) → (A ∈ S ∧ (B ∈ S ∧ C ∈ S)))
14		3anass 938	. . . . 5 ⊢ ((A ∈ S ∧ B ∈ S ∧ C ∈ S) ↔ (A ∈ S ∧ (B ∈ S ∧ C ∈ S)))
15	13, 14	sylibr 203	. . . 4 ⊢ ((A ∈ S ∧ (BFC) ∈ S) → (A ∈ S ∧ B ∈ S ∧ C ∈ S))
16	15	con3i 127	. . 3 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ¬ (A ∈ S ∧ (BFC) ∈ S))
17		ovex 5551	. . . 4 ⊢ (BFC) ∈ V
18	17, 2	ndmov 5615	. . 3 ⊢ (¬ (A ∈ S ∧ (BFC) ∈ S) → (AF(BFC)) = ∅)
19	16, 18	syl 15	. 2 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → (AF(BFC)) = ∅)
20	11, 19	eqtr4d 2388	1 ⊢ (¬ (A ∈ S ∧ B ∈ S ∧ C ∈ S) → ((AFB)FC) = (AF(BFC)))

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)