Theorem assintopcllaw 41848

Description: The closure low holds for an associative (closed internal binary) operation for a set. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)

Assertion

Ref	Expression
assintopcllaw	|- ( .o. e. ( assIntOp ` M ) -> .o. clLaw M )

Proof of Theorem assintopcllaw

Dummy variable o is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6221	. 2 |- ( .o. e. ( assIntOp ` M ) -> M e. _V )
2		assintopval 41841	. . . . 5 |- ( M e. _V -> ( assIntOp ` M ) = { o e. ( clIntOp ` M ) | o assLaw M } )
3	2	eleq2d 2687	. . . 4 |- ( M e. _V -> ( .o. e. ( assIntOp ` M ) <-> .o. e. { o e. ( clIntOp ` M ) | o assLaw M } ) )
4		breq1 4656	. . . . 5 |- ( o = .o. -> ( o assLaw M <-> .o. assLaw M ) )
5	4	elrab 3363	. . . 4 |- ( .o. e. { o e. ( clIntOp ` M ) | o assLaw M } <-> ( .o. e. ( clIntOp ` M ) /\ .o. assLaw M ) )
6	3, 5	syl6bb 276	. . 3 |- ( M e. _V -> ( .o. e. ( assIntOp ` M ) <-> ( .o. e. ( clIntOp ` M ) /\ .o. assLaw M ) ) )
7		clintopcllaw 41847	. . . 4 |- ( .o. e. ( clIntOp ` M ) -> .o. clLaw M )
8	7	adantr 481	. . 3 |- ( ( .o. e. ( clIntOp ` M ) /\ .o. assLaw M ) -> .o. clLaw M )
9	6, 8	syl6bi 243	. 2 |- ( M e. _V -> ( .o. e. ( assIntOp ` M ) -> .o. clLaw M ) )
10	1, 9	mpcom 38	1 |- ( .o. e. ( assIntOp ` M ) -> .o. clLaw M )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   {crab 2916   _Vcvv 3200   class class class wbr 4653   ` cfv 5888   clLaw ccllaw 41819   assLaw casslaw 41820   clIntOp cclintop 41833   assIntOp cassintop 41834

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-cllaw 41822  df-intop 41835  df-clintop 41836  df-assintop 41837

This theorem is referenced by: (None)