Theorem cntzcmn 18245

Description: The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)

Hypotheses

Ref	Expression
cntzcmn.b	|- B = ( Base ` G )
cntzcmn.z	|- Z = (Cntz ` G )

Assertion

Ref	Expression
cntzcmn	|- ( ( G e. CMnd /\ S C_ B ) -> ( Z ` S ) = B )

Proof of Theorem cntzcmn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cntzcmn.b	. . . 4 |- B = ( Base ` G )
2		cntzcmn.z	. . . 4 |- Z = (Cntz ` G )
3	1, 2	cntzssv 17761	. . 3 |- ( Z ` S ) C_ B
4	3	a1i 11	. 2 |- ( ( G e. CMnd /\ S C_ B ) -> ( Z ` S ) C_ B )
5		simpl1 1064	. . . . . . 7 |- ( ( ( G e. CMnd /\ S C_ B /\ x e. B ) /\ y e. S ) -> G e. CMnd )
6		simpl3 1066	. . . . . . 7 |- ( ( ( G e. CMnd /\ S C_ B /\ x e. B ) /\ y e. S ) -> x e. B )
7		simp2 1062	. . . . . . . 8 |- ( ( G e. CMnd /\ S C_ B /\ x e. B ) -> S C_ B )
8	7	sselda 3603	. . . . . . 7 |- ( ( ( G e. CMnd /\ S C_ B /\ x e. B ) /\ y e. S ) -> y e. B )
9		eqid 2622	. . . . . . . 8 |- ( +g ` G ) = ( +g ` G )
10	1, 9	cmncom 18209	. . . . . . 7 |- ( ( G e. CMnd /\ x e. B /\ y e. B ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
11	5, 6, 8, 10	syl3anc 1326	. . . . . 6 |- ( ( ( G e. CMnd /\ S C_ B /\ x e. B ) /\ y e. S ) -> ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
12	11	ralrimiva 2966	. . . . 5 |- ( ( G e. CMnd /\ S C_ B /\ x e. B ) -> A. y e. S ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) )
13	1, 9, 2	cntzel 17756	. . . . . 6 |- ( ( S C_ B /\ x e. B ) -> ( x e. ( Z ` S ) <-> A. y e. S ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
14	13	3adant1 1079	. . . . 5 |- ( ( G e. CMnd /\ S C_ B /\ x e. B ) -> ( x e. ( Z ` S ) <-> A. y e. S ( x ( +g ` G ) y ) = ( y ( +g ` G ) x ) ) )
15	12, 14	mpbird 247	. . . 4 |- ( ( G e. CMnd /\ S C_ B /\ x e. B ) -> x e. ( Z ` S ) )
16	15	3expia 1267	. . 3 |- ( ( G e. CMnd /\ S C_ B ) -> ( x e. B -> x e. ( Z ` S ) ) )
17	16	ssrdv 3609	. 2 |- ( ( G e. CMnd /\ S C_ B ) -> B C_ ( Z ` S ) )
18	4, 17	eqssd 3620	1 |- ( ( G e. CMnd /\ S C_ B ) -> ( Z ` S ) = B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   ` cfv 5888  (class class class)co 6650   Basecbs 15857   +g cplusg 15941  Cntzccntz 17748  CMndccmn 18193

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-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

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-reu 2919  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-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-cntz 17750  df-cmn 18195

This theorem is referenced by:  cntzcmnss  18246  cntzcmnf  18248  ablcntzd  18260  gsumadd  18323