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Theorem cntzspan 15137
Description: If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
Hypotheses
Ref Expression
cntzspan.z  |-  Z  =  (Cntz `  G )
cntzspan.k  |-  K  =  (mrCls `  (SubMnd `  G
) )
cntzspan.h  |-  H  =  ( Gs  ( K `  S ) )
Assertion
Ref Expression
cntzspan  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  ->  H  e. CMnd )

Proof of Theorem cntzspan
StepHypRef Expression
1 eqid 2283 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
21submacs 14442 . . . . 5  |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS
`  ( Base `  G
) ) )
32adantr 451 . . . 4  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
(SubMnd `  G )  e.  (ACS `  ( Base `  G ) ) )
4 acsmre 13554 . . . 4  |-  ( (SubMnd `  G )  e.  (ACS
`  ( Base `  G
) )  ->  (SubMnd `  G )  e.  (Moore `  ( Base `  G
) ) )
53, 4syl 15 . . 3  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
(SubMnd `  G )  e.  (Moore `  ( Base `  G ) ) )
6 simpr 447 . . . . 5  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  ->  S  C_  ( Z `  S ) )
7 cntzspan.z . . . . . . . 8  |-  Z  =  (Cntz `  G )
81, 7cntzssv 14804 . . . . . . 7  |-  ( Z `
 S )  C_  ( Base `  G )
96, 8syl6ss 3191 . . . . . 6  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  ->  S  C_  ( Base `  G
) )
101, 7cntzsubm 14811 . . . . . 6  |-  ( ( G  e.  Mnd  /\  S  C_  ( Base `  G
) )  ->  ( Z `  S )  e.  (SubMnd `  G )
)
119, 10syldan 456 . . . . 5  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
( Z `  S
)  e.  (SubMnd `  G ) )
12 cntzspan.k . . . . . 6  |-  K  =  (mrCls `  (SubMnd `  G
) )
1312mrcsscl 13522 . . . . 5  |-  ( ( (SubMnd `  G )  e.  (Moore `  ( Base `  G ) )  /\  S  C_  ( Z `  S )  /\  ( Z `  S )  e.  (SubMnd `  G )
)  ->  ( K `  S )  C_  ( Z `  S )
)
145, 6, 11, 13syl3anc 1182 . . . 4  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
( K `  S
)  C_  ( Z `  S ) )
1512mrccl 13513 . . . . . . 7  |-  ( ( (SubMnd `  G )  e.  (Moore `  ( Base `  G ) )  /\  S  C_  ( Base `  G
) )  ->  ( K `  S )  e.  (SubMnd `  G )
)
165, 9, 15syl2anc 642 . . . . . 6  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
( K `  S
)  e.  (SubMnd `  G ) )
171submss 14427 . . . . . 6  |-  ( ( K `  S )  e.  (SubMnd `  G
)  ->  ( K `  S )  C_  ( Base `  G ) )
1816, 17syl 15 . . . . 5  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
( K `  S
)  C_  ( Base `  G ) )
191, 7cntzrec 14809 . . . . 5  |-  ( ( ( K `  S
)  C_  ( Base `  G )  /\  S  C_  ( Base `  G
) )  ->  (
( K `  S
)  C_  ( Z `  S )  <->  S  C_  ( Z `  ( K `  S ) ) ) )
2018, 9, 19syl2anc 642 . . . 4  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
( ( K `  S )  C_  ( Z `  S )  <->  S 
C_  ( Z `  ( K `  S ) ) ) )
2114, 20mpbid 201 . . 3  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  ->  S  C_  ( Z `  ( K `  S ) ) )
221, 7cntzsubm 14811 . . . 4  |-  ( ( G  e.  Mnd  /\  ( K `  S ) 
C_  ( Base `  G
) )  ->  ( Z `  ( K `  S ) )  e.  (SubMnd `  G )
)
2318, 22syldan 456 . . 3  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
( Z `  ( K `  S )
)  e.  (SubMnd `  G ) )
2412mrcsscl 13522 . . 3  |-  ( ( (SubMnd `  G )  e.  (Moore `  ( Base `  G ) )  /\  S  C_  ( Z `  ( K `  S ) )  /\  ( Z `
 ( K `  S ) )  e.  (SubMnd `  G )
)  ->  ( K `  S )  C_  ( Z `  ( K `  S ) ) )
255, 21, 23, 24syl3anc 1182 . 2  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
( K `  S
)  C_  ( Z `  ( K `  S
) ) )
26 cntzspan.h . . . 4  |-  H  =  ( Gs  ( K `  S ) )
2726, 7submcmn2 15135 . . 3  |-  ( ( K `  S )  e.  (SubMnd `  G
)  ->  ( H  e. CMnd  <-> 
( K `  S
)  C_  ( Z `  ( K `  S
) ) ) )
2816, 27syl 15 . 2  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  -> 
( H  e. CMnd  <->  ( K `  S )  C_  ( Z `  ( K `  S ) ) ) )
2925, 28mpbird 223 1  |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  ->  H  e. CMnd )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   ` cfv 5255  (class class class)co 5858   Basecbs 13148   ↾s cress 13149  Moorecmre 13484  mrClscmrc 13485  ACScacs 13487   Mndcmnd 14361  SubMndcsubmnd 14414  Cntzccntz 14791  CMndccmn 15089
This theorem is referenced by:  gsumzsplit  15206  gsumzoppg  15216  gsumpt  15222
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-0g 13404  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-cntz 14793  df-cmn 15091
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