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Theorem zlmtset 27812
Description: Topology in a  ZZ-module (if present). (Contributed by Thierry Arnoux, 8-Nov-2017.)
Hypotheses
Ref Expression
zlmlem2.1  |-  W  =  ( ZMod `  G
)
zlmtset.1  |-  J  =  (TopSet `  G )
Assertion
Ref Expression
zlmtset  |-  ( G  e.  V  ->  J  =  (TopSet `  W )
)

Proof of Theorem zlmtset
StepHypRef Expression
1 zlmlem2.1 . . . 4  |-  W  =  ( ZMod `  G
)
2 eqid 2441 . . . 4  |-  (.g `  G
)  =  (.g `  G
)
31, 2zlmval 18420 . . 3  |-  ( G  e.  V  ->  W  =  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  G
) >. ) )
43fveq2d 5856 . 2  |-  ( G  e.  V  ->  (TopSet `  W )  =  (TopSet `  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  G ) >.
) ) )
5 zlmtset.1 . . 3  |-  J  =  (TopSet `  G )
6 tsetid 14657 . . . 4  |- TopSet  = Slot  (TopSet ` 
ndx )
7 5re 10615 . . . . . 6  |-  5  e.  RR
8 5lt9 10734 . . . . . 6  |-  5  <  9
97, 8gtneii 9694 . . . . 5  |-  9  =/=  5
10 tsetndx 14656 . . . . . 6  |-  (TopSet `  ndx )  =  9
11 scandx 14629 . . . . . 6  |-  (Scalar `  ndx )  =  5
1210, 11neeq12i 2730 . . . . 5  |-  ( (TopSet `  ndx )  =/=  (Scalar ` 
ndx )  <->  9  =/=  5 )
139, 12mpbir 209 . . . 4  |-  (TopSet `  ndx )  =/=  (Scalar ` 
ndx )
146, 13setsnid 14546 . . 3  |-  (TopSet `  G )  =  (TopSet `  ( G sSet  <. (Scalar ` 
ndx ) ,ring >. ) )
15 6re 10617 . . . . . 6  |-  6  e.  RR
16 6lt9 10733 . . . . . 6  |-  6  <  9
1715, 16gtneii 9694 . . . . 5  |-  9  =/=  6
18 vscandx 14631 . . . . . 6  |-  ( .s
`  ndx )  =  6
1910, 18neeq12i 2730 . . . . 5  |-  ( (TopSet `  ndx )  =/=  ( .s `  ndx )  <->  9  =/=  6 )
2017, 19mpbir 209 . . . 4  |-  (TopSet `  ndx )  =/=  ( .s `  ndx )
216, 20setsnid 14546 . . 3  |-  (TopSet `  ( G sSet  <. (Scalar `  ndx ) ,ring >. ) )  =  (TopSet `  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  G ) >.
) )
225, 14, 213eqtri 2474 . 2  |-  J  =  (TopSet `  ( ( G sSet  <. (Scalar `  ndx ) ,ring >. ) sSet  <. ( .s `  ndx ) ,  (.g `  G ) >.
) )
234, 22syl6reqr 2501 1  |-  ( G  e.  V  ->  J  =  (TopSet `  W )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802    =/= wne 2636   <.cop 4016   ` cfv 5574  (class class class)co 6277   5c5 10589   6c6 10590   9c9 10593   ndxcnx 14501   sSet csts 14502  Scalarcsca 14572   .scvsca 14573  TopSetcts 14575  .gcmg 15925  ℤringzring 18356   ZModczlm 18405
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-er 7309  df-en 7515  df-dom 7516  df-sdom 7517  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-2 10595  df-3 10596  df-4 10597  df-5 10598  df-6 10599  df-7 10600  df-8 10601  df-9 10602  df-ndx 14507  df-slot 14508  df-sets 14510  df-sca 14585  df-vsca 14586  df-tset 14588  df-zlm 18409
This theorem is referenced by:  zhmnrg  27814
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