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Theorem zlmtset 24341
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 2435 . . . 4  |-  (flds  ZZ )  =  (flds  ZZ )
3 eqid 2435 . . . 4  |-  (.g `  G
)  =  (.g `  G
)
41, 2, 3zlmval 16789 . . 3  |-  ( G  e.  V  ->  W  =  ( ( G sSet  <. (Scalar `  ndx ) ,  (flds  ZZ ) >. ) sSet  <. ( .s `  ndx ) ,  (.g `  G ) >.
) )
54fveq2d 5724 . 2  |-  ( G  e.  V  ->  (TopSet `  W )  =  (TopSet `  ( ( G sSet  <. (Scalar `  ndx ) ,  (flds  ZZ )
>. ) sSet  <. ( .s
`  ndx ) ,  (.g `  G ) >. )
) )
6 zlmtset.1 . . 3  |-  J  =  (TopSet `  G )
7 tsetid 13607 . . . 4  |- TopSet  = Slot  (TopSet ` 
ndx )
8 5re 10067 . . . . . 6  |-  5  e.  RR
9 5lt9 10165 . . . . . 6  |-  5  <  9
108, 9gtneii 9177 . . . . 5  |-  9  =/=  5
11 tsetndx 13606 . . . . . 6  |-  (TopSet `  ndx )  =  9
12 scandx 13581 . . . . . 6  |-  (Scalar `  ndx )  =  5
1311, 12neeq12i 2610 . . . . 5  |-  ( (TopSet `  ndx )  =/=  (Scalar ` 
ndx )  <->  9  =/=  5 )
1410, 13mpbir 201 . . . 4  |-  (TopSet `  ndx )  =/=  (Scalar ` 
ndx )
157, 14setsnid 13501 . . 3  |-  (TopSet `  G )  =  (TopSet `  ( G sSet  <. (Scalar ` 
ndx ) ,  (flds  ZZ )
>. ) )
16 6re 10068 . . . . . 6  |-  6  e.  RR
17 6lt9 10164 . . . . . 6  |-  6  <  9
1816, 17gtneii 9177 . . . . 5  |-  9  =/=  6
19 vscandx 13583 . . . . . 6  |-  ( .s
`  ndx )  =  6
2011, 19neeq12i 2610 . . . . 5  |-  ( (TopSet `  ndx )  =/=  ( .s `  ndx )  <->  9  =/=  6 )
2118, 20mpbir 201 . . . 4  |-  (TopSet `  ndx )  =/=  ( .s `  ndx )
227, 21setsnid 13501 . . 3  |-  (TopSet `  ( G sSet  <. (Scalar `  ndx ) ,  (flds  ZZ ) >. )
)  =  (TopSet `  ( ( G sSet  <. (Scalar `  ndx ) ,  (flds  ZZ )
>. ) sSet  <. ( .s
`  ndx ) ,  (.g `  G ) >. )
)
236, 15, 223eqtri 2459 . 2  |-  J  =  (TopSet `  ( ( G sSet  <. (Scalar `  ndx ) ,  (flds  ZZ ) >. ) sSet  <.
( .s `  ndx ) ,  (.g `  G
) >. ) )
245, 23syl6reqr 2486 1  |-  ( G  e.  V  ->  J  =  (TopSet `  W )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2598   <.cop 3809   ` cfv 5446  (class class class)co 6073   5c5 10044   6c6 10045   9c9 10048   ZZcz 10274   ndxcnx 13458   sSet csts 13459   ↾s cress 13462  Scalarcsca 13524   .scvsca 13525  TopSetcts 13527  .gcmg 14681  ℂfldccnfld 16695   ZModczlm 16771
This theorem is referenced by:  zhmnrg  24343
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-ndx 13464  df-slot 13465  df-sets 13467  df-sca 13537  df-vsca 13538  df-tset 13540  df-zlm 16775
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