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Theorem lsmmod2 15001
 Description: Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
Hypothesis
Ref Expression
lsmmod.p
Assertion
Ref Expression
lsmmod2 SubGrp SubGrp SubGrp

Proof of Theorem lsmmod2
StepHypRef Expression
1 simpl3 960 . . . . . 6 SubGrp SubGrp SubGrp SubGrp
2 eqid 2296 . . . . . . 7 oppg oppg
32oppgsubg 14852 . . . . . 6 SubGrp SubGrpoppg
41, 3syl6eleq 2386 . . . . 5 SubGrp SubGrp SubGrp SubGrpoppg
5 simpl2 959 . . . . . 6 SubGrp SubGrp SubGrp SubGrp
65, 3syl6eleq 2386 . . . . 5 SubGrp SubGrp SubGrp SubGrpoppg
7 simpl1 958 . . . . . 6 SubGrp SubGrp SubGrp SubGrp
87, 3syl6eleq 2386 . . . . 5 SubGrp SubGrp SubGrp SubGrpoppg
9 simpr 447 . . . . 5 SubGrp SubGrp SubGrp
10 eqid 2296 . . . . . 6 oppg oppg
1110lsmmod 15000 . . . . 5 SubGrpoppg SubGrpoppg SubGrpoppg oppg oppg
124, 6, 8, 9, 11syl31anc 1185 . . . 4 SubGrp SubGrp SubGrp oppg oppg
1312eqcomd 2301 . . 3 SubGrp SubGrp SubGrp oppg oppg
14 incom 3374 . . 3 oppg oppg
15 incom 3374 . . . 4
1615oveq2i 5885 . . 3 oppg oppg
1713, 14, 163eqtr3g 2351 . 2 SubGrp SubGrp SubGrp oppg oppg
18 lsmmod.p . . . 4
192, 18oppglsm 14969 . . 3 oppg
2019ineq2i 3380 . 2 oppg
212, 18oppglsm 14969 . 2 oppg
2217, 20, 213eqtr3g 2351 1 SubGrp SubGrp SubGrp
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   w3a 934   wceq 1632   wcel 1696   cin 3164   wss 3165  cfv 5271  (class class class)co 5874  SubGrpcsubg 14631  oppgcoppg 14834  clsm 14961 This theorem is referenced by:  lcvexchlem3  29848  lcfrlem23  32377 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830 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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-tpos 6250  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-ndx 13167  df-slot 13168  df-base 13169  df-sets 13170  df-ress 13171  df-plusg 13237  df-0g 13420  df-mre 13504  df-mrc 13505  df-acs 13507  df-mnd 14383  df-submnd 14432  df-grp 14505  df-minusg 14506  df-subg 14634  df-oppg 14835  df-lsm 14963
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