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Theorem lsmmod2 14820
 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 965 . . . . . 6 SubGrp SubGrp SubGrp SubGrp
2 eqid 2253 . . . . . . 7 oppg oppg
32oppgsubg 14671 . . . . . 6 SubGrp SubGrpoppg
41, 3syl6eleq 2343 . . . . 5 SubGrp SubGrp SubGrp SubGrpoppg
5 simpl2 964 . . . . . 6 SubGrp SubGrp SubGrp SubGrp
65, 3syl6eleq 2343 . . . . 5 SubGrp SubGrp SubGrp SubGrpoppg
7 simpl1 963 . . . . . 6 SubGrp SubGrp SubGrp SubGrp
87, 3syl6eleq 2343 . . . . 5 SubGrp SubGrp SubGrp SubGrpoppg
9 simpr 449 . . . . 5 SubGrp SubGrp SubGrp
10 eqid 2253 . . . . . 6 oppg oppg
1110lsmmod 14819 . . . . 5 SubGrpoppg SubGrpoppg SubGrpoppg oppg oppg
124, 6, 8, 9, 11syl31anc 1190 . . . 4 SubGrp SubGrp SubGrp oppg oppg
1312eqcomd 2258 . . 3 SubGrp SubGrp SubGrp oppg oppg
14 incom 3269 . . 3 oppg oppg
15 incom 3269 . . . 4
1615oveq2i 5721 . . 3 oppg oppg
1713, 14, 163eqtr3g 2308 . 2 SubGrp SubGrp SubGrp oppg oppg
18 lsmmod.p . . . 4
192, 18oppglsm 14788 . . 3 oppg
2019ineq2i 3275 . 2 oppg
212, 18oppglsm 14788 . 2 oppg
2217, 20, 213eqtr3g 2308 1 SubGrp SubGrp SubGrp
 Colors of variables: wff set class Syntax hints:   wi 6   wa 360   w3a 939   wceq 1619   wcel 1621   cin 3077   wss 3078  cfv 4592  (class class class)co 5710  SubGrpcsubg 14450  oppgcoppg 14653  clsm 14780 This theorem is referenced by:  lcvexchlem3  27915  lcfrlem23  30444 This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-0g 13278  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-submnd 14251  df-grp 14324  df-minusg 14325  df-subg 14453  df-oppg 14654  df-lsm 14782
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