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Theorem lsmcom2 15281
 Description: Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
Hypotheses
Ref Expression
lsmsubg.p
lsmsubg.z Cntz
Assertion
Ref Expression
lsmcom2 SubGrp SubGrp

Proof of Theorem lsmcom2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 959 . . . . . . . . 9 SubGrp SubGrp
21sselda 3340 . . . . . . . 8 SubGrp SubGrp
32adantrr 698 . . . . . . 7 SubGrp SubGrp
4 simprr 734 . . . . . . 7 SubGrp SubGrp
5 eqid 2435 . . . . . . . 8
6 lsmsubg.z . . . . . . . 8 Cntz
75, 6cntzi 15120 . . . . . . 7
83, 4, 7syl2anc 643 . . . . . 6 SubGrp SubGrp
98eqeq2d 2446 . . . . 5 SubGrp SubGrp
1092rexbidva 2738 . . . 4 SubGrp SubGrp
11 rexcom 2861 . . . 4
1210, 11syl6bb 253 . . 3 SubGrp SubGrp
13 lsmsubg.p . . . . 5
145, 13lsmelval 15275 . . . 4 SubGrp SubGrp
15143adant3 977 . . 3 SubGrp SubGrp
165, 13lsmelval 15275 . . . . 5 SubGrp SubGrp
1716ancoms 440 . . . 4 SubGrp SubGrp
18173adant3 977 . . 3 SubGrp SubGrp
1912, 15, 183bitr4d 277 . 2 SubGrp SubGrp
2019eqrdv 2433 1 SubGrp SubGrp
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wrex 2698   wss 3312  cfv 5446  (class class class)co 6073   cplusg 13521  SubGrpcsubg 14930  Cntzccntz 15106  clsm 15260 This theorem is referenced by:  lsmdisj3  15307  lsmdisj3r  15310  lsmdisj3a  15313  lsmdisj3b  15314  pj2f  15322  pj1id  15323 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-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  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-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-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  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-1st 6341  df-2nd 6342  df-subg 14933  df-cntz 15108  df-lsm 15262
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