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Theorem lsmdisj2b 17924
Description: Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
Hypotheses
Ref Expression
lsmcntz.p = (LSSum‘𝐺)
lsmcntz.s (𝜑𝑆 ∈ (SubGrp‘𝐺))
lsmcntz.t (𝜑𝑇 ∈ (SubGrp‘𝐺))
lsmcntz.u (𝜑𝑈 ∈ (SubGrp‘𝐺))
lsmdisj.o 0 = (0g𝐺)
Assertion
Ref Expression
lsmdisj2b (𝜑 → ((((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 }) ↔ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })))

Proof of Theorem lsmdisj2b
StepHypRef Expression
1 incom 3767 . . . 4 (𝑆 ∩ (𝑇 𝑈)) = ((𝑇 𝑈) ∩ 𝑆)
2 lsmcntz.p . . . . 5 = (LSSum‘𝐺)
3 lsmcntz.t . . . . . 6 (𝜑𝑇 ∈ (SubGrp‘𝐺))
43adantr 480 . . . . 5 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → 𝑇 ∈ (SubGrp‘𝐺))
5 lsmcntz.s . . . . . 6 (𝜑𝑆 ∈ (SubGrp‘𝐺))
65adantr 480 . . . . 5 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → 𝑆 ∈ (SubGrp‘𝐺))
7 lsmcntz.u . . . . . 6 (𝜑𝑈 ∈ (SubGrp‘𝐺))
87adantr 480 . . . . 5 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → 𝑈 ∈ (SubGrp‘𝐺))
9 lsmdisj.o . . . . 5 0 = (0g𝐺)
10 incom 3767 . . . . . 6 (𝑇 ∩ (𝑆 𝑈)) = ((𝑆 𝑈) ∩ 𝑇)
11 simprl 790 . . . . . 6 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → ((𝑆 𝑈) ∩ 𝑇) = { 0 })
1210, 11syl5eq 2656 . . . . 5 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → (𝑇 ∩ (𝑆 𝑈)) = { 0 })
13 simprr 792 . . . . 5 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → (𝑆𝑈) = { 0 })
142, 4, 6, 8, 9, 12, 13lsmdisj2r 17921 . . . 4 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → ((𝑇 𝑈) ∩ 𝑆) = { 0 })
151, 14syl5eq 2656 . . 3 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → (𝑆 ∩ (𝑇 𝑈)) = { 0 })
16 incom 3767 . . . 4 (𝑇𝑈) = (𝑈𝑇)
172, 6, 8, 4, 9, 11lsmdisj 17917 . . . . 5 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → ((𝑆𝑇) = { 0 } ∧ (𝑈𝑇) = { 0 }))
1817simprd 478 . . . 4 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → (𝑈𝑇) = { 0 })
1916, 18syl5eq 2656 . . 3 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → (𝑇𝑈) = { 0 })
2015, 19jca 553 . 2 ((𝜑 ∧ (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 })) → ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 }))
215adantr 480 . . . 4 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → 𝑆 ∈ (SubGrp‘𝐺))
223adantr 480 . . . 4 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → 𝑇 ∈ (SubGrp‘𝐺))
237adantr 480 . . . 4 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → 𝑈 ∈ (SubGrp‘𝐺))
24 simprl 790 . . . 4 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → (𝑆 ∩ (𝑇 𝑈)) = { 0 })
25 simprr 792 . . . 4 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → (𝑇𝑈) = { 0 })
262, 21, 22, 23, 9, 24, 25lsmdisj2r 17921 . . 3 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → ((𝑆 𝑈) ∩ 𝑇) = { 0 })
272, 21, 22, 23, 9, 24lsmdisjr 17920 . . . 4 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → ((𝑆𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 }))
2827simprd 478 . . 3 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → (𝑆𝑈) = { 0 })
2926, 28jca 553 . 2 ((𝜑 ∧ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })) → (((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 }))
3020, 29impbida 873 1 (𝜑 → ((((𝑆 𝑈) ∩ 𝑇) = { 0 } ∧ (𝑆𝑈) = { 0 }) ↔ ((𝑆 ∩ (𝑇 𝑈)) = { 0 } ∧ (𝑇𝑈) = { 0 })))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  cin 3539  {csn 4125  cfv 5804  (class class class)co 6549  0gc0g 15923  SubGrpcsubg 17411  LSSumclsm 17872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-oppg 17599  df-lsm 17874
This theorem is referenced by:  lsmdisj3b  17926
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