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Theorem subgdisj1 17927
 Description: Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
subgdisj.p + = (+g𝐺)
subgdisj.o 0 = (0g𝐺)
subgdisj.z 𝑍 = (Cntz‘𝐺)
subgdisj.t (𝜑𝑇 ∈ (SubGrp‘𝐺))
subgdisj.u (𝜑𝑈 ∈ (SubGrp‘𝐺))
subgdisj.i (𝜑 → (𝑇𝑈) = { 0 })
subgdisj.s (𝜑𝑇 ⊆ (𝑍𝑈))
subgdisj.a (𝜑𝐴𝑇)
subgdisj.c (𝜑𝐶𝑇)
subgdisj.b (𝜑𝐵𝑈)
subgdisj.d (𝜑𝐷𝑈)
subgdisj.j (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))
Assertion
Ref Expression
subgdisj1 (𝜑𝐴 = 𝐶)

Proof of Theorem subgdisj1
StepHypRef Expression
1 subgdisj.t . . . . . 6 (𝜑𝑇 ∈ (SubGrp‘𝐺))
2 subgdisj.a . . . . . 6 (𝜑𝐴𝑇)
3 subgdisj.c . . . . . 6 (𝜑𝐶𝑇)
4 eqid 2610 . . . . . . 7 (-g𝐺) = (-g𝐺)
54subgsubcl 17428 . . . . . 6 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝐴𝑇𝐶𝑇) → (𝐴(-g𝐺)𝐶) ∈ 𝑇)
61, 2, 3, 5syl3anc 1318 . . . . 5 (𝜑 → (𝐴(-g𝐺)𝐶) ∈ 𝑇)
7 subgdisj.j . . . . . . . . 9 (𝜑 → (𝐴 + 𝐵) = (𝐶 + 𝐷))
8 subgdisj.s . . . . . . . . . . 11 (𝜑𝑇 ⊆ (𝑍𝑈))
98, 3sseldd 3569 . . . . . . . . . 10 (𝜑𝐶 ∈ (𝑍𝑈))
10 subgdisj.b . . . . . . . . . 10 (𝜑𝐵𝑈)
11 subgdisj.p . . . . . . . . . . 11 + = (+g𝐺)
12 subgdisj.z . . . . . . . . . . 11 𝑍 = (Cntz‘𝐺)
1311, 12cntzi 17585 . . . . . . . . . 10 ((𝐶 ∈ (𝑍𝑈) ∧ 𝐵𝑈) → (𝐶 + 𝐵) = (𝐵 + 𝐶))
149, 10, 13syl2anc 691 . . . . . . . . 9 (𝜑 → (𝐶 + 𝐵) = (𝐵 + 𝐶))
157, 14oveq12d 6567 . . . . . . . 8 (𝜑 → ((𝐴 + 𝐵)(-g𝐺)(𝐶 + 𝐵)) = ((𝐶 + 𝐷)(-g𝐺)(𝐵 + 𝐶)))
16 subgrcl 17422 . . . . . . . . . 10 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
171, 16syl 17 . . . . . . . . 9 (𝜑𝐺 ∈ Grp)
18 eqid 2610 . . . . . . . . . . . . 13 (Base‘𝐺) = (Base‘𝐺)
1918subgss 17418 . . . . . . . . . . . 12 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
201, 19syl 17 . . . . . . . . . . 11 (𝜑𝑇 ⊆ (Base‘𝐺))
2120, 2sseldd 3569 . . . . . . . . . 10 (𝜑𝐴 ∈ (Base‘𝐺))
22 subgdisj.u . . . . . . . . . . . 12 (𝜑𝑈 ∈ (SubGrp‘𝐺))
2318subgss 17418 . . . . . . . . . . . 12 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
2422, 23syl 17 . . . . . . . . . . 11 (𝜑𝑈 ⊆ (Base‘𝐺))
2524, 10sseldd 3569 . . . . . . . . . 10 (𝜑𝐵 ∈ (Base‘𝐺))
2618, 11grpcl 17253 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → (𝐴 + 𝐵) ∈ (Base‘𝐺))
2717, 21, 25, 26syl3anc 1318 . . . . . . . . 9 (𝜑 → (𝐴 + 𝐵) ∈ (Base‘𝐺))
2820, 3sseldd 3569 . . . . . . . . 9 (𝜑𝐶 ∈ (Base‘𝐺))
2918, 11, 4grpsubsub4 17331 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ ((𝐴 + 𝐵) ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺) ∧ 𝐶 ∈ (Base‘𝐺))) → (((𝐴 + 𝐵)(-g𝐺)𝐵)(-g𝐺)𝐶) = ((𝐴 + 𝐵)(-g𝐺)(𝐶 + 𝐵)))
3017, 27, 25, 28, 29syl13anc 1320 . . . . . . . 8 (𝜑 → (((𝐴 + 𝐵)(-g𝐺)𝐵)(-g𝐺)𝐶) = ((𝐴 + 𝐵)(-g𝐺)(𝐶 + 𝐵)))
317, 27eqeltrrd 2689 . . . . . . . . 9 (𝜑 → (𝐶 + 𝐷) ∈ (Base‘𝐺))
3218, 11, 4grpsubsub4 17331 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ ((𝐶 + 𝐷) ∈ (Base‘𝐺) ∧ 𝐶 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺))) → (((𝐶 + 𝐷)(-g𝐺)𝐶)(-g𝐺)𝐵) = ((𝐶 + 𝐷)(-g𝐺)(𝐵 + 𝐶)))
3317, 31, 28, 25, 32syl13anc 1320 . . . . . . . 8 (𝜑 → (((𝐶 + 𝐷)(-g𝐺)𝐶)(-g𝐺)𝐵) = ((𝐶 + 𝐷)(-g𝐺)(𝐵 + 𝐶)))
3415, 30, 333eqtr4d 2654 . . . . . . 7 (𝜑 → (((𝐴 + 𝐵)(-g𝐺)𝐵)(-g𝐺)𝐶) = (((𝐶 + 𝐷)(-g𝐺)𝐶)(-g𝐺)𝐵))
3518, 11, 4grppncan 17329 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝐴 ∈ (Base‘𝐺) ∧ 𝐵 ∈ (Base‘𝐺)) → ((𝐴 + 𝐵)(-g𝐺)𝐵) = 𝐴)
3617, 21, 25, 35syl3anc 1318 . . . . . . . 8 (𝜑 → ((𝐴 + 𝐵)(-g𝐺)𝐵) = 𝐴)
3736oveq1d 6564 . . . . . . 7 (𝜑 → (((𝐴 + 𝐵)(-g𝐺)𝐵)(-g𝐺)𝐶) = (𝐴(-g𝐺)𝐶))
38 subgdisj.d . . . . . . . . . . 11 (𝜑𝐷𝑈)
3911, 12cntzi 17585 . . . . . . . . . . 11 ((𝐶 ∈ (𝑍𝑈) ∧ 𝐷𝑈) → (𝐶 + 𝐷) = (𝐷 + 𝐶))
409, 38, 39syl2anc 691 . . . . . . . . . 10 (𝜑 → (𝐶 + 𝐷) = (𝐷 + 𝐶))
4140oveq1d 6564 . . . . . . . . 9 (𝜑 → ((𝐶 + 𝐷)(-g𝐺)𝐶) = ((𝐷 + 𝐶)(-g𝐺)𝐶))
4224, 38sseldd 3569 . . . . . . . . . 10 (𝜑𝐷 ∈ (Base‘𝐺))
4318, 11, 4grppncan 17329 . . . . . . . . . 10 ((𝐺 ∈ Grp ∧ 𝐷 ∈ (Base‘𝐺) ∧ 𝐶 ∈ (Base‘𝐺)) → ((𝐷 + 𝐶)(-g𝐺)𝐶) = 𝐷)
4417, 42, 28, 43syl3anc 1318 . . . . . . . . 9 (𝜑 → ((𝐷 + 𝐶)(-g𝐺)𝐶) = 𝐷)
4541, 44eqtrd 2644 . . . . . . . 8 (𝜑 → ((𝐶 + 𝐷)(-g𝐺)𝐶) = 𝐷)
4645oveq1d 6564 . . . . . . 7 (𝜑 → (((𝐶 + 𝐷)(-g𝐺)𝐶)(-g𝐺)𝐵) = (𝐷(-g𝐺)𝐵))
4734, 37, 463eqtr3d 2652 . . . . . 6 (𝜑 → (𝐴(-g𝐺)𝐶) = (𝐷(-g𝐺)𝐵))
484subgsubcl 17428 . . . . . . 7 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝐷𝑈𝐵𝑈) → (𝐷(-g𝐺)𝐵) ∈ 𝑈)
4922, 38, 10, 48syl3anc 1318 . . . . . 6 (𝜑 → (𝐷(-g𝐺)𝐵) ∈ 𝑈)
5047, 49eqeltrd 2688 . . . . 5 (𝜑 → (𝐴(-g𝐺)𝐶) ∈ 𝑈)
516, 50elind 3760 . . . 4 (𝜑 → (𝐴(-g𝐺)𝐶) ∈ (𝑇𝑈))
52 subgdisj.i . . . 4 (𝜑 → (𝑇𝑈) = { 0 })
5351, 52eleqtrd 2690 . . 3 (𝜑 → (𝐴(-g𝐺)𝐶) ∈ { 0 })
54 elsni 4142 . . 3 ((𝐴(-g𝐺)𝐶) ∈ { 0 } → (𝐴(-g𝐺)𝐶) = 0 )
5553, 54syl 17 . 2 (𝜑 → (𝐴(-g𝐺)𝐶) = 0 )
56 subgdisj.o . . . 4 0 = (0g𝐺)
5718, 56, 4grpsubeq0 17324 . . 3 ((𝐺 ∈ Grp ∧ 𝐴 ∈ (Base‘𝐺) ∧ 𝐶 ∈ (Base‘𝐺)) → ((𝐴(-g𝐺)𝐶) = 0𝐴 = 𝐶))
5817, 21, 28, 57syl3anc 1318 . 2 (𝜑 → ((𝐴(-g𝐺)𝐶) = 0𝐴 = 𝐶))
5955, 58mpbid 221 1 (𝜑𝐴 = 𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  {csn 4125  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923  Grpcgrp 17245  -gcsg 17247  SubGrpcsubg 17411  Cntzccntz 17571 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-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-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-cntz 17573 This theorem is referenced by:  subgdisj2  17928  subgdisjb  17929  lvecindp  18959
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