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Theorem subglsm 17909
 Description: The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
Hypotheses
Ref Expression
subglsm.h 𝐻 = (𝐺s 𝑆)
subglsm.s = (LSSum‘𝐺)
subglsm.a 𝐴 = (LSSum‘𝐻)
Assertion
Ref Expression
subglsm ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) = (𝑇𝐴𝑈))

Proof of Theorem subglsm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1084 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) ∧ 𝑥𝑇𝑦𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
2 subglsm.h . . . . . . 7 𝐻 = (𝐺s 𝑆)
3 eqid 2610 . . . . . . 7 (+g𝐺) = (+g𝐺)
42, 3ressplusg 15818 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
51, 4syl 17 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) ∧ 𝑥𝑇𝑦𝑈) → (+g𝐺) = (+g𝐻))
65oveqd 6566 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) ∧ 𝑥𝑇𝑦𝑈) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
76mpt2eq3dva 6617 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐺)𝑦)) = (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐻)𝑦)))
87rneqd 5274 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐺)𝑦)) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐻)𝑦)))
9 subgrcl 17422 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1093ad2ant1 1075 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝐺 ∈ Grp)
11 simp2 1055 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑇𝑆)
12 eqid 2610 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
1312subgss 17418 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
14133ad2ant1 1075 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑆 ⊆ (Base‘𝐺))
1511, 14sstrd 3578 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑇 ⊆ (Base‘𝐺))
16 simp3 1056 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑈𝑆)
1716, 14sstrd 3578 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑈 ⊆ (Base‘𝐺))
18 subglsm.s . . . 4 = (LSSum‘𝐺)
1912, 3, 18lsmvalx 17877 . . 3 ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐺)𝑦)))
2010, 15, 17, 19syl3anc 1318 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐺)𝑦)))
212subggrp 17420 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
22213ad2ant1 1075 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝐻 ∈ Grp)
232subgbas 17421 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
24233ad2ant1 1075 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑆 = (Base‘𝐻))
2511, 24sseqtrd 3604 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑇 ⊆ (Base‘𝐻))
2616, 24sseqtrd 3604 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑈 ⊆ (Base‘𝐻))
27 eqid 2610 . . . 4 (Base‘𝐻) = (Base‘𝐻)
28 eqid 2610 . . . 4 (+g𝐻) = (+g𝐻)
29 subglsm.a . . . 4 𝐴 = (LSSum‘𝐻)
3027, 28, 29lsmvalx 17877 . . 3 ((𝐻 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐻) ∧ 𝑈 ⊆ (Base‘𝐻)) → (𝑇𝐴𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐻)𝑦)))
3122, 25, 26, 30syl3anc 1318 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇𝐴𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐻)𝑦)))
328, 20, 313eqtr4d 2654 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) = (𝑇𝐴𝑈))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  Basecbs 15695   ↾s cress 15696  +gcplusg 15768  Grpcgrp 17245  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-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-subg 17414  df-lsm 17874 This theorem is referenced by:  pgpfaclem1  18303
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