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Theorem subglsm 18026
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 1089 . . . . . 6 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) ∧ 𝑥𝑇𝑦𝑈) → 𝑆 ∈ (SubGrp‘𝐺))
2 subglsm.h . . . . . . 7 𝐻 = (𝐺s 𝑆)
3 eqid 2621 . . . . . . 7 (+g𝐺) = (+g𝐺)
42, 3ressplusg 15933 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → (+g𝐺) = (+g𝐻))
51, 4syl 17 . . . . 5 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) ∧ 𝑥𝑇𝑦𝑈) → (+g𝐺) = (+g𝐻))
65oveqd 6632 . . . 4 (((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) ∧ 𝑥𝑇𝑦𝑈) → (𝑥(+g𝐺)𝑦) = (𝑥(+g𝐻)𝑦))
76mpt2eq3dva 6684 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐺)𝑦)) = (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐻)𝑦)))
87rneqd 5323 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐺)𝑦)) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐻)𝑦)))
9 subgrcl 17539 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
1093ad2ant1 1080 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝐺 ∈ Grp)
11 simp2 1060 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑇𝑆)
12 eqid 2621 . . . . . 6 (Base‘𝐺) = (Base‘𝐺)
1312subgss 17535 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺))
14133ad2ant1 1080 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑆 ⊆ (Base‘𝐺))
1511, 14sstrd 3598 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑇 ⊆ (Base‘𝐺))
16 simp3 1061 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑈𝑆)
1716, 14sstrd 3598 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑈 ⊆ (Base‘𝐺))
18 subglsm.s . . . 4 = (LSSum‘𝐺)
1912, 3, 18lsmvalx 17994 . . 3 ((𝐺 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐺) ∧ 𝑈 ⊆ (Base‘𝐺)) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐺)𝑦)))
2010, 15, 17, 19syl3anc 1323 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐺)𝑦)))
212subggrp 17537 . . . 4 (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp)
22213ad2ant1 1080 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝐻 ∈ Grp)
232subgbas 17538 . . . . 5 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻))
24233ad2ant1 1080 . . . 4 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑆 = (Base‘𝐻))
2511, 24sseqtrd 3626 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑇 ⊆ (Base‘𝐻))
2616, 24sseqtrd 3626 . . 3 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → 𝑈 ⊆ (Base‘𝐻))
27 eqid 2621 . . . 4 (Base‘𝐻) = (Base‘𝐻)
28 eqid 2621 . . . 4 (+g𝐻) = (+g𝐻)
29 subglsm.a . . . 4 𝐴 = (LSSum‘𝐻)
3027, 28, 29lsmvalx 17994 . . 3 ((𝐻 ∈ Grp ∧ 𝑇 ⊆ (Base‘𝐻) ∧ 𝑈 ⊆ (Base‘𝐻)) → (𝑇𝐴𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐻)𝑦)))
3122, 25, 26, 30syl3anc 1323 . 2 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇𝐴𝑈) = ran (𝑥𝑇, 𝑦𝑈 ↦ (𝑥(+g𝐻)𝑦)))
328, 20, 313eqtr4d 2665 1 ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑇𝑆𝑈𝑆) → (𝑇 𝑈) = (𝑇𝐴𝑈))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  wss 3560  ran crn 5085  cfv 5857  (class class class)co 6615  cmpt2 6617  Basecbs 15800  s cress 15801  +gcplusg 15881  Grpcgrp 17362  SubGrpcsubg 17528  LSSumclsm 17989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-subg 17531  df-lsm 17991
This theorem is referenced by:  pgpfaclem1  18420
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