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Theorem grplsmid 31113
 Description: The direct sum of an element 𝑋 of a subgroup 𝐴 is the subgroup itself. (Contributed by Thierry Arnoux, 27-Jul-2024.)
Hypothesis
Ref Expression
grplsmid.p = (LSSum‘𝐺)
Assertion
Ref Expression
grplsmid ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ({𝑋} 𝐴) = 𝐴)

Proof of Theorem grplsmid
Dummy variables 𝑥 𝑎 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subgrcl 18351 . . . . 5 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 484 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝐺 ∈ Grp)
3 eqid 2758 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
43subgss 18347 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺))
54sselda 3892 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝑋 ∈ (Base‘𝐺))
65snssd 4699 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → {𝑋} ⊆ (Base‘𝐺))
74adantr 484 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝐴 ⊆ (Base‘𝐺))
8 eqid 2758 . . . . 5 (+g𝐺) = (+g𝐺)
9 grplsmid.p . . . . 5 = (LSSum‘𝐺)
103, 8, 9lsmelvalx 18832 . . . 4 ((𝐺 ∈ Grp ∧ {𝑋} ⊆ (Base‘𝐺) ∧ 𝐴 ⊆ (Base‘𝐺)) → (𝑥 ∈ ({𝑋} 𝐴) ↔ ∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎)))
112, 6, 7, 10syl3anc 1368 . . 3 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (𝑥 ∈ ({𝑋} 𝐴) ↔ ∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎)))
12 oveq1 7157 . . . . . . 7 (𝑜 = 𝑋 → (𝑜(+g𝐺)𝑎) = (𝑋(+g𝐺)𝑎))
1312eqeq2d 2769 . . . . . 6 (𝑜 = 𝑋 → (𝑥 = (𝑜(+g𝐺)𝑎) ↔ 𝑥 = (𝑋(+g𝐺)𝑎)))
1413rexbidv 3221 . . . . 5 (𝑜 = 𝑋 → (∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎) ↔ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)))
1514rexsng 4571 . . . 4 (𝑋𝐴 → (∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎) ↔ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)))
1615adantl 485 . . 3 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎) ↔ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)))
17 simpr 488 . . . . . 6 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑎𝐴) ∧ 𝑥 = (𝑋(+g𝐺)𝑎)) → 𝑥 = (𝑋(+g𝐺)𝑎))
188subgcl 18356 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴𝑎𝐴) → (𝑋(+g𝐺)𝑎) ∈ 𝐴)
1918ad4ant123 1169 . . . . . 6 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑎𝐴) ∧ 𝑥 = (𝑋(+g𝐺)𝑎)) → (𝑋(+g𝐺)𝑎) ∈ 𝐴)
2017, 19eqeltrd 2852 . . . . 5 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑎𝐴) ∧ 𝑥 = (𝑋(+g𝐺)𝑎)) → 𝑥𝐴)
2120r19.29an 3212 . . . 4 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)) → 𝑥𝐴)
22 simpll 766 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝐴 ∈ (SubGrp‘𝐺))
23 eqid 2758 . . . . . . . 8 (invg𝐺) = (invg𝐺)
2423subginvcl 18355 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ((invg𝐺)‘𝑋) ∈ 𝐴)
2524adantr 484 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → ((invg𝐺)‘𝑋) ∈ 𝐴)
26 simpr 488 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑥𝐴)
278subgcl 18356 . . . . . 6 ((𝐴 ∈ (SubGrp‘𝐺) ∧ ((invg𝐺)‘𝑋) ∈ 𝐴𝑥𝐴) → (((invg𝐺)‘𝑋)(+g𝐺)𝑥) ∈ 𝐴)
2822, 25, 26, 27syl3anc 1368 . . . . 5 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → (((invg𝐺)‘𝑋)(+g𝐺)𝑥) ∈ 𝐴)
29 oveq2 7158 . . . . . . 7 (𝑎 = (((invg𝐺)‘𝑋)(+g𝐺)𝑥) → (𝑋(+g𝐺)𝑎) = (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)))
3029eqeq2d 2769 . . . . . 6 (𝑎 = (((invg𝐺)‘𝑋)(+g𝐺)𝑥) → (𝑥 = (𝑋(+g𝐺)𝑎) ↔ 𝑥 = (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥))))
3130adantl 485 . . . . 5 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) ∧ 𝑎 = (((invg𝐺)‘𝑋)(+g𝐺)𝑥)) → (𝑥 = (𝑋(+g𝐺)𝑎) ↔ 𝑥 = (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥))))
322adantr 484 . . . . . . 7 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝐺 ∈ Grp)
335adantr 484 . . . . . . 7 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑋 ∈ (Base‘𝐺))
347sselda 3892 . . . . . . 7 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑥 ∈ (Base‘𝐺))
353, 8, 23grpasscan1 18229 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)) = 𝑥)
3632, 33, 34, 35syl3anc 1368 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)) = 𝑥)
3736eqcomd 2764 . . . . 5 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑥 = (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)))
3828, 31, 37rspcedvd 3544 . . . 4 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎))
3921, 38impbida 800 . . 3 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎) ↔ 𝑥𝐴))
4011, 16, 393bitrd 308 . 2 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (𝑥 ∈ ({𝑋} 𝐴) ↔ 𝑥𝐴))
4140eqrdv 2756 1 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ({𝑋} 𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ⊆ wss 3858  {csn 4522  ‘cfv 6335  (class class class)co 7150  Basecbs 16541  +gcplusg 16623  Grpcgrp 18169  invgcminusg 18170  SubGrpcsubg 18340  LSSumclsm 18826 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-1cn 10633  ax-icn 10634  ax-addcl 10635  ax-addrcl 10636  ax-mulcl 10637  ax-mulrcl 10638  ax-mulcom 10639  ax-addass 10640  ax-mulass 10641  ax-distr 10642  ax-i2m1 10643  ax-1ne0 10644  ax-1rid 10645  ax-rnegex 10646  ax-rrecex 10647  ax-cnre 10648  ax-pre-lttri 10649  ax-pre-lttrn 10650  ax-pre-ltadd 10651  ax-pre-mulgt0 10652 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-pss 3877  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-tp 4527  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-tr 5139  df-id 5430  df-eprel 5435  df-po 5443  df-so 5444  df-fr 5483  df-we 5485  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7580  df-1st 7693  df-2nd 7694  df-wrecs 7957  df-recs 8018  df-rdg 8056  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718  df-le 10719  df-sub 10910  df-neg 10911  df-nn 11675  df-2 11737  df-ndx 16544  df-slot 16545  df-base 16547  df-sets 16548  df-ress 16549  df-plusg 16636  df-0g 16773  df-mgm 17918  df-sgrp 17967  df-mnd 17978  df-grp 18172  df-minusg 18173  df-subg 18343  df-lsm 18828 This theorem is referenced by:  nsgmgc  31118  nsgqusf1olem2  31120  nsgqusf1olem3  31121
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