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Theorem grplsmid 33412
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 19162 . . . . 5 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 480 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝐺 ∈ Grp)
3 eqid 2735 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
43subgss 19158 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺))
54sselda 3995 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝑋 ∈ (Base‘𝐺))
65snssd 4814 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → {𝑋} ⊆ (Base‘𝐺))
74adantr 480 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝐴 ⊆ (Base‘𝐺))
8 eqid 2735 . . . . 5 (+g𝐺) = (+g𝐺)
9 grplsmid.p . . . . 5 = (LSSum‘𝐺)
103, 8, 9lsmelvalx 19673 . . . 4 ((𝐺 ∈ Grp ∧ {𝑋} ⊆ (Base‘𝐺) ∧ 𝐴 ⊆ (Base‘𝐺)) → (𝑥 ∈ ({𝑋} 𝐴) ↔ ∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎)))
112, 6, 7, 10syl3anc 1370 . . 3 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (𝑥 ∈ ({𝑋} 𝐴) ↔ ∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎)))
12 oveq1 7438 . . . . . . 7 (𝑜 = 𝑋 → (𝑜(+g𝐺)𝑎) = (𝑋(+g𝐺)𝑎))
1312eqeq2d 2746 . . . . . 6 (𝑜 = 𝑋 → (𝑥 = (𝑜(+g𝐺)𝑎) ↔ 𝑥 = (𝑋(+g𝐺)𝑎)))
1413rexbidv 3177 . . . . 5 (𝑜 = 𝑋 → (∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎) ↔ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)))
1514rexsng 4681 . . . 4 (𝑋𝐴 → (∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎) ↔ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)))
1615adantl 481 . . 3 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎) ↔ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)))
17 simpr 484 . . . . . 6 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑎𝐴) ∧ 𝑥 = (𝑋(+g𝐺)𝑎)) → 𝑥 = (𝑋(+g𝐺)𝑎))
188subgcl 19167 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴𝑎𝐴) → (𝑋(+g𝐺)𝑎) ∈ 𝐴)
1918ad4ant123 1171 . . . . . 6 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑎𝐴) ∧ 𝑥 = (𝑋(+g𝐺)𝑎)) → (𝑋(+g𝐺)𝑎) ∈ 𝐴)
2017, 19eqeltrd 2839 . . . . 5 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑎𝐴) ∧ 𝑥 = (𝑋(+g𝐺)𝑎)) → 𝑥𝐴)
2120r19.29an 3156 . . . 4 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)) → 𝑥𝐴)
22 simpll 767 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝐴 ∈ (SubGrp‘𝐺))
23 eqid 2735 . . . . . . . 8 (invg𝐺) = (invg𝐺)
2423subginvcl 19166 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ((invg𝐺)‘𝑋) ∈ 𝐴)
2524adantr 480 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → ((invg𝐺)‘𝑋) ∈ 𝐴)
26 simpr 484 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑥𝐴)
278subgcl 19167 . . . . . 6 ((𝐴 ∈ (SubGrp‘𝐺) ∧ ((invg𝐺)‘𝑋) ∈ 𝐴𝑥𝐴) → (((invg𝐺)‘𝑋)(+g𝐺)𝑥) ∈ 𝐴)
2822, 25, 26, 27syl3anc 1370 . . . . 5 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → (((invg𝐺)‘𝑋)(+g𝐺)𝑥) ∈ 𝐴)
29 oveq2 7439 . . . . . . 7 (𝑎 = (((invg𝐺)‘𝑋)(+g𝐺)𝑥) → (𝑋(+g𝐺)𝑎) = (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)))
3029eqeq2d 2746 . . . . . 6 (𝑎 = (((invg𝐺)‘𝑋)(+g𝐺)𝑥) → (𝑥 = (𝑋(+g𝐺)𝑎) ↔ 𝑥 = (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥))))
3130adantl 481 . . . . 5 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) ∧ 𝑎 = (((invg𝐺)‘𝑋)(+g𝐺)𝑥)) → (𝑥 = (𝑋(+g𝐺)𝑎) ↔ 𝑥 = (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥))))
322adantr 480 . . . . . . 7 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝐺 ∈ Grp)
335adantr 480 . . . . . . 7 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑋 ∈ (Base‘𝐺))
347sselda 3995 . . . . . . 7 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑥 ∈ (Base‘𝐺))
353, 8, 23grpasscan1 19032 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)) = 𝑥)
3632, 33, 34, 35syl3anc 1370 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)) = 𝑥)
3736eqcomd 2741 . . . . 5 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑥 = (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)))
3828, 31, 37rspcedvd 3624 . . . 4 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎))
3921, 38impbida 801 . . 3 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎) ↔ 𝑥𝐴))
4011, 16, 393bitrd 305 . 2 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (𝑥 ∈ ({𝑋} 𝐴) ↔ 𝑥𝐴))
4140eqrdv 2733 1 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ({𝑋} 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  wss 3963  {csn 4631  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Grpcgrp 18964  invgcminusg 18965  SubGrpcsubg 19151  LSSumclsm 19667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967  df-minusg 18968  df-subg 19154  df-lsm 19669
This theorem is referenced by:  nsgmgc  33420  nsgqusf1olem2  33422  nsgqusf1olem3  33423
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