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Theorem grplsmid 33432
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 19149 . . . . 5 (𝐴 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
21adantr 480 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝐺 ∈ Grp)
3 eqid 2737 . . . . . . 7 (Base‘𝐺) = (Base‘𝐺)
43subgss 19145 . . . . . 6 (𝐴 ∈ (SubGrp‘𝐺) → 𝐴 ⊆ (Base‘𝐺))
54sselda 3983 . . . . 5 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝑋 ∈ (Base‘𝐺))
65snssd 4809 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → {𝑋} ⊆ (Base‘𝐺))
74adantr 480 . . . 4 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → 𝐴 ⊆ (Base‘𝐺))
8 eqid 2737 . . . . 5 (+g𝐺) = (+g𝐺)
9 grplsmid.p . . . . 5 = (LSSum‘𝐺)
103, 8, 9lsmelvalx 19658 . . . 4 ((𝐺 ∈ Grp ∧ {𝑋} ⊆ (Base‘𝐺) ∧ 𝐴 ⊆ (Base‘𝐺)) → (𝑥 ∈ ({𝑋} 𝐴) ↔ ∃𝑜 ∈ {𝑋}∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎)))
112, 6, 7, 10syl3anc 1373 . . 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 2748 . . . . . 6 (𝑜 = 𝑋 → (𝑥 = (𝑜(+g𝐺)𝑎) ↔ 𝑥 = (𝑋(+g𝐺)𝑎)))
1413rexbidv 3179 . . . . 5 (𝑜 = 𝑋 → (∃𝑎𝐴 𝑥 = (𝑜(+g𝐺)𝑎) ↔ ∃𝑎𝐴 𝑥 = (𝑋(+g𝐺)𝑎)))
1514rexsng 4676 . . . 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 19154 . . . . . . 7 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴𝑎𝐴) → (𝑋(+g𝐺)𝑎) ∈ 𝐴)
1918ad4ant123 1173 . . . . . 6 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑎𝐴) ∧ 𝑥 = (𝑋(+g𝐺)𝑎)) → (𝑋(+g𝐺)𝑎) ∈ 𝐴)
2017, 19eqeltrd 2841 . . . . 5 ((((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑎𝐴) ∧ 𝑥 = (𝑋(+g𝐺)𝑎)) → 𝑥𝐴)
2120r19.29an 3158 . . . 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 2737 . . . . . . . 8 (invg𝐺) = (invg𝐺)
2423subginvcl 19153 . . . . . . 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 19154 . . . . . 6 ((𝐴 ∈ (SubGrp‘𝐺) ∧ ((invg𝐺)‘𝑋) ∈ 𝐴𝑥𝐴) → (((invg𝐺)‘𝑋)(+g𝐺)𝑥) ∈ 𝐴)
2822, 25, 26, 27syl3anc 1373 . . . . 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 2748 . . . . . 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 3983 . . . . . . 7 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → 𝑥 ∈ (Base‘𝐺))
353, 8, 23grpasscan1 19019 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑋 ∈ (Base‘𝐺) ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)) = 𝑥)
3632, 33, 34, 35syl3anc 1373 . . . . . 6 (((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) ∧ 𝑥𝐴) → (𝑋(+g𝐺)(((invg𝐺)‘𝑋)(+g𝐺)𝑥)) = 𝑥)
3736eqcomd 2743 . . . . 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 2735 1 ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → ({𝑋} 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  wss 3951  {csn 4626  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Grpcgrp 18951  invgcminusg 18952  SubGrpcsubg 19138  LSSumclsm 19652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-subg 19141  df-lsm 19654
This theorem is referenced by:  nsgmgc  33440  nsgqusf1olem2  33442  nsgqusf1olem3  33443
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