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Theorem quslsm 31118
 Description: Express the image by the quotient map in terms of direct sum. (Contributed by Thierry Arnoux, 27-Jul-2024.)
Hypotheses
Ref Expression
quslsm.b 𝐵 = (Base‘𝐺)
quslsm.p = (LSSum‘𝐺)
quslsm.n (𝜑𝑆 ∈ (SubGrp‘𝐺))
quslsm.s (𝜑𝑋𝐵)
Assertion
Ref Expression
quslsm (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} 𝑆))

Proof of Theorem quslsm
Dummy variables 𝑖 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 quslsm.n . . . . . 6 (𝜑𝑆 ∈ (SubGrp‘𝐺))
2 subgrcl 18356 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (𝜑𝐺 ∈ Grp)
4 quslsm.b . . . . . . 7 𝐵 = (Base‘𝐺)
54subgss 18352 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝐵)
61, 5syl 17 . . . . 5 (𝜑𝑆𝐵)
7 eqid 2758 . . . . . 6 (invg𝐺) = (invg𝐺)
8 eqid 2758 . . . . . 6 (+g𝐺) = (+g𝐺)
9 eqid 2758 . . . . . 6 (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
104, 7, 8, 9eqgfval 18400 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑆𝐵) → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)})
113, 6, 10syl2anc 587 . . . 4 (𝜑 → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)})
12 simpr 488 . . . . . . . 8 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
13 oveq2 7163 . . . . . . . . . 10 (𝑘 = (((invg𝐺)‘𝑖)(+g𝐺)𝑗) → (𝑖(+g𝐺)𝑘) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
1413eqeq1d 2760 . . . . . . . . 9 (𝑘 = (((invg𝐺)‘𝑖)(+g𝐺)𝑗) → ((𝑖(+g𝐺)𝑘) = 𝑗 ↔ (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)) = 𝑗))
1514adantl 485 . . . . . . . 8 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) ∧ 𝑘 = (((invg𝐺)‘𝑖)(+g𝐺)𝑗)) → ((𝑖(+g𝐺)𝑘) = 𝑗 ↔ (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)) = 𝑗))
163adantr 484 . . . . . . . . . . . 12 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝐺 ∈ Grp)
17 vex 3413 . . . . . . . . . . . . . . . 16 𝑖 ∈ V
18 vex 3413 . . . . . . . . . . . . . . . 16 𝑗 ∈ V
1917, 18prss 4713 . . . . . . . . . . . . . . 15 ((𝑖𝐵𝑗𝐵) ↔ {𝑖, 𝑗} ⊆ 𝐵)
2019bicomi 227 . . . . . . . . . . . . . 14 ({𝑖, 𝑗} ⊆ 𝐵 ↔ (𝑖𝐵𝑗𝐵))
2120simplbi 501 . . . . . . . . . . . . 13 ({𝑖, 𝑗} ⊆ 𝐵𝑖𝐵)
2221adantl 485 . . . . . . . . . . . 12 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑖𝐵)
23 eqid 2758 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
244, 8, 23, 7grprinv 18225 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → (𝑖(+g𝐺)((invg𝐺)‘𝑖)) = (0g𝐺))
2516, 22, 24syl2anc 587 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → (𝑖(+g𝐺)((invg𝐺)‘𝑖)) = (0g𝐺))
2625oveq1d 7170 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = ((0g𝐺)(+g𝐺)𝑗))
274, 7grpinvcl 18223 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
2816, 22, 27syl2anc 587 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
2920simprbi 500 . . . . . . . . . . . 12 ({𝑖, 𝑗} ⊆ 𝐵𝑗𝐵)
3029adantl 485 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑗𝐵)
314, 8grpass 18183 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑖𝐵 ∧ ((invg𝐺)‘𝑖) ∈ 𝐵𝑗𝐵)) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
3216, 22, 28, 30, 31syl13anc 1369 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
334, 8, 23grplid 18205 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑗𝐵) → ((0g𝐺)(+g𝐺)𝑗) = 𝑗)
3416, 30, 33syl2anc 587 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((0g𝐺)(+g𝐺)𝑗) = 𝑗)
3526, 32, 343eqtr3d 2801 . . . . . . . . 9 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)) = 𝑗)
3635adantr 484 . . . . . . . 8 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)) = 𝑗)
3712, 15, 36rspcedvd 3546 . . . . . . 7 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)
38 oveq2 7163 . . . . . . . . . . 11 ((𝑖(+g𝐺)𝑘) = 𝑗 → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = (((invg𝐺)‘𝑖)(+g𝐺)𝑗))
3938adantl 485 . . . . . . . . . 10 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = (((invg𝐺)‘𝑖)(+g𝐺)𝑗))
40 simpll 766 . . . . . . . . . . . 12 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → 𝜑)
4122adantr 484 . . . . . . . . . . . 12 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → 𝑖𝐵)
426adantr 484 . . . . . . . . . . . . 13 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑆𝐵)
4342sselda 3894 . . . . . . . . . . . 12 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → 𝑘𝐵)
4433ad2ant1 1130 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐵𝑘𝐵) → 𝐺 ∈ Grp)
45 simp2 1134 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐵𝑘𝐵) → 𝑖𝐵)
464, 8, 23, 7grplinv 18224 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)𝑖) = (0g𝐺))
4744, 45, 46syl2anc 587 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)𝑖) = (0g𝐺))
4847oveq1d 7170 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = ((0g𝐺)(+g𝐺)𝑘))
4944, 45, 27syl2anc 587 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
50 simp3 1135 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → 𝑘𝐵)
514, 8grpass 18183 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑖) ∈ 𝐵𝑖𝐵𝑘𝐵)) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)))
5244, 49, 45, 50, 51syl13anc 1369 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)))
534, 8, 23grplid 18205 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑘𝐵) → ((0g𝐺)(+g𝐺)𝑘) = 𝑘)
5444, 50, 53syl2anc 587 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((0g𝐺)(+g𝐺)𝑘) = 𝑘)
5548, 52, 543eqtr3d 2801 . . . . . . . . . . . 12 ((𝜑𝑖𝐵𝑘𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = 𝑘)
5640, 41, 43, 55syl3anc 1368 . . . . . . . . . . 11 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = 𝑘)
5756adantr 484 . . . . . . . . . 10 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = 𝑘)
5839, 57eqtr3d 2795 . . . . . . . . 9 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) = 𝑘)
59 simplr 768 . . . . . . . . 9 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → 𝑘𝑆)
6058, 59eqeltrd 2852 . . . . . . . 8 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
6160r19.29an 3212 . . . . . . 7 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
6237, 61impbida 800 . . . . . 6 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆 ↔ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗))
6362pm5.32da 582 . . . . 5 (𝜑 → (({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) ↔ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)))
6463opabbidv 5101 . . . 4 (𝜑 → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
6511, 64eqtrd 2793 . . 3 (𝜑 → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
6665eceq2d 8346 . 2 (𝜑 → [𝑋](𝐺 ~QG 𝑆) = [𝑋]{⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
67 quslsm.p . . 3 = (LSSum‘𝐺)
68 eqid 2758 . . 3 {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)}
693grpmndd 18186 . . 3 (𝜑𝐺 ∈ Mnd)
70 quslsm.s . . 3 (𝜑𝑋𝐵)
714, 8, 67, 68, 69, 6, 70lsmsnorb2 31105 . 2 (𝜑 → ({𝑋} 𝑆) = [𝑋]{⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
7266, 71eqtr4d 2796 1 (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3071   ⊆ wss 3860  {csn 4525  {cpr 4527  {copab 5097  ‘cfv 6339  (class class class)co 7155  [cec 8302  Basecbs 16546  +gcplusg 16628  0gc0g 16776  Grpcgrp 18174  invgcminusg 18175  SubGrpcsubg 18345   ~QG cqg 18347  LSSumclsm 18831 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 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 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 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-1st 7698  df-2nd 7699  df-tpos 7907  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-er 8304  df-ec 8306  df-en 8533  df-dom 8534  df-sdom 8535  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-ndx 16549  df-slot 16550  df-base 16552  df-sets 16553  df-plusg 16641  df-0g 16778  df-mgm 17923  df-sgrp 17972  df-mnd 17983  df-grp 18177  df-minusg 18178  df-subg 18348  df-eqg 18350  df-oppg 18546  df-lsm 18833 This theorem is referenced by:  qusima  31119  nsgqus0  31120  nsgmgclem  31121  nsgqusf1olem1  31123  nsgqusf1olem2  31124  nsgqusf1olem3  31125
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