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Theorem quslsm 32516
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 19011 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (𝜑𝐺 ∈ Grp)
4 quslsm.b . . . . . . 7 𝐵 = (Base‘𝐺)
54subgss 19007 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝐵)
61, 5syl 17 . . . . 5 (𝜑𝑆𝐵)
7 eqid 2733 . . . . . 6 (invg𝐺) = (invg𝐺)
8 eqid 2733 . . . . . 6 (+g𝐺) = (+g𝐺)
9 eqid 2733 . . . . . 6 (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
104, 7, 8, 9eqgfval 19056 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑆𝐵) → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)})
113, 6, 10syl2anc 585 . . . 4 (𝜑 → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)})
12 simpr 486 . . . . . . . 8 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
13 oveq2 7417 . . . . . . . . . 10 (𝑘 = (((invg𝐺)‘𝑖)(+g𝐺)𝑗) → (𝑖(+g𝐺)𝑘) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
1413eqeq1d 2735 . . . . . . . . 9 (𝑘 = (((invg𝐺)‘𝑖)(+g𝐺)𝑗) → ((𝑖(+g𝐺)𝑘) = 𝑗 ↔ (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)) = 𝑗))
1514adantl 483 . . . . . . . 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 482 . . . . . . . . . . . 12 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝐺 ∈ Grp)
17 vex 3479 . . . . . . . . . . . . . . . 16 𝑖 ∈ V
18 vex 3479 . . . . . . . . . . . . . . . 16 𝑗 ∈ V
1917, 18prss 4824 . . . . . . . . . . . . . . 15 ((𝑖𝐵𝑗𝐵) ↔ {𝑖, 𝑗} ⊆ 𝐵)
2019bicomi 223 . . . . . . . . . . . . . 14 ({𝑖, 𝑗} ⊆ 𝐵 ↔ (𝑖𝐵𝑗𝐵))
2120simplbi 499 . . . . . . . . . . . . 13 ({𝑖, 𝑗} ⊆ 𝐵𝑖𝐵)
2221adantl 483 . . . . . . . . . . . 12 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑖𝐵)
23 eqid 2733 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
244, 8, 23, 7grprinv 18875 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → (𝑖(+g𝐺)((invg𝐺)‘𝑖)) = (0g𝐺))
2516, 22, 24syl2anc 585 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → (𝑖(+g𝐺)((invg𝐺)‘𝑖)) = (0g𝐺))
2625oveq1d 7424 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = ((0g𝐺)(+g𝐺)𝑗))
274, 7grpinvcl 18872 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
2816, 22, 27syl2anc 585 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
2920simprbi 498 . . . . . . . . . . . 12 ({𝑖, 𝑗} ⊆ 𝐵𝑗𝐵)
3029adantl 483 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑗𝐵)
314, 8grpass 18828 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑖𝐵 ∧ ((invg𝐺)‘𝑖) ∈ 𝐵𝑗𝐵)) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
3216, 22, 28, 30, 31syl13anc 1373 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
334, 8, 23grplid 18852 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑗𝐵) → ((0g𝐺)(+g𝐺)𝑗) = 𝑗)
3416, 30, 33syl2anc 585 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((0g𝐺)(+g𝐺)𝑗) = 𝑗)
3526, 32, 343eqtr3d 2781 . . . . . . . . 9 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)) = 𝑗)
3635adantr 482 . . . . . . . 8 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)) = 𝑗)
3712, 15, 36rspcedvd 3615 . . . . . . 7 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)
38 oveq2 7417 . . . . . . . . . . 11 ((𝑖(+g𝐺)𝑘) = 𝑗 → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = (((invg𝐺)‘𝑖)(+g𝐺)𝑗))
3938adantl 483 . . . . . . . . . 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 482 . . . . . . . . . . . 12 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → 𝑖𝐵)
426adantr 482 . . . . . . . . . . . . 13 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑆𝐵)
4342sselda 3983 . . . . . . . . . . . 12 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → 𝑘𝐵)
4433ad2ant1 1134 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐵𝑘𝐵) → 𝐺 ∈ Grp)
45 simp2 1138 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐵𝑘𝐵) → 𝑖𝐵)
464, 8, 23, 7grplinv 18874 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)𝑖) = (0g𝐺))
4744, 45, 46syl2anc 585 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)𝑖) = (0g𝐺))
4847oveq1d 7424 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = ((0g𝐺)(+g𝐺)𝑘))
4944, 45, 27syl2anc 585 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
50 simp3 1139 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → 𝑘𝐵)
514, 8grpass 18828 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑖) ∈ 𝐵𝑖𝐵𝑘𝐵)) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)))
5244, 49, 45, 50, 51syl13anc 1373 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)))
534, 8, 23grplid 18852 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑘𝐵) → ((0g𝐺)(+g𝐺)𝑘) = 𝑘)
5444, 50, 53syl2anc 585 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((0g𝐺)(+g𝐺)𝑘) = 𝑘)
5548, 52, 543eqtr3d 2781 . . . . . . . . . . . 12 ((𝜑𝑖𝐵𝑘𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = 𝑘)
5640, 41, 43, 55syl3anc 1372 . . . . . . . . . . 11 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = 𝑘)
5756adantr 482 . . . . . . . . . 10 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = 𝑘)
5839, 57eqtr3d 2775 . . . . . . . . 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 2834 . . . . . . . 8 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
6160r19.29an 3159 . . . . . . 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 580 . . . . 5 (𝜑 → (({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) ↔ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)))
6463opabbidv 5215 . . . 4 (𝜑 → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
6511, 64eqtrd 2773 . . 3 (𝜑 → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
6665eceq2d 8745 . 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 2733 . . 3 {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)}
693grpmndd 18832 . . 3 (𝜑𝐺 ∈ Mnd)
70 quslsm.s . . 3 (𝜑𝑋𝐵)
714, 8, 67, 68, 69, 6, 70lsmsnorb2 32502 . 2 (𝜑 → ({𝑋} 𝑆) = [𝑋]{⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
7266, 71eqtr4d 2776 1 (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wrex 3071  wss 3949  {csn 4629  {cpr 4631  {copab 5211  cfv 6544  (class class class)co 7409  [cec 8701  Basecbs 17144  +gcplusg 17197  0gc0g 17385  Grpcgrp 18819  invgcminusg 18820  SubGrpcsubg 19000   ~QG cqg 19002  LSSumclsm 19502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-ec 8705  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-0g 17387  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-subg 19003  df-eqg 19005  df-oppg 19210  df-lsm 19504
This theorem is referenced by:  qusbas2  32517  qus0g  32518  qusima  32519  nsgqus0  32521  nsgmgclem  32522  nsgqusf1olem1  32524  nsgqusf1olem2  32525  nsgqusf1olem3  32526
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