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Theorem quslsm 31638
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 18805 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (𝜑𝐺 ∈ Grp)
4 quslsm.b . . . . . . 7 𝐵 = (Base‘𝐺)
54subgss 18801 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝑆𝐵)
61, 5syl 17 . . . . 5 (𝜑𝑆𝐵)
7 eqid 2736 . . . . . 6 (invg𝐺) = (invg𝐺)
8 eqid 2736 . . . . . 6 (+g𝐺) = (+g𝐺)
9 eqid 2736 . . . . . 6 (𝐺 ~QG 𝑆) = (𝐺 ~QG 𝑆)
104, 7, 8, 9eqgfval 18849 . . . . 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 7315 . . . . . . . . . 10 (𝑘 = (((invg𝐺)‘𝑖)(+g𝐺)𝑗) → (𝑖(+g𝐺)𝑘) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
1413eqeq1d 2738 . . . . . . . . 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 3441 . . . . . . . . . . . . . . . 16 𝑖 ∈ V
18 vex 3441 . . . . . . . . . . . . . . . 16 𝑗 ∈ V
1917, 18prss 4759 . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
244, 8, 23, 7grprinv 18674 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → (𝑖(+g𝐺)((invg𝐺)‘𝑖)) = (0g𝐺))
2516, 22, 24syl2anc 585 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → (𝑖(+g𝐺)((invg𝐺)‘𝑖)) = (0g𝐺))
2625oveq1d 7322 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = ((0g𝐺)(+g𝐺)𝑗))
274, 7grpinvcl 18672 . . . . . . . . . . . 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 18631 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ (𝑖𝐵 ∧ ((invg𝐺)‘𝑖) ∈ 𝐵𝑗𝐵)) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
3216, 22, 28, 30, 31syl13anc 1372 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)))
334, 8, 23grplid 18654 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑗𝐵) → ((0g𝐺)(+g𝐺)𝑗) = 𝑗)
3416, 30, 33syl2anc 585 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((0g𝐺)(+g𝐺)𝑗) = 𝑗)
3526, 32, 343eqtr3d 2784 . . . . . . . . 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 3568 . . . . . . 7 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)
38 oveq2 7315 . . . . . . . . . . 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 765 . . . . . . . . . . . 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 3926 . . . . . . . . . . . 12 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → 𝑘𝐵)
4433ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐵𝑘𝐵) → 𝐺 ∈ Grp)
45 simp2 1137 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐵𝑘𝐵) → 𝑖𝐵)
464, 8, 23, 7grplinv 18673 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)𝑖) = (0g𝐺))
4744, 45, 46syl2anc 585 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)𝑖) = (0g𝐺))
4847oveq1d 7322 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = ((0g𝐺)(+g𝐺)𝑘))
4944, 45, 27syl2anc 585 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
50 simp3 1138 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → 𝑘𝐵)
514, 8grpass 18631 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ (((invg𝐺)‘𝑖) ∈ 𝐵𝑖𝐵𝑘𝐵)) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)))
5244, 49, 45, 50, 51syl13anc 1372 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)))
534, 8, 23grplid 18654 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑘𝐵) → ((0g𝐺)(+g𝐺)𝑘) = 𝑘)
5444, 50, 53syl2anc 585 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((0g𝐺)(+g𝐺)𝑘) = 𝑘)
5548, 52, 543eqtr3d 2784 . . . . . . . . . . . 12 ((𝜑𝑖𝐵𝑘𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = 𝑘)
5640, 41, 43, 55syl3anc 1371 . . . . . . . . . . 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 2778 . . . . . . . . 9 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) = 𝑘)
59 simplr 767 . . . . . . . . 9 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → 𝑘𝑆)
6058, 59eqeltrd 2837 . . . . . . . 8 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
6160r19.29an 3152 . . . . . . 7 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
6237, 61impbida 799 . . . . . 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 5147 . . . 4 (𝜑 → {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
6511, 64eqtrd 2776 . . 3 (𝜑 → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
6665eceq2d 8571 . 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 2736 . . 3 {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)} = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)}
693grpmndd 18634 . . 3 (𝜑𝐺 ∈ Mnd)
70 quslsm.s . . 3 (𝜑𝑋𝐵)
714, 8, 67, 68, 69, 6, 70lsmsnorb2 31625 . 2 (𝜑 → ({𝑋} 𝑆) = [𝑋]{⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)})
7266, 71eqtr4d 2779 1 (𝜑 → [𝑋](𝐺 ~QG 𝑆) = ({𝑋} 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1539  wcel 2104  wrex 3071  wss 3892  {csn 4565  {cpr 4567  {copab 5143  cfv 6458  (class class class)co 7307  [cec 8527  Basecbs 16957  +gcplusg 17007  0gc0g 17195  Grpcgrp 18622  invgcminusg 18623  SubGrpcsubg 18794   ~QG cqg 18796  LSSumclsm 19284
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620  ax-cnex 10973  ax-resscn 10974  ax-1cn 10975  ax-icn 10976  ax-addcl 10977  ax-addrcl 10978  ax-mulcl 10979  ax-mulrcl 10980  ax-mulcom 10981  ax-addass 10982  ax-mulass 10983  ax-distr 10984  ax-i2m1 10985  ax-1ne0 10986  ax-1rid 10987  ax-rnegex 10988  ax-rrecex 10989  ax-cnre 10990  ax-pre-lttri 10991  ax-pre-lttrn 10992  ax-pre-ltadd 10993  ax-pre-mulgt0 10994
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3285  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-tr 5199  df-id 5500  df-eprel 5506  df-po 5514  df-so 5515  df-fr 5555  df-we 5557  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-pred 6217  df-ord 6284  df-on 6285  df-lim 6286  df-suc 6287  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-riota 7264  df-ov 7310  df-oprab 7311  df-mpo 7312  df-om 7745  df-1st 7863  df-2nd 7864  df-tpos 8073  df-frecs 8128  df-wrecs 8159  df-recs 8233  df-rdg 8272  df-er 8529  df-ec 8531  df-en 8765  df-dom 8766  df-sdom 8767  df-pnf 11057  df-mnf 11058  df-xr 11059  df-ltxr 11060  df-le 11061  df-sub 11253  df-neg 11254  df-nn 12020  df-2 12082  df-sets 16910  df-slot 16928  df-ndx 16940  df-base 16958  df-plusg 17020  df-0g 17197  df-mgm 18371  df-sgrp 18420  df-mnd 18431  df-grp 18625  df-minusg 18626  df-subg 18797  df-eqg 18799  df-oppg 18995  df-lsm 19286
This theorem is referenced by:  qusima  31639  nsgqus0  31640  nsgmgclem  31641  nsgqusf1olem1  31643  nsgqusf1olem2  31644  nsgqusf1olem3  31645
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