Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  quslsm Structured version   Visualization version   GIF version

Theorem quslsm 32186
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 18933 . . . . . 6 (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
31, 2syl 17 . . . . 5 (𝜑𝐺 ∈ Grp)
4 quslsm.b . . . . . . 7 𝐵 = (Base‘𝐺)
54subgss 18929 . . . . . 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 18978 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑆𝐵) → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)})
113, 6, 10syl2anc 584 . . . 4 (𝜑 → (𝐺 ~QG 𝑆) = {⟨𝑖, 𝑗⟩ ∣ ({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)})
12 simpr 485 . . . . . . . 8 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
13 oveq2 7365 . . . . . . . . . 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 482 . . . . . . . 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 481 . . . . . . . . . . . 12 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝐺 ∈ Grp)
17 vex 3449 . . . . . . . . . . . . . . . 16 𝑖 ∈ V
18 vex 3449 . . . . . . . . . . . . . . . 16 𝑗 ∈ V
1917, 18prss 4780 . . . . . . . . . . . . . . 15 ((𝑖𝐵𝑗𝐵) ↔ {𝑖, 𝑗} ⊆ 𝐵)
2019bicomi 223 . . . . . . . . . . . . . 14 ({𝑖, 𝑗} ⊆ 𝐵 ↔ (𝑖𝐵𝑗𝐵))
2120simplbi 498 . . . . . . . . . . . . 13 ({𝑖, 𝑗} ⊆ 𝐵𝑖𝐵)
2221adantl 482 . . . . . . . . . . . 12 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑖𝐵)
23 eqid 2736 . . . . . . . . . . . . 13 (0g𝐺) = (0g𝐺)
244, 8, 23, 7grprinv 18801 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → (𝑖(+g𝐺)((invg𝐺)‘𝑖)) = (0g𝐺))
2516, 22, 24syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → (𝑖(+g𝐺)((invg𝐺)‘𝑖)) = (0g𝐺))
2625oveq1d 7372 . . . . . . . . . 10 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((𝑖(+g𝐺)((invg𝐺)‘𝑖))(+g𝐺)𝑗) = ((0g𝐺)(+g𝐺)𝑗))
274, 7grpinvcl 18798 . . . . . . . . . . . 12 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
2816, 22, 27syl2anc 584 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
2920simprbi 497 . . . . . . . . . . . 12 ({𝑖, 𝑗} ⊆ 𝐵𝑗𝐵)
3029adantl 482 . . . . . . . . . . 11 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑗𝐵)
314, 8grpass 18757 . . . . . . . . . . 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 18780 . . . . . . . . . . 11 ((𝐺 ∈ Grp ∧ 𝑗𝐵) → ((0g𝐺)(+g𝐺)𝑗) = 𝑗)
3416, 30, 33syl2anc 584 . . . . . . . . . 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 481 . . . . . . . 8 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → (𝑖(+g𝐺)(((invg𝐺)‘𝑖)(+g𝐺)𝑗)) = 𝑗)
3712, 15, 36rspcedvd 3583 . . . . . . 7 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) → ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)
38 oveq2 7365 . . . . . . . . . . 11 ((𝑖(+g𝐺)𝑘) = 𝑗 → (((invg𝐺)‘𝑖)(+g𝐺)(𝑖(+g𝐺)𝑘)) = (((invg𝐺)‘𝑖)(+g𝐺)𝑗))
3938adantl 482 . . . . . . . . . 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 481 . . . . . . . . . . . 12 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → 𝑖𝐵)
426adantr 481 . . . . . . . . . . . . 13 ((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) → 𝑆𝐵)
4342sselda 3944 . . . . . . . . . . . 12 (((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) → 𝑘𝐵)
4433ad2ant1 1133 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐵𝑘𝐵) → 𝐺 ∈ Grp)
45 simp2 1137 . . . . . . . . . . . . . . 15 ((𝜑𝑖𝐵𝑘𝐵) → 𝑖𝐵)
464, 8, 23, 7grplinv 18800 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝑖𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)𝑖) = (0g𝐺))
4744, 45, 46syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → (((invg𝐺)‘𝑖)(+g𝐺)𝑖) = (0g𝐺))
4847oveq1d 7372 . . . . . . . . . . . . 13 ((𝜑𝑖𝐵𝑘𝐵) → ((((invg𝐺)‘𝑖)(+g𝐺)𝑖)(+g𝐺)𝑘) = ((0g𝐺)(+g𝐺)𝑘))
4944, 45, 27syl2anc 584 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → ((invg𝐺)‘𝑖) ∈ 𝐵)
50 simp3 1138 . . . . . . . . . . . . . 14 ((𝜑𝑖𝐵𝑘𝐵) → 𝑘𝐵)
514, 8grpass 18757 . . . . . . . . . . . . . 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 18780 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝑘𝐵) → ((0g𝐺)(+g𝐺)𝑘) = 𝑘)
5444, 50, 53syl2anc 584 . . . . . . . . . . . . 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 481 . . . . . . . . . 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 2838 . . . . . . . 8 ((((𝜑 ∧ {𝑖, 𝑗} ⊆ 𝐵) ∧ 𝑘𝑆) ∧ (𝑖(+g𝐺)𝑘) = 𝑗) → (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆)
6160r19.29an 3155 . . . . . . 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 579 . . . . 5 (𝜑 → (({𝑖, 𝑗} ⊆ 𝐵 ∧ (((invg𝐺)‘𝑖)(+g𝐺)𝑗) ∈ 𝑆) ↔ ({𝑖, 𝑗} ⊆ 𝐵 ∧ ∃𝑘𝑆 (𝑖(+g𝐺)𝑘) = 𝑗)))
6463opabbidv 5171 . . . 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 8690 . 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 18760 . . 3 (𝜑𝐺 ∈ Mnd)
70 quslsm.s . . 3 (𝜑𝑋𝐵)
714, 8, 67, 68, 69, 6, 70lsmsnorb2 32173 . 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 396  w3a 1087   = wceq 1541  wcel 2106  wrex 3073  wss 3910  {csn 4586  {cpr 4588  {copab 5167  cfv 6496  (class class class)co 7357  [cec 8646  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Grpcgrp 18748  invgcminusg 18749  SubGrpcsubg 18922   ~QG cqg 18924  LSSumclsm 19416
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-tpos 8157  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-ec 8650  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-plusg 17146  df-0g 17323  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-grp 18751  df-minusg 18752  df-subg 18925  df-eqg 18927  df-oppg 19124  df-lsm 19418
This theorem is referenced by:  qusima  32187  nsgqus0  32188  nsgmgclem  32189  nsgqusf1olem1  32191  nsgqusf1olem2  32192  nsgqusf1olem3  32193
  Copyright terms: Public domain W3C validator