Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  lsmelvalm Structured version   Visualization version   GIF version

Theorem lsmelvalm 18006
 Description: Subgroup sum membership analogue of lsmelval 18004 using vector subtraction. TODO: any way to shorten proof? (Contributed by NM, 16-Mar-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
Hypotheses
Ref Expression
lsmelvalm.m = (-g𝐺)
lsmelvalm.p = (LSSum‘𝐺)
lsmelvalm.t (𝜑𝑇 ∈ (SubGrp‘𝐺))
lsmelvalm.u (𝜑𝑈 ∈ (SubGrp‘𝐺))
Assertion
Ref Expression
lsmelvalm (𝜑 → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 𝑧)))
Distinct variable groups:   𝑦,𝑧,   𝑦,𝐺,𝑧   𝜑,𝑦,𝑧   𝑦,𝑇,𝑧   𝑦,𝑈,𝑧   𝑦,𝑋,𝑧
Allowed substitution hints:   (𝑦,𝑧)

Proof of Theorem lsmelvalm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 lsmelvalm.t . . 3 (𝜑𝑇 ∈ (SubGrp‘𝐺))
2 lsmelvalm.u . . 3 (𝜑𝑈 ∈ (SubGrp‘𝐺))
3 eqid 2621 . . . 4 (+g𝐺) = (+g𝐺)
4 lsmelvalm.p . . . 4 = (LSSum‘𝐺)
53, 4lsmelval 18004 . . 3 ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑥𝑈 𝑋 = (𝑦(+g𝐺)𝑥)))
61, 2, 5syl2anc 692 . 2 (𝜑 → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑥𝑈 𝑋 = (𝑦(+g𝐺)𝑥)))
72adantr 481 . . . . . . . 8 ((𝜑𝑦𝑇) → 𝑈 ∈ (SubGrp‘𝐺))
8 eqid 2621 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
98subginvcl 17543 . . . . . . . 8 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑥𝑈) → ((invg𝐺)‘𝑥) ∈ 𝑈)
107, 9sylan 488 . . . . . . 7 (((𝜑𝑦𝑇) ∧ 𝑥𝑈) → ((invg𝐺)‘𝑥) ∈ 𝑈)
11 eqid 2621 . . . . . . . . 9 (Base‘𝐺) = (Base‘𝐺)
12 lsmelvalm.m . . . . . . . . 9 = (-g𝐺)
13 subgrcl 17539 . . . . . . . . . . 11 (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp)
141, 13syl 17 . . . . . . . . . 10 (𝜑𝐺 ∈ Grp)
1514ad2antrr 761 . . . . . . . . 9 (((𝜑𝑦𝑇) ∧ 𝑥𝑈) → 𝐺 ∈ Grp)
1611subgss 17535 . . . . . . . . . . . 12 (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺))
171, 16syl 17 . . . . . . . . . . 11 (𝜑𝑇 ⊆ (Base‘𝐺))
1817sselda 3588 . . . . . . . . . 10 ((𝜑𝑦𝑇) → 𝑦 ∈ (Base‘𝐺))
1918adantr 481 . . . . . . . . 9 (((𝜑𝑦𝑇) ∧ 𝑥𝑈) → 𝑦 ∈ (Base‘𝐺))
2011subgss 17535 . . . . . . . . . . 11 (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺))
217, 20syl 17 . . . . . . . . . 10 ((𝜑𝑦𝑇) → 𝑈 ⊆ (Base‘𝐺))
2221sselda 3588 . . . . . . . . 9 (((𝜑𝑦𝑇) ∧ 𝑥𝑈) → 𝑥 ∈ (Base‘𝐺))
2311, 3, 12, 8, 15, 19, 22grpsubinv 17428 . . . . . . . 8 (((𝜑𝑦𝑇) ∧ 𝑥𝑈) → (𝑦 ((invg𝐺)‘𝑥)) = (𝑦(+g𝐺)𝑥))
2423eqcomd 2627 . . . . . . 7 (((𝜑𝑦𝑇) ∧ 𝑥𝑈) → (𝑦(+g𝐺)𝑥) = (𝑦 ((invg𝐺)‘𝑥)))
25 oveq2 6623 . . . . . . . . 9 (𝑧 = ((invg𝐺)‘𝑥) → (𝑦 𝑧) = (𝑦 ((invg𝐺)‘𝑥)))
2625eqeq2d 2631 . . . . . . . 8 (𝑧 = ((invg𝐺)‘𝑥) → ((𝑦(+g𝐺)𝑥) = (𝑦 𝑧) ↔ (𝑦(+g𝐺)𝑥) = (𝑦 ((invg𝐺)‘𝑥))))
2726rspcev 3299 . . . . . . 7 ((((invg𝐺)‘𝑥) ∈ 𝑈 ∧ (𝑦(+g𝐺)𝑥) = (𝑦 ((invg𝐺)‘𝑥))) → ∃𝑧𝑈 (𝑦(+g𝐺)𝑥) = (𝑦 𝑧))
2810, 24, 27syl2anc 692 . . . . . 6 (((𝜑𝑦𝑇) ∧ 𝑥𝑈) → ∃𝑧𝑈 (𝑦(+g𝐺)𝑥) = (𝑦 𝑧))
29 eqeq1 2625 . . . . . . 7 (𝑋 = (𝑦(+g𝐺)𝑥) → (𝑋 = (𝑦 𝑧) ↔ (𝑦(+g𝐺)𝑥) = (𝑦 𝑧)))
3029rexbidv 3047 . . . . . 6 (𝑋 = (𝑦(+g𝐺)𝑥) → (∃𝑧𝑈 𝑋 = (𝑦 𝑧) ↔ ∃𝑧𝑈 (𝑦(+g𝐺)𝑥) = (𝑦 𝑧)))
3128, 30syl5ibrcom 237 . . . . 5 (((𝜑𝑦𝑇) ∧ 𝑥𝑈) → (𝑋 = (𝑦(+g𝐺)𝑥) → ∃𝑧𝑈 𝑋 = (𝑦 𝑧)))
3231rexlimdva 3026 . . . 4 ((𝜑𝑦𝑇) → (∃𝑥𝑈 𝑋 = (𝑦(+g𝐺)𝑥) → ∃𝑧𝑈 𝑋 = (𝑦 𝑧)))
338subginvcl 17543 . . . . . . . 8 ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑧𝑈) → ((invg𝐺)‘𝑧) ∈ 𝑈)
347, 33sylan 488 . . . . . . 7 (((𝜑𝑦𝑇) ∧ 𝑧𝑈) → ((invg𝐺)‘𝑧) ∈ 𝑈)
3518adantr 481 . . . . . . . 8 (((𝜑𝑦𝑇) ∧ 𝑧𝑈) → 𝑦 ∈ (Base‘𝐺))
3621sselda 3588 . . . . . . . 8 (((𝜑𝑦𝑇) ∧ 𝑧𝑈) → 𝑧 ∈ (Base‘𝐺))
3711, 3, 8, 12grpsubval 17405 . . . . . . . 8 ((𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦 𝑧) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
3835, 36, 37syl2anc 692 . . . . . . 7 (((𝜑𝑦𝑇) ∧ 𝑧𝑈) → (𝑦 𝑧) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
39 oveq2 6623 . . . . . . . . 9 (𝑥 = ((invg𝐺)‘𝑧) → (𝑦(+g𝐺)𝑥) = (𝑦(+g𝐺)((invg𝐺)‘𝑧)))
4039eqeq2d 2631 . . . . . . . 8 (𝑥 = ((invg𝐺)‘𝑧) → ((𝑦 𝑧) = (𝑦(+g𝐺)𝑥) ↔ (𝑦 𝑧) = (𝑦(+g𝐺)((invg𝐺)‘𝑧))))
4140rspcev 3299 . . . . . . 7 ((((invg𝐺)‘𝑧) ∈ 𝑈 ∧ (𝑦 𝑧) = (𝑦(+g𝐺)((invg𝐺)‘𝑧))) → ∃𝑥𝑈 (𝑦 𝑧) = (𝑦(+g𝐺)𝑥))
4234, 38, 41syl2anc 692 . . . . . 6 (((𝜑𝑦𝑇) ∧ 𝑧𝑈) → ∃𝑥𝑈 (𝑦 𝑧) = (𝑦(+g𝐺)𝑥))
43 eqeq1 2625 . . . . . . 7 (𝑋 = (𝑦 𝑧) → (𝑋 = (𝑦(+g𝐺)𝑥) ↔ (𝑦 𝑧) = (𝑦(+g𝐺)𝑥)))
4443rexbidv 3047 . . . . . 6 (𝑋 = (𝑦 𝑧) → (∃𝑥𝑈 𝑋 = (𝑦(+g𝐺)𝑥) ↔ ∃𝑥𝑈 (𝑦 𝑧) = (𝑦(+g𝐺)𝑥)))
4542, 44syl5ibrcom 237 . . . . 5 (((𝜑𝑦𝑇) ∧ 𝑧𝑈) → (𝑋 = (𝑦 𝑧) → ∃𝑥𝑈 𝑋 = (𝑦(+g𝐺)𝑥)))
4645rexlimdva 3026 . . . 4 ((𝜑𝑦𝑇) → (∃𝑧𝑈 𝑋 = (𝑦 𝑧) → ∃𝑥𝑈 𝑋 = (𝑦(+g𝐺)𝑥)))
4732, 46impbid 202 . . 3 ((𝜑𝑦𝑇) → (∃𝑥𝑈 𝑋 = (𝑦(+g𝐺)𝑥) ↔ ∃𝑧𝑈 𝑋 = (𝑦 𝑧)))
4847rexbidva 3044 . 2 (𝜑 → (∃𝑦𝑇𝑥𝑈 𝑋 = (𝑦(+g𝐺)𝑥) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 𝑧)))
496, 48bitrd 268 1 (𝜑 → (𝑋 ∈ (𝑇 𝑈) ↔ ∃𝑦𝑇𝑧𝑈 𝑋 = (𝑦 𝑧)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2909   ⊆ wss 3560  ‘cfv 5857  (class class class)co 6615  Basecbs 15800  +gcplusg 15881  Grpcgrp 17362  invgcminusg 17363  -gcsg 17364  SubGrpcsubg 17528  LSSumclsm 17989 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-0g 16042  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-grp 17365  df-minusg 17366  df-sbg 17367  df-subg 17531  df-lsm 17991 This theorem is referenced by:  lsmelvalmi  18007  pgpfac1lem2  18414  pgpfac1lem3  18416  pgpfac1lem4  18417  mapdpglem3  36483
 Copyright terms: Public domain W3C validator