Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfgrp2 Structured version   Visualization version   GIF version

Theorem dfgrp2 18127
 Description: Alternate definition of a group as semigroup with a left identity and a left inverse for each element. This "definition" is weaker than df-grp 18105, based on the definition of a monoid which provides a left and a right identity. (Contributed by AV, 28-Aug-2021.)
Hypotheses
Ref Expression
dfgrp2.b 𝐵 = (Base‘𝐺)
dfgrp2.p + = (+g𝐺)
Assertion
Ref Expression
dfgrp2 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
Distinct variable groups:   𝐵,𝑖,𝑛,𝑥   𝑖,𝐺,𝑛,𝑥   + ,𝑖,𝑛,𝑥

Proof of Theorem dfgrp2
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 grpsgrp 18126 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2 grpmnd 18109 . . . . 5 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3 dfgrp2.b . . . . . 6 𝐵 = (Base‘𝐺)
4 eqid 2821 . . . . . 6 (0g𝐺) = (0g𝐺)
53, 4mndidcl 17925 . . . . 5 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
62, 5syl 17 . . . 4 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
7 oveq1 7162 . . . . . . . 8 (𝑛 = (0g𝐺) → (𝑛 + 𝑥) = ((0g𝐺) + 𝑥))
87eqeq1d 2823 . . . . . . 7 (𝑛 = (0g𝐺) → ((𝑛 + 𝑥) = 𝑥 ↔ ((0g𝐺) + 𝑥) = 𝑥))
9 eqeq2 2833 . . . . . . . 8 (𝑛 = (0g𝐺) → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑥) = (0g𝐺)))
109rexbidv 3297 . . . . . . 7 (𝑛 = (0g𝐺) → (∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
118, 10anbi12d 632 . . . . . 6 (𝑛 = (0g𝐺) → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))))
1211ralbidv 3197 . . . . 5 (𝑛 = (0g𝐺) → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ∀𝑥𝐵 (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))))
1312adantl 484 . . . 4 ((𝐺 ∈ Grp ∧ 𝑛 = (0g𝐺)) → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ∀𝑥𝐵 (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))))
14 dfgrp2.p . . . . . . . 8 + = (+g𝐺)
153, 14, 4mndlid 17930 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ((0g𝐺) + 𝑥) = 𝑥)
162, 15sylan 582 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((0g𝐺) + 𝑥) = 𝑥)
173, 14, 4grpinvex 18112 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))
1816, 17jca 514 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
1918ralrimiva 3182 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
206, 13, 19rspcedvd 3625 . . 3 (𝐺 ∈ Grp → ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛))
211, 20jca 514 . 2 (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
223a1i 11 . . . . . 6 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐵 = (Base‘𝐺))
2314a1i 11 . . . . . 6 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → + = (+g𝐺))
24 sgrpmgm 17905 . . . . . . . 8 (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
2524adantl 484 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Mgm)
263, 14mgmcl 17854 . . . . . . 7 ((𝐺 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
2725, 26syl3an1 1159 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
283, 14sgrpass 17906 . . . . . . 7 ((𝐺 ∈ Smgrp ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
2928adantll 712 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
30 simpll 765 . . . . . 6 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝑛𝐵)
31 oveq2 7163 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑛 + 𝑥) = (𝑛 + 𝑎))
32 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
3331, 32eqeq12d 2837 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝑛 + 𝑥) = 𝑥 ↔ (𝑛 + 𝑎) = 𝑎))
34 oveq2 7163 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑖 + 𝑥) = (𝑖 + 𝑎))
3534eqeq1d 2823 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑎) = 𝑛))
3635rexbidv 3297 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛))
3733, 36anbi12d 632 . . . . . . . . . 10 (𝑥 = 𝑎 → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛)))
3837rspcv 3617 . . . . . . . . 9 (𝑎𝐵 → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛)))
39 simpl 485 . . . . . . . . 9 (((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛) → (𝑛 + 𝑎) = 𝑎)
4038, 39syl6com 37 . . . . . . . 8 (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → (𝑎𝐵 → (𝑛 + 𝑎) = 𝑎))
4140ad2antlr 725 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎𝐵 → (𝑛 + 𝑎) = 𝑎))
4241imp 409 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵) → (𝑛 + 𝑎) = 𝑎)
43 oveq1 7162 . . . . . . . . . . . . 13 (𝑖 = 𝑏 → (𝑖 + 𝑎) = (𝑏 + 𝑎))
4443eqeq1d 2823 . . . . . . . . . . . 12 (𝑖 = 𝑏 → ((𝑖 + 𝑎) = 𝑛 ↔ (𝑏 + 𝑎) = 𝑛))
4544cbvrexvw 3450 . . . . . . . . . . 11 (∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛 ↔ ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
4645biimpi 218 . . . . . . . . . 10 (∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛 → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
4746adantl 484 . . . . . . . . 9 (((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛) → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
4838, 47syl6com 37 . . . . . . . 8 (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → (𝑎𝐵 → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛))
4948ad2antlr 725 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎𝐵 → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛))
5049imp 409 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵) → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
5122, 23, 27, 29, 30, 42, 50isgrpde 18123 . . . . 5 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Grp)
5251ex 415 . . . 4 ((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
5352rexlimiva 3281 . . 3 (∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
5453impcom 410 . 2 ((𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) → 𝐺 ∈ Grp)
5521, 54impbii 211 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  ‘cfv 6354  (class class class)co 7155  Basecbs 16482  +gcplusg 16564  0gc0g 16712  Mgmcmgm 17849  Smgrpcsgrp 17899  Mndcmnd 17910  Grpcgrp 18102 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-riota 7113  df-ov 7158  df-0g 16714  df-mgm 17851  df-sgrp 17900  df-mnd 17911  df-grp 18105 This theorem is referenced by:  dfgrp2e  18128  dfgrp3  18197
 Copyright terms: Public domain W3C validator