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Theorem dfgrp2 18980
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 18954, 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 18978 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2 grpmnd 18958 . . . . 5 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3 dfgrp2.b . . . . . 6 𝐵 = (Base‘𝐺)
4 eqid 2737 . . . . . 6 (0g𝐺) = (0g𝐺)
53, 4mndidcl 18762 . . . . 5 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
62, 5syl 17 . . . 4 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
7 oveq1 7438 . . . . . . . 8 (𝑛 = (0g𝐺) → (𝑛 + 𝑥) = ((0g𝐺) + 𝑥))
87eqeq1d 2739 . . . . . . 7 (𝑛 = (0g𝐺) → ((𝑛 + 𝑥) = 𝑥 ↔ ((0g𝐺) + 𝑥) = 𝑥))
9 eqeq2 2749 . . . . . . . 8 (𝑛 = (0g𝐺) → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑥) = (0g𝐺)))
109rexbidv 3179 . . . . . . 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 3178 . . . . 5 (𝑛 = (0g𝐺) → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ∀𝑥𝐵 (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))))
1312adantl 481 . . . 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 18767 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ((0g𝐺) + 𝑥) = 𝑥)
162, 15sylan 580 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((0g𝐺) + 𝑥) = 𝑥)
173, 14, 4grpinvex 18961 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))
1816, 17jca 511 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
1918ralrimiva 3146 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
206, 13, 19rspcedvd 3624 . . 3 (𝐺 ∈ Grp → ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛))
211, 20jca 511 . 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 18737 . . . . . . . 8 (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
2524adantl 481 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Mgm)
263, 14mgmcl 18656 . . . . . . 7 ((𝐺 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
2725, 26syl3an1 1164 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
283, 14sgrpass 18738 . . . . . . 7 ((𝐺 ∈ Smgrp ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
2928adantll 714 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
30 simpll 767 . . . . . 6 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝑛𝐵)
31 oveq2 7439 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑛 + 𝑥) = (𝑛 + 𝑎))
32 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
3331, 32eqeq12d 2753 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝑛 + 𝑥) = 𝑥 ↔ (𝑛 + 𝑎) = 𝑎))
34 oveq2 7439 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑖 + 𝑥) = (𝑖 + 𝑎))
3534eqeq1d 2739 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑎) = 𝑛))
3635rexbidv 3179 . . . . . . . . . . 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 3618 . . . . . . . . 9 (𝑎𝐵 → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛)))
39 simpl 482 . . . . . . . . 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 727 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎𝐵 → (𝑛 + 𝑎) = 𝑎))
4241imp 406 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵) → (𝑛 + 𝑎) = 𝑎)
43 oveq1 7438 . . . . . . . . . . . . 13 (𝑖 = 𝑏 → (𝑖 + 𝑎) = (𝑏 + 𝑎))
4443eqeq1d 2739 . . . . . . . . . . . 12 (𝑖 = 𝑏 → ((𝑖 + 𝑎) = 𝑛 ↔ (𝑏 + 𝑎) = 𝑛))
4544cbvrexvw 3238 . . . . . . . . . . 11 (∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛 ↔ ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
4645biimpi 216 . . . . . . . . . 10 (∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛 → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
4746adantl 481 . . . . . . . . 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 727 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎𝐵 → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛))
5049imp 406 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵) → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
5122, 23, 27, 29, 30, 42, 50isgrpde 18975 . . . . 5 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Grp)
5251ex 412 . . . 4 ((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
5352rexlimiva 3147 . . 3 (∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
5453impcom 407 . 2 ((𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) → 𝐺 ∈ Grp)
5521, 54impbii 209 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Mgmcmgm 18651  Smgrpcsgrp 18731  Mndcmnd 18747  Grpcgrp 18951
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954
This theorem is referenced by:  dfgrp2e  18981  dfgrp3  19057
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