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Theorem dfgrp2 18604
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 18580, 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 18603 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2 grpmnd 18584 . . . . 5 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3 dfgrp2.b . . . . . 6 𝐵 = (Base‘𝐺)
4 eqid 2738 . . . . . 6 (0g𝐺) = (0g𝐺)
53, 4mndidcl 18400 . . . . 5 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
62, 5syl 17 . . . 4 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
7 oveq1 7282 . . . . . . . 8 (𝑛 = (0g𝐺) → (𝑛 + 𝑥) = ((0g𝐺) + 𝑥))
87eqeq1d 2740 . . . . . . 7 (𝑛 = (0g𝐺) → ((𝑛 + 𝑥) = 𝑥 ↔ ((0g𝐺) + 𝑥) = 𝑥))
9 eqeq2 2750 . . . . . . . 8 (𝑛 = (0g𝐺) → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑥) = (0g𝐺)))
109rexbidv 3226 . . . . . . 7 (𝑛 = (0g𝐺) → (∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
118, 10anbi12d 631 . . . . . 6 (𝑛 = (0g𝐺) → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))))
1211ralbidv 3112 . . . . 5 (𝑛 = (0g𝐺) → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ∀𝑥𝐵 (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))))
1312adantl 482 . . . 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 18405 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ((0g𝐺) + 𝑥) = 𝑥)
162, 15sylan 580 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((0g𝐺) + 𝑥) = 𝑥)
173, 14, 4grpinvex 18587 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))
1816, 17jca 512 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
1918ralrimiva 3103 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
206, 13, 19rspcedvd 3563 . . 3 (𝐺 ∈ Grp → ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛))
211, 20jca 512 . 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 18380 . . . . . . . 8 (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
2524adantl 482 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Mgm)
263, 14mgmcl 18329 . . . . . . 7 ((𝐺 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
2725, 26syl3an1 1162 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
283, 14sgrpass 18381 . . . . . . 7 ((𝐺 ∈ Smgrp ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
2928adantll 711 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
30 simpll 764 . . . . . 6 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝑛𝐵)
31 oveq2 7283 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑛 + 𝑥) = (𝑛 + 𝑎))
32 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
3331, 32eqeq12d 2754 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝑛 + 𝑥) = 𝑥 ↔ (𝑛 + 𝑎) = 𝑎))
34 oveq2 7283 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑖 + 𝑥) = (𝑖 + 𝑎))
3534eqeq1d 2740 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑎) = 𝑛))
3635rexbidv 3226 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛))
3733, 36anbi12d 631 . . . . . . . . . 10 (𝑥 = 𝑎 → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛)))
3837rspcv 3557 . . . . . . . . 9 (𝑎𝐵 → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛)))
39 simpl 483 . . . . . . . . 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 724 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎𝐵 → (𝑛 + 𝑎) = 𝑎))
4241imp 407 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵) → (𝑛 + 𝑎) = 𝑎)
43 oveq1 7282 . . . . . . . . . . . . 13 (𝑖 = 𝑏 → (𝑖 + 𝑎) = (𝑏 + 𝑎))
4443eqeq1d 2740 . . . . . . . . . . . 12 (𝑖 = 𝑏 → ((𝑖 + 𝑎) = 𝑛 ↔ (𝑏 + 𝑎) = 𝑛))
4544cbvrexvw 3384 . . . . . . . . . . 11 (∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛 ↔ ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
4645biimpi 215 . . . . . . . . . 10 (∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛 → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
4746adantl 482 . . . . . . . . 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 724 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎𝐵 → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛))
5049imp 407 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎𝐵) → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
5122, 23, 27, 29, 30, 42, 50isgrpde 18600 . . . . 5 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Grp)
5251ex 413 . . . 4 ((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
5352rexlimiva 3210 . . 3 (∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
5453impcom 408 . 2 ((𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) → 𝐺 ∈ Grp)
5521, 54impbii 208 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  wrex 3065  cfv 6433  (class class class)co 7275  Basecbs 16912  +gcplusg 16962  0gc0g 17150  Mgmcmgm 18324  Smgrpcsgrp 18374  Mndcmnd 18385  Grpcgrp 18577
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580
This theorem is referenced by:  dfgrp2e  18605  dfgrp3  18674
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