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Theorem dfgrp2 17379
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 17357, 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 ∈ SGrp ∧ ∃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 17378 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ SGrp)
2 grpmnd 17361 . . . . 5 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3 dfgrp2.b . . . . . 6 𝐵 = (Base‘𝐺)
4 eqid 2621 . . . . . 6 (0g𝐺) = (0g𝐺)
53, 4mndidcl 17240 . . . . 5 (𝐺 ∈ Mnd → (0g𝐺) ∈ 𝐵)
62, 5syl 17 . . . 4 (𝐺 ∈ Grp → (0g𝐺) ∈ 𝐵)
7 oveq1 6617 . . . . . . . 8 (𝑛 = (0g𝐺) → (𝑛 + 𝑥) = ((0g𝐺) + 𝑥))
87eqeq1d 2623 . . . . . . 7 (𝑛 = (0g𝐺) → ((𝑛 + 𝑥) = 𝑥 ↔ ((0g𝐺) + 𝑥) = 𝑥))
9 eqeq2 2632 . . . . . . . 8 (𝑛 = (0g𝐺) → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑥) = (0g𝐺)))
109rexbidv 3046 . . . . . . 7 (𝑛 = (0g𝐺) → (∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
118, 10anbi12d 746 . . . . . 6 (𝑛 = (0g𝐺) → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))))
1211ralbidv 2981 . . . . 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 17243 . . . . . . 7 ((𝐺 ∈ Mnd ∧ 𝑥𝐵) → ((0g𝐺) + 𝑥) = 𝑥)
162, 15sylan 488 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((0g𝐺) + 𝑥) = 𝑥)
173, 14, 4grpinvex 17364 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺))
1816, 17jca 554 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
1918ralrimiva 2961 . . . 4 (𝐺 ∈ Grp → ∀𝑥𝐵 (((0g𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = (0g𝐺)))
206, 13, 19rspcedvd 3305 . . 3 (𝐺 ∈ Grp → ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛))
211, 20jca 554 . 2 (𝐺 ∈ Grp → (𝐺 ∈ SGrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
223a1i 11 . . . . . 6 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) → 𝐵 = (Base‘𝐺))
2314a1i 11 . . . . . 6 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) → + = (+g𝐺))
24 sgrpmgm 17221 . . . . . . . 8 (𝐺 ∈ SGrp → 𝐺 ∈ Mgm)
2524adantl 482 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) → 𝐺 ∈ Mgm)
263, 14mgmcl 17177 . . . . . . 7 ((𝐺 ∈ Mgm ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
2725, 26syl3an1 1356 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) ∧ 𝑎𝐵𝑏𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
283, 14sgrpass 17222 . . . . . . 7 ((𝐺 ∈ SGrp ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
2928adantll 749 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) ∧ (𝑎𝐵𝑏𝐵𝑐𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
30 simpll 789 . . . . . 6 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) → 𝑛𝐵)
31 oveq2 6618 . . . . . . . . . . . 12 (𝑥 = 𝑎 → (𝑛 + 𝑥) = (𝑛 + 𝑎))
32 id 22 . . . . . . . . . . . 12 (𝑥 = 𝑎𝑥 = 𝑎)
3331, 32eqeq12d 2636 . . . . . . . . . . 11 (𝑥 = 𝑎 → ((𝑛 + 𝑥) = 𝑥 ↔ (𝑛 + 𝑎) = 𝑎))
34 oveq2 6618 . . . . . . . . . . . . 13 (𝑥 = 𝑎 → (𝑖 + 𝑥) = (𝑖 + 𝑎))
3534eqeq1d 2623 . . . . . . . . . . . 12 (𝑥 = 𝑎 → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑎) = 𝑛))
3635rexbidv 3046 . . . . . . . . . . 11 (𝑥 = 𝑎 → (∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛))
3733, 36anbi12d 746 . . . . . . . . . 10 (𝑥 = 𝑎 → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛)))
3837rspcv 3294 . . . . . . . . 9 (𝑎𝐵 → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛)))
39 simpl 473 . . . . . . . . 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 762 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) → (𝑎𝐵 → (𝑛 + 𝑎) = 𝑎))
4241imp 445 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) ∧ 𝑎𝐵) → (𝑛 + 𝑎) = 𝑎)
43 oveq1 6617 . . . . . . . . . . . . 13 (𝑖 = 𝑏 → (𝑖 + 𝑎) = (𝑏 + 𝑎))
4443eqeq1d 2623 . . . . . . . . . . . 12 (𝑖 = 𝑏 → ((𝑖 + 𝑎) = 𝑛 ↔ (𝑏 + 𝑎) = 𝑛))
4544cbvrexv 3163 . . . . . . . . . . 11 (∃𝑖𝐵 (𝑖 + 𝑎) = 𝑛 ↔ ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
4645biimpi 206 . . . . . . . . . 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 762 . . . . . . 7 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) → (𝑎𝐵 → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛))
5049imp 445 . . . . . 6 ((((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) ∧ 𝑎𝐵) → ∃𝑏𝐵 (𝑏 + 𝑎) = 𝑛)
5122, 23, 27, 29, 30, 42, 50isgrpde 17375 . . . . 5 (((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ SGrp) → 𝐺 ∈ Grp)
5251ex 450 . . . 4 ((𝑛𝐵 ∧ ∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) → (𝐺 ∈ SGrp → 𝐺 ∈ Grp))
5352rexlimiva 3022 . . 3 (∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ SGrp → 𝐺 ∈ Grp))
5453impcom 446 . 2 ((𝐺 ∈ SGrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) → 𝐺 ∈ Grp)
5521, 54impbii 199 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  0gc0g 16032  Mgmcmgm 17172  SGrpcsgrp 17215  Mndcmnd 17226  Grpcgrp 17354
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-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-riota 6571  df-ov 6613  df-0g 16034  df-mgm 17174  df-sgrp 17216  df-mnd 17227  df-grp 17357
This theorem is referenced by:  dfgrp2e  17380  dfgrp3  17446
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