Theorem isgrpde 19019

Description: Deduce a group from its properties. In this version of isgrpd 19020, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 6-Jan-2015.)

Hypotheses

Ref	Expression
isgrpd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
isgrpd.p	⊢ (𝜑 → + = (+g‘𝐺))
isgrpd.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
isgrpd.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
isgrpd.z	⊢ (𝜑 → 0 ∈ 𝐵)
isgrpd.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
isgrpde.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )

Assertion

Ref	Expression
isgrpde	⊢ (𝜑 → 𝐺 ∈ Grp)

Distinct variable groups:   𝑥,𝑦,𝑧, +   𝑥, 0 ,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧

Proof of Theorem isgrpde

Step	Hyp	Ref	Expression
1		isgrpd.b	. 2 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
2		isgrpd.p	. 2 ⊢ (𝜑 → + = (+g‘𝐺))
3		isgrpd.z	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
4		isgrpd.i	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
5		isgrpd.c	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
6		isgrpd.a	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
7		isgrpde.n	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
8	5, 3, 4, 6, 7	grprida 18728	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
9	1, 2, 3, 4, 8	grpidd 18724	. 2 ⊢ (𝜑 → 0 = (0g‘𝐺))
10	1, 2, 5, 6, 3, 4, 8	ismndd 18809	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
11	1, 2, 9, 10, 7	isgrpd2e 19017	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∃wrex 3089  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  +_gcplusg 17305  Grpcgrp 18995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-iota 6492  df-fun 6538  df-fv 6544  df-riota 7367  df-ov 7413  df-0g 17489  df-mgm 18693  df-sgrp 18772  df-mnd 18788  df-grp 18998

This theorem is referenced by:  isgrpd  19020  dfgrp2  19024  imasgrp2  19116  unitgrp  20461  isdrng3lem2  20852