Theorem isgrpde 18867

Description: Deduce a group from its properties. In this version of isgrpd 18868, 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 18580	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
9	1, 2, 3, 4, 8	grpidd 18576	. 2 ⊢ (𝜑 → 0 = (0g‘𝐺))
10	1, 2, 5, 6, 3, 4, 8	ismndd 18661	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
11	1, 2, 9, 10, 7	isgrpd2e 18865	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∃wrex 3056  ‘cfv 6481  (class class class)co 7346  Basecbs 17117  +_gcplusg 17158  Grpcgrp 18843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-iota 6437  df-fun 6483  df-fv 6489  df-riota 7303  df-ov 7349  df-0g 17342  df-mgm 18545  df-sgrp 18624  df-mnd 18640  df-grp 18846

This theorem is referenced by:  isgrpd  18868  dfgrp2  18872  imasgrp2  18965  unitgrp  20299