Theorem isgrpd2e 13626

Description: Deduce a group from its properties. In this version of isgrpd2 13627, we don't assume there is an expression for the inverse of 𝑥. (Contributed by NM, 10-Aug-2013.)

Hypotheses

Ref	Expression
isgrpd2.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
isgrpd2.p	⊢ (𝜑 → + = (+g‘𝐺))
isgrpd2.z	⊢ (𝜑 → 0 = (0g‘𝐺))
isgrpd2.g	⊢ (𝜑 → 𝐺 ∈ Mnd)
isgrpd2e.n	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )

Assertion

Ref	Expression
isgrpd2e	⊢ (𝜑 → 𝐺 ∈ Grp)

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

Allowed substitution hint:   0 (𝑥)

Proof of Theorem isgrpd2e

Step	Hyp	Ref	Expression
1		isgrpd2.g	. 2 ⊢ (𝜑 → 𝐺 ∈ Mnd)
2		isgrpd2e.n	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
3	2	ralrimiva 2604	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )
4		isgrpd2.b	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
5		isgrpd2.p	. . . . . . 7 ⊢ (𝜑 → + = (+g‘𝐺))
6	5	oveqd 6040	. . . . . 6 ⊢ (𝜑 → (𝑦 + 𝑥) = (𝑦(+g‘𝐺)𝑥))
7		isgrpd2.z	. . . . . 6 ⊢ (𝜑 → 0 = (0g‘𝐺))
8	6, 7	eqeq12d 2245	. . . . 5 ⊢ (𝜑 → ((𝑦 + 𝑥) = 0 ↔ (𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
9	4, 8	rexeqbidv 2746	. . . 4 ⊢ (𝜑 → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
10	4, 9	raleqbidv 2745	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
11	3, 10	mpbid 147	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺))
12		eqid 2230	. . 3 ⊢ (Base‘𝐺) = (Base‘𝐺)
13		eqid 2230	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
14		eqid 2230	. . 3 ⊢ (0g‘𝐺) = (0g‘𝐺)
15	12, 13, 14	isgrp 13612	. 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∃𝑦 ∈ (Base‘𝐺)(𝑦(+g‘𝐺)𝑥) = (0g‘𝐺)))
16	1, 11, 15	sylanbrc 417	1 ⊢ (𝜑 → 𝐺 ∈ Grp)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  ∀wral 2509  ∃wrex 2510  ‘cfv 5328  (class class class)co 6023  Basecbs 13105  +_gcplusg 13183  0_gc0g 13362  Mndcmnd 13522  Grpcgrp 13606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2212

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-rab 2518  df-v 2803  df-un 3203  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-br 4090  df-iota 5288  df-fv 5336  df-ov 6026  df-grp 13609

This theorem is referenced by:  isgrpd2  13627  isgrpde  13628