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Theorem isgrpd2 17650
Description: Deduce a group from its properties. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). Note: normally we don't use a 𝜑 antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2771, but we make an exception for theorems such as isgrpd2 17650, ismndd 17521, and islmodd 19079 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
Hypotheses
Ref Expression
isgrpd2.b (𝜑𝐵 = (Base‘𝐺))
isgrpd2.p (𝜑+ = (+g𝐺))
isgrpd2.z (𝜑0 = (0g𝐺))
isgrpd2.g (𝜑𝐺 ∈ Mnd)
isgrpd2.n ((𝜑𝑥𝐵) → 𝑁𝐵)
isgrpd2.j ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )
Assertion
Ref Expression
isgrpd2 (𝜑𝐺 ∈ Grp)
Distinct variable groups:   𝑥, +   𝑥,𝐵   𝑥,𝐺   𝜑,𝑥
Allowed substitution hints:   𝑁(𝑥)   0 (𝑥)

Proof of Theorem isgrpd2
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 isgrpd2.b . 2 (𝜑𝐵 = (Base‘𝐺))
2 isgrpd2.p . 2 (𝜑+ = (+g𝐺))
3 isgrpd2.z . 2 (𝜑0 = (0g𝐺))
4 isgrpd2.g . 2 (𝜑𝐺 ∈ Mnd)
5 isgrpd2.n . . 3 ((𝜑𝑥𝐵) → 𝑁𝐵)
6 isgrpd2.j . . 3 ((𝜑𝑥𝐵) → (𝑁 + 𝑥) = 0 )
7 oveq1 6803 . . . . 5 (𝑦 = 𝑁 → (𝑦 + 𝑥) = (𝑁 + 𝑥))
87eqeq1d 2773 . . . 4 (𝑦 = 𝑁 → ((𝑦 + 𝑥) = 0 ↔ (𝑁 + 𝑥) = 0 ))
98rspcev 3460 . . 3 ((𝑁𝐵 ∧ (𝑁 + 𝑥) = 0 ) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
105, 6, 9syl2anc 573 . 2 ((𝜑𝑥𝐵) → ∃𝑦𝐵 (𝑦 + 𝑥) = 0 )
111, 2, 3, 4, 10isgrpd2e 17649 1 (𝜑𝐺 ∈ Grp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wrex 3062  cfv 6030  (class class class)co 6796  Basecbs 16064  +gcplusg 16149  0gc0g 16308  Mndcmnd 17502  Grpcgrp 17630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-iota 5993  df-fv 6038  df-ov 6799  df-grp 17633
This theorem is referenced by:  prdsgrpd  17733  oppggrp  17994
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