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Theorem isgrpi 12905
Description: Properties that determine a group. 𝑁 (negative) is normally dependent on 𝑥 i.e. read it as 𝑁(𝑥). (Contributed by NM, 3-Sep-2011.)
Hypotheses
Ref Expression
isgrpi.b 𝐵 = (Base‘𝐺)
isgrpi.p + = (+g𝐺)
isgrpi.c ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
isgrpi.a ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
isgrpi.z 0𝐵
isgrpi.i (𝑥𝐵 → ( 0 + 𝑥) = 𝑥)
isgrpi.n (𝑥𝐵𝑁𝐵)
isgrpi.j (𝑥𝐵 → (𝑁 + 𝑥) = 0 )
Assertion
Ref Expression
isgrpi 𝐺 ∈ Grp
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝐺,𝑦,𝑧   𝑦,𝑁   𝑥, + ,𝑦,𝑧   𝑥, 0 ,𝑦,𝑧
Allowed substitution hints:   𝑁(𝑥,𝑧)

Proof of Theorem isgrpi
StepHypRef Expression
1 isgrpi.b . . . 4 𝐵 = (Base‘𝐺)
21a1i 9 . . 3 (⊤ → 𝐵 = (Base‘𝐺))
3 isgrpi.p . . . 4 + = (+g𝐺)
43a1i 9 . . 3 (⊤ → + = (+g𝐺))
5 isgrpi.c . . . 4 ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
653adant1 1015 . . 3 ((⊤ ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
7 isgrpi.a . . . 4 ((𝑥𝐵𝑦𝐵𝑧𝐵) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
87adantl 277 . . 3 ((⊤ ∧ (𝑥𝐵𝑦𝐵𝑧𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
9 isgrpi.z . . . 4 0𝐵
109a1i 9 . . 3 (⊤ → 0𝐵)
11 isgrpi.i . . . 4 (𝑥𝐵 → ( 0 + 𝑥) = 𝑥)
1211adantl 277 . . 3 ((⊤ ∧ 𝑥𝐵) → ( 0 + 𝑥) = 𝑥)
13 isgrpi.n . . . 4 (𝑥𝐵𝑁𝐵)
1413adantl 277 . . 3 ((⊤ ∧ 𝑥𝐵) → 𝑁𝐵)
15 isgrpi.j . . . 4 (𝑥𝐵 → (𝑁 + 𝑥) = 0 )
1615adantl 277 . . 3 ((⊤ ∧ 𝑥𝐵) → (𝑁 + 𝑥) = 0 )
172, 4, 6, 8, 10, 12, 14, 16isgrpd 12904 . 2 (⊤ → 𝐺 ∈ Grp)
1817mptru 1362 1 𝐺 ∈ Grp
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wtru 1354  wcel 2148  cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  Grpcgrp 12882
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mgm 12780  df-sgrp 12813  df-mnd 12823  df-grp 12885
This theorem is referenced by:  cncrng  13548
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