MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  issgrp Structured version   Visualization version   GIF version

Theorem issgrp 17493
Description: The predicate "is a semigroup". (Contributed by FL, 2-Nov-2009.) (Revised by by AV, 6-Jan-2020.)
Hypotheses
Ref Expression
issgrp.b 𝐵 = (Base‘𝑀)
issgrp.o = (+g𝑀)
Assertion
Ref Expression
issgrp (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Distinct variable groups:   𝑥,𝐵,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥, ,𝑦,𝑧

Proof of Theorem issgrp
Dummy variables 𝑏 𝑔 𝑜 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6344 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) ∈ V)
2 fveq2 6332 . . . 4 (𝑔 = 𝑀 → (Base‘𝑔) = (Base‘𝑀))
3 issgrp.b . . . 4 𝐵 = (Base‘𝑀)
42, 3syl6eqr 2823 . . 3 (𝑔 = 𝑀 → (Base‘𝑔) = 𝐵)
5 fvexd 6344 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) ∈ V)
6 fveq2 6332 . . . . . 6 (𝑔 = 𝑀 → (+g𝑔) = (+g𝑀))
76adantr 466 . . . . 5 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = (+g𝑀))
8 issgrp.o . . . . 5 = (+g𝑀)
97, 8syl6eqr 2823 . . . 4 ((𝑔 = 𝑀𝑏 = 𝐵) → (+g𝑔) = )
10 simplr 744 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → 𝑏 = 𝐵)
11 id 22 . . . . . . . . . 10 (𝑜 = 𝑜 = )
12 oveq 6799 . . . . . . . . . 10 (𝑜 = → (𝑥𝑜𝑦) = (𝑥 𝑦))
13 eqidd 2772 . . . . . . . . . 10 (𝑜 = 𝑧 = 𝑧)
1411, 12, 13oveq123d 6814 . . . . . . . . 9 (𝑜 = → ((𝑥𝑜𝑦)𝑜𝑧) = ((𝑥 𝑦) 𝑧))
15 eqidd 2772 . . . . . . . . . 10 (𝑜 = 𝑥 = 𝑥)
16 oveq 6799 . . . . . . . . . 10 (𝑜 = → (𝑦𝑜𝑧) = (𝑦 𝑧))
1711, 15, 16oveq123d 6814 . . . . . . . . 9 (𝑜 = → (𝑥𝑜(𝑦𝑜𝑧)) = (𝑥 (𝑦 𝑧)))
1814, 17eqeq12d 2786 . . . . . . . 8 (𝑜 = → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
1918adantl 467 . . . . . . 7 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2010, 19raleqbidv 3301 . . . . . 6 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2110, 20raleqbidv 3301 . . . . 5 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
2210, 21raleqbidv 3301 . . . 4 (((𝑔 = 𝑀𝑏 = 𝐵) ∧ 𝑜 = ) → (∀𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
235, 9, 22sbcied2 3625 . . 3 ((𝑔 = 𝑀𝑏 = 𝐵) → ([(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
241, 4, 23sbcied2 3625 . 2 (𝑔 = 𝑀 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧)) ↔ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
25 df-sgrp 17492 . 2 SGrp = {𝑔 ∈ Mgm ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑜]𝑥𝑏𝑦𝑏𝑧𝑏 ((𝑥𝑜𝑦)𝑜𝑧) = (𝑥𝑜(𝑦𝑜𝑧))}
2624, 25elrab2 3518 1 (𝑀 ∈ SGrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥𝐵𝑦𝐵𝑧𝐵 ((𝑥 𝑦) 𝑧) = (𝑥 (𝑦 𝑧))))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  [wsbc 3587  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  Mgmcmgm 17448  SGrpcsgrp 17491
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  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  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-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039  df-ov 6796  df-sgrp 17492
This theorem is referenced by:  issgrpv  17494  issgrpn0  17495  isnsgrp  17496  sgrpmgm  17497  sgrpass  17498  sgrp0  17499  sgrp0b  17500  sgrp1  17501  sgrp2nmndlem4  17623  copissgrp  42336  nnsgrp  42345  sgrpplusgaopALT  42359  sgrp2sgrp  42392  lidlmsgrp  42454  2zrngasgrp  42468  2zrngmsgrp  42475
  Copyright terms: Public domain W3C validator