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Theorem ismnddef 17119
 Description: The predicate "is a monoid", corresponding 1-to-1 to the definition. (Contributed by FL, 2-Nov-2009.) (Revised by AV, 1-Feb-2020.)
Hypotheses
Ref Expression
ismnddef.b 𝐵 = (Base‘𝐺)
ismnddef.p + = (+g𝐺)
Assertion
Ref Expression
ismnddef (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
Distinct variable groups:   𝐵,𝑎,𝑒   + ,𝑎,𝑒
Allowed substitution hints:   𝐺(𝑒,𝑎)

Proof of Theorem ismnddef
Dummy variables 𝑏 𝑔 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvex 6113 . . 3 (Base‘𝑔) ∈ V
2 fvex 6113 . . 3 (+g𝑔) ∈ V
3 fveq2 6103 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 ismnddef.b . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4syl6eqr 2662 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
65eqeq2d 2620 . . . . 5 (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
7 fveq2 6103 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 ismnddef.p . . . . . . 7 + = (+g𝐺)
97, 8syl6eqr 2662 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
109eqeq2d 2620 . . . . 5 (𝑔 = 𝐺 → (𝑝 = (+g𝑔) ↔ 𝑝 = + ))
116, 10anbi12d 743 . . . 4 (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g𝑔)) ↔ (𝑏 = 𝐵𝑝 = + )))
12 simpl 472 . . . . 5 ((𝑏 = 𝐵𝑝 = + ) → 𝑏 = 𝐵)
13 oveq 6555 . . . . . . . . 9 (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
1413eqeq1d 2612 . . . . . . . 8 (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
15 oveq 6555 . . . . . . . . 9 (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
1615eqeq1d 2612 . . . . . . . 8 (𝑝 = + → ((𝑎𝑝𝑒) = 𝑎 ↔ (𝑎 + 𝑒) = 𝑎))
1714, 16anbi12d 743 . . . . . . 7 (𝑝 = + → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
1817adantl 481 . . . . . 6 ((𝑏 = 𝐵𝑝 = + ) → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
1912, 18raleqbidv 3129 . . . . 5 ((𝑏 = 𝐵𝑝 = + ) → (∀𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
2012, 19rexeqbidv 3130 . . . 4 ((𝑏 = 𝐵𝑝 = + ) → (∃𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
2111, 20syl6bi 242 . . 3 (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g𝑔)) → (∃𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))))
221, 2, 21sbc2iedv 3473 . 2 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
23 df-mnd 17118 . 2 Mnd = {𝑔 ∈ SGrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
2422, 23elrab2 3333 1 (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  [wsbc 3402  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  SGrpcsgrp 17106  Mndcmnd 17117 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-mnd 17118 This theorem is referenced by:  ismnd  17120  isnmnd  17121  mndsgrp  17122  mnd1  17154  isringrng  41671  2zrngamnd  41731
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