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Theorem ismnddef 18762
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 ∈ Smgrp ∧ ∃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 6920 . . 3 (Base‘𝑔) ∈ V
2 fvex 6920 . . 3 (+g𝑔) ∈ V
3 fveq2 6907 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 ismnddef.b . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2793 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
65eqeq2d 2746 . . . . 5 (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
7 fveq2 6907 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 ismnddef.p . . . . . . 7 + = (+g𝐺)
97, 8eqtr4di 2793 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
109eqeq2d 2746 . . . . 5 (𝑔 = 𝐺 → (𝑝 = (+g𝑔) ↔ 𝑝 = + ))
116, 10anbi12d 632 . . . 4 (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g𝑔)) ↔ (𝑏 = 𝐵𝑝 = + )))
12 simpl 482 . . . . 5 ((𝑏 = 𝐵𝑝 = + ) → 𝑏 = 𝐵)
13 oveq 7437 . . . . . . . . 9 (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
1413eqeq1d 2737 . . . . . . . 8 (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
15 oveq 7437 . . . . . . . . 9 (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
1615eqeq1d 2737 . . . . . . . 8 (𝑝 = + → ((𝑎𝑝𝑒) = 𝑎 ↔ (𝑎 + 𝑒) = 𝑎))
1714, 16anbi12d 632 . . . . . . 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 3344 . . . . 5 ((𝑏 = 𝐵𝑝 = + ) → (∀𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
2012, 19rexeqbidv 3345 . . . 4 ((𝑏 = 𝐵𝑝 = + ) → (∃𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
2111, 20biimtrdi 253 . . 3 (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g𝑔)) → (∃𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))))
221, 2, 21sbc2iedv 3876 . 2 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
23 df-mnd 18761 . 2 Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
2422, 23elrab2 3698 1 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  wrex 3068  [wsbc 3791  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  Smgrpcsgrp 18744  Mndcmnd 18760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-iota 6516  df-fv 6571  df-ov 7434  df-mnd 18761
This theorem is referenced by:  ismnd  18763  isnmnd  18764  sgrpidmnd  18765  mndsgrp  18766  mnd1  18805  efmndmnd  18915  smndex1mnd  18936  isringrng  20301  2zrngamnd  48091
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