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Theorem ismnddef 18724
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 6913 . . 3 (Base‘𝑔) ∈ V
2 fvex 6913 . . 3 (+g𝑔) ∈ V
3 fveq2 6900 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 ismnddef.b . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4eqtr4di 2783 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
65eqeq2d 2736 . . . . 5 (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
7 fveq2 6900 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 ismnddef.p . . . . . . 7 + = (+g𝐺)
97, 8eqtr4di 2783 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
109eqeq2d 2736 . . . . 5 (𝑔 = 𝐺 → (𝑝 = (+g𝑔) ↔ 𝑝 = + ))
116, 10anbi12d 630 . . . 4 (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g𝑔)) ↔ (𝑏 = 𝐵𝑝 = + )))
12 simpl 481 . . . . 5 ((𝑏 = 𝐵𝑝 = + ) → 𝑏 = 𝐵)
13 oveq 7429 . . . . . . . . 9 (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
1413eqeq1d 2727 . . . . . . . 8 (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
15 oveq 7429 . . . . . . . . 9 (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
1615eqeq1d 2727 . . . . . . . 8 (𝑝 = + → ((𝑎𝑝𝑒) = 𝑎 ↔ (𝑎 + 𝑒) = 𝑎))
1714, 16anbi12d 630 . . . . . . 7 (𝑝 = + → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
1817adantl 480 . . . . . 6 ((𝑏 = 𝐵𝑝 = + ) → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
1912, 18raleqbidv 3329 . . . . 5 ((𝑏 = 𝐵𝑝 = + ) → (∀𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
2012, 19rexeqbidv 3330 . . . 4 ((𝑏 = 𝐵𝑝 = + ) → (∃𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
2111, 20biimtrdi 252 . . 3 (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g𝑔)) → (∃𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))))
221, 2, 21sbc2iedv 3860 . 2 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
23 df-mnd 18723 . 2 Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
2422, 23elrab2 3683 1 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059  [wsbc 3775  cfv 6553  (class class class)co 7423  Basecbs 17208  +gcplusg 17261  Smgrpcsgrp 18706  Mndcmnd 18722
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-nul 5310
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-dif 3949  df-un 3951  df-ss 3963  df-nul 4325  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7426  df-mnd 18723
This theorem is referenced by:  ismnd  18725  isnmnd  18726  sgrpidmnd  18727  mndsgrp  18728  mnd1  18764  efmndmnd  18874  smndex1mnd  18895  isringrng  20261  2zrngamnd  47561
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