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Theorem ismnddef 17504
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 6342 . . 3 (Base‘𝑔) ∈ V
2 fvex 6342 . . 3 (+g𝑔) ∈ V
3 fveq2 6332 . . . . . . 7 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4 ismnddef.b . . . . . . 7 𝐵 = (Base‘𝐺)
53, 4syl6eqr 2823 . . . . . 6 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
65eqeq2d 2781 . . . . 5 (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
7 fveq2 6332 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 ismnddef.p . . . . . . 7 + = (+g𝐺)
97, 8syl6eqr 2823 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
109eqeq2d 2781 . . . . 5 (𝑔 = 𝐺 → (𝑝 = (+g𝑔) ↔ 𝑝 = + ))
116, 10anbi12d 616 . . . 4 (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g𝑔)) ↔ (𝑏 = 𝐵𝑝 = + )))
12 simpl 468 . . . . 5 ((𝑏 = 𝐵𝑝 = + ) → 𝑏 = 𝐵)
13 oveq 6799 . . . . . . . . 9 (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
1413eqeq1d 2773 . . . . . . . 8 (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
15 oveq 6799 . . . . . . . . 9 (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
1615eqeq1d 2773 . . . . . . . 8 (𝑝 = + → ((𝑎𝑝𝑒) = 𝑎 ↔ (𝑎 + 𝑒) = 𝑎))
1714, 16anbi12d 616 . . . . . . 7 (𝑝 = + → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
1817adantl 467 . . . . . 6 ((𝑏 = 𝐵𝑝 = + ) → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
1912, 18raleqbidv 3301 . . . . 5 ((𝑏 = 𝐵𝑝 = + ) → (∀𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
2012, 19rexeqbidv 3302 . . . 4 ((𝑏 = 𝐵𝑝 = + ) → (∃𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
2111, 20syl6bi 243 . . 3 (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g𝑔)) → (∃𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))))
221, 2, 21sbc2iedv 3656 . 2 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
23 df-mnd 17503 . 2 Mnd = {𝑔 ∈ SGrp ∣ [(Base‘𝑔) / 𝑏][(+g𝑔) / 𝑝]𝑒𝑏𝑎𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
2422, 23elrab2 3518 1 (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒𝐵𝑎𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wcel 2145  wral 3061  wrex 3062  [wsbc 3587  cfv 6031  (class class class)co 6793  Basecbs 16064  +gcplusg 16149  SGrpcsgrp 17491  Mndcmnd 17502
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 837  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-mnd 17503
This theorem is referenced by:  ismnd  17505  isnmnd  17506  mndsgrp  17507  mnd1  17539  isringrng  42409  2zrngamnd  42469
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