Theorem ismnddef 18826

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	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
ismnddef	⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((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.

Step	Hyp	Ref	Expression
1		fvex 6898	. . 3 ⊢ (Base‘𝑔) ∈ V
2		fvex 6898	. . 3 ⊢ (+g‘𝑔) ∈ V
3		fveq2 6885	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		ismnddef.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2818	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	eqeq2d 2776	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
7		fveq2 6885	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		ismnddef.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2818	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	eqeq2d 2776	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑝 = (+g‘𝑔) ↔ 𝑝 = + ))
11	6, 10	anbi12d 644	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g‘𝑔)) ↔ (𝑏 = 𝐵 ∧ 𝑝 = + )))
12		simpl 488	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → 𝑏 = 𝐵)
13		oveq 7425	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
14	13	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
15		oveq 7425	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
16	15	eqeq1d 2767	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑎𝑝𝑒) = 𝑎 ↔ (𝑎 + 𝑒) = 𝑎))
17	14, 16	anbi12d 644	. . . . . . 7 ⊢ (𝑝 = + → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
18	17	adantl 487	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
19	12, 18	raleqbidv 3340	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
20	12, 19	rexeqbidv 3341	. . . 4 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
21	11, 20	biimtrdi 256	. . 3 ⊢ (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g‘𝑔)) → (∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎))))
22	1, 2, 21	sbc2iedv 3822	. 2 ⊢ (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
23		df-mnd 18825	. 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
24	22, 23	elrab2 3656	1 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  [wsbc 3746  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  Smgrpcsgrp 18808  Mndcmnd 18824

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-nul 5271

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7422  df-mnd 18825

This theorem is used by:  ismnd  18827  isnmnd  18828  sgrpidmnd  18829  mndsgrp  18830  mnd1  18874  efmndmnd  18985  smndex1mnd  19009  isringrng  20415  2zrngamnd  49069