Theorem ismnddef 18629

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 6904	. . 3 ⊢ (Base‘𝑔) ∈ V
2		fvex 6904	. . 3 ⊢ (+g‘𝑔) ∈ V
3		fveq2 6891	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		ismnddef.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	eqtr4di 2790	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	eqeq2d 2743	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
7		fveq2 6891	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		ismnddef.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	7, 8	eqtr4di 2790	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	eqeq2d 2743	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑝 = (+g‘𝑔) ↔ 𝑝 = + ))
11	6, 10	anbi12d 631	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g‘𝑔)) ↔ (𝑏 = 𝐵 ∧ 𝑝 = + )))
12		simpl 483	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → 𝑏 = 𝐵)
13		oveq 7417	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
14	13	eqeq1d 2734	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
15		oveq 7417	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
16	15	eqeq1d 2734	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑎𝑝𝑒) = 𝑎 ↔ (𝑎 + 𝑒) = 𝑎))
17	14, 16	anbi12d 631	. . . . . . 7 ⊢ (𝑝 = + → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
18	17	adantl 482	. . . . . 6 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
19	12, 18	raleqbidv 3342	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
20	12, 19	rexeqbidv 3343	. . . 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	syl6bi 252	. . 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 3862	. 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 18628	. 2 ⊢ Mnd = {𝑔 ∈ Smgrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
24	22, 23	elrab2 3686	1 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070  [wsbc 3777  ‘cfv 6543  (class class class)co 7411  Basecbs 17146  +_gcplusg 17199  Smgrpcsgrp 18611  Mndcmnd 18627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-nul 5306

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-iota 6495  df-fv 6551  df-ov 7414  df-mnd 18628

This theorem is referenced by:  ismnd  18630  isnmnd  18631  sgrpidmnd  18632  mndsgrp  18633  mnd1  18669  efmndmnd  18772  smndex1mnd  18793  isringrng  46736  2zrngamnd  46918