Theorem ismnddef 17296

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 ∈ SGrp ∧ ∃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 6201	. . 3 ⊢ (Base‘𝑔) ∈ V
2		fvex 6201	. . 3 ⊢ (+g‘𝑔) ∈ V
3		fveq2 6191	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
4		ismnddef.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝐺)
5	3, 4	syl6eqr 2674	. . . . . 6 ⊢ (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
6	5	eqeq2d 2632	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑏 = (Base‘𝑔) ↔ 𝑏 = 𝐵))
7		fveq2 6191	. . . . . . 7 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = (+g‘𝐺))
8		ismnddef.p	. . . . . . 7 ⊢ + = (+g‘𝐺)
9	7, 8	syl6eqr 2674	. . . . . 6 ⊢ (𝑔 = 𝐺 → (+g‘𝑔) = + )
10	9	eqeq2d 2632	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑝 = (+g‘𝑔) ↔ 𝑝 = + ))
11	6, 10	anbi12d 747	. . . 4 ⊢ (𝑔 = 𝐺 → ((𝑏 = (Base‘𝑔) ∧ 𝑝 = (+g‘𝑔)) ↔ (𝑏 = 𝐵 ∧ 𝑝 = + )))
12		simpl 473	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → 𝑏 = 𝐵)
13		oveq 6656	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑒𝑝𝑎) = (𝑒 + 𝑎))
14	13	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑒𝑝𝑎) = 𝑎 ↔ (𝑒 + 𝑎) = 𝑎))
15		oveq 6656	. . . . . . . . 9 ⊢ (𝑝 = + → (𝑎𝑝𝑒) = (𝑎 + 𝑒))
16	15	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑝 = + → ((𝑎𝑝𝑒) = 𝑎 ↔ (𝑎 + 𝑒) = 𝑎))
17	14, 16	anbi12d 747	. . . . . . 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 3152	. . . . 5 ⊢ ((𝑏 = 𝐵 ∧ 𝑝 = + ) → (∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎) ↔ ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
20	12, 19	rexeqbidv 3153	. . . 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 243	. . 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 3506	. 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 17295	. 2 ⊢ Mnd = {𝑔 ∈ SGrp ∣ [(Base‘𝑔) / 𝑏][(+g‘𝑔) / 𝑝]∃𝑒 ∈ 𝑏 ∀𝑎 ∈ 𝑏 ((𝑒𝑝𝑎) = 𝑎 ∧ (𝑎𝑝𝑒) = 𝑎)}
24	22, 23	elrab2 3366	1 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ SGrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  [wsbc 3435  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  SGrpcsgrp 17283  Mndcmnd 17294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653  df-mnd 17295

This theorem is referenced by:  ismnd  17297  isnmnd  17298  mndsgrp  17299  mnd1  17331  isringrng  41881  2zrngamnd  41941