Theorem sgrpidmnd 18960

Description: A semigroup with an identity element which is not the empty set is a monoid. Of course there could be monoids with the empty set as identity element (see, for example, the monoid of the power set of a class under union, pwmnd 19175 and pwmndid 19174), but these cannot be proven to be monoids with this theorem. (Contributed by AV, 29-Jan-2024.)

Hypotheses

Ref	Expression
sgrpidmnd.b	⊢ 𝐵 = (Base‘𝐺)
sgrpidmnd.0	⊢ 0 = (0g‘𝐺)

Assertion

Ref	Expression
sgrpidmnd	⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)

Distinct variable groups:   𝐵,𝑒   𝑒,𝐺

Allowed substitution hint:   0 (𝑒)

Proof of Theorem sgrpidmnd

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sgrpidmnd.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2763	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		sgrpidmnd.0	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grpidval 18872	. . . . . . 7 ⊢ 0 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))
5	4	eqeq2i 2776	. . . . . 6 ⊢ (𝑒 = 0 ↔ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))
6		eleq1w 2847	. . . . . . . . 9 ⊢ (𝑦 = 𝑒 → (𝑦 ∈ 𝐵 ↔ 𝑒 ∈ 𝐵))
7		oveq1 7453	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑒 → (𝑦(+g‘𝐺)𝑥) = (𝑒(+g‘𝐺)𝑥))
8	7	eqeq1d 2765	. . . . . . . . . 10 ⊢ (𝑦 = 𝑒 → ((𝑦(+g‘𝐺)𝑥) = 𝑥 ↔ (𝑒(+g‘𝐺)𝑥) = 𝑥))
9	8	ovanraleqv 7470	. . . . . . . . 9 ⊢ (𝑦 = 𝑒 → (∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥) ↔ ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
10	6, 9	anbi12d 644	. . . . . . . 8 ⊢ (𝑦 = 𝑒 → ((𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)) ↔ (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))))
11	10	iotan0 6550	. . . . . . 7 ⊢ ((𝑒 ≠ ∅ ∧ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))) → (𝑒 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
12		rsp 3254	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥) → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
13	11, 12	simpl2im 513	. . . . . 6 ⊢ ((𝑒 ≠ ∅ ∧ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))) → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
14	5, 13	sylan2b 606	. . . . 5 ⊢ ((𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
15	14	ralrimiv 3157	. . . 4 ⊢ ((𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))
16	15	reximi 3104	. . 3 ⊢ (∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))
17	16	anim2i 629	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
18	1, 2	ismnddef 18957	. 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
19	17, 18	sylibr 237	1 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  ∅c0 4281  ℩cio 6513  ‘cfv 6561  (class class class)co 7446  Basecbs 17419  +_gcplusg 17460  0_gc0g 17642  Smgrpcsgrp 18939  Mndcmnd 18955

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

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-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-0g 17644  df-mnd 18956

This theorem is used by: (None)