Theorem sgrpidmnd 18934

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 19149 and pwmndid 19148), 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 2760	. . . . . . . 8 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		sgrpidmnd.0	. . . . . . . 8 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grpidval 18846	. . . . . . 7 ⊢ 0 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))
5	4	eqeq2i 2773	. . . . . 6 ⊢ (𝑒 = 0 ↔ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))
6		eleq1w 2843	. . . . . . . . 9 ⊢ (𝑦 = 𝑒 → (𝑦 ∈ 𝐵 ↔ 𝑒 ∈ 𝐵))
7		oveq1 7428	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑒 → (𝑦(+g‘𝐺)𝑥) = (𝑒(+g‘𝐺)𝑥))
8	7	eqeq1d 2762	. . . . . . . . . 10 ⊢ (𝑦 = 𝑒 → ((𝑦(+g‘𝐺)𝑥) = 𝑥 ↔ (𝑒(+g‘𝐺)𝑥) = 𝑥))
9	8	ovanraleqv 7445	. . . . . . . . 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 6528	. . . . . . 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 3250	. . . . . . 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 3153	. . . 4 ⊢ ((𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))
16	15	reximi 3100	. . 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 18931	. 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  ∅c0 4278  ℩cio 6492  ‘cfv 6539  (class class class)co 7421  Basecbs 17393  +_gcplusg 17434  0_gc0g 17616  Smgrpcsgrp 18913  Mndcmnd 18929

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6541  df-fv 6547  df-ov 7424  df-0g 17618  df-mnd 18930

This theorem is used by: (None)