Theorem sgrpidmndm 13001

Description: A semigroup with an identity element which is inhabited is a monoid. Of course there could be monoids with the empty set as identity element, 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
sgrpidmndm	⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)

Distinct variable groups:   𝐵,𝑒,𝑤   𝑒,𝐺,𝑤   𝑤, 0   𝑤,𝑒

Allowed substitution hint:   0 (𝑒)

Proof of Theorem sgrpidmndm

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

Step	Hyp	Ref	Expression
1		simp-4r 542	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 ∈ 𝐵)
2		simpllr 534	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑤 ∈ 𝑒)
3	2	19.8ad 1602	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → ∃𝑤 𝑤 ∈ 𝑒)
4		simplr 528	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 = 0 )
5		sgrpidmnd.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐺)
6		eqid 2193	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
7		sgrpidmnd.0	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝐺)
8	5, 6, 7	grpidvalg 12956	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Smgrp → 0 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))
9	8	eqeq2d 2205	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Smgrp → (𝑒 = 0 ↔ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))))
10	9	ad4antr 494	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → (𝑒 = 0 ↔ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))))
11	4, 10	mpbid 147	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))
12	1, 3, 11	3jca 1179	. . . . . . . . 9 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → (𝑒 ∈ 𝐵 ∧ ∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))))
13		simpr 110	. . . . . . . . 9 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
14		eleq1w 2254	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑒 → (𝑦 ∈ 𝐵 ↔ 𝑒 ∈ 𝐵))
15		oveq1 5925	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑒 → (𝑦(+g‘𝐺)𝑥) = (𝑒(+g‘𝐺)𝑥))
16	15	eqeq1d 2202	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑒 → ((𝑦(+g‘𝐺)𝑥) = 𝑥 ↔ (𝑒(+g‘𝐺)𝑥) = 𝑥))
17	16	ovanraleqv 5942	. . . . . . . . . . . 12 ⊢ (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)))
18	14, 17	anbi12d 473	. . . . . . . . . . 11 ⊢ (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))))
19	18	iotam 5246	. . . . . . . . . 10 ⊢ ((𝑒 ∈ 𝐵 ∧ ∃𝑤 𝑤 ∈ 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)))
20		rsp 2541	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥) → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
21	19, 20	simpl2im 386	. . . . . . . . 9 ⊢ ((𝑒 ∈ 𝐵 ∧ ∃𝑤 𝑤 ∈ 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)))
22	12, 13, 21	sylc 62	. . . . . . . 8 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))
23	22	ralrimiva 2567	. . . . . . 7 ⊢ ((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))
24	23	exp31 364	. . . . . 6 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) → (𝑤 ∈ 𝑒 → (𝑒 = 0 → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))))
25	24	exlimdv 1830	. . . . 5 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) → (∃𝑤 𝑤 ∈ 𝑒 → (𝑒 = 0 → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))))
26	25	impd 254	. . . 4 ⊢ ((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) → ((∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 ) → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
27	26	reximdva 2596	. . 3 ⊢ (𝐺 ∈ Smgrp → (∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 ) → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
28	27	imdistani 445	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 )) → (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
29	5, 6	ismnddef 12999	. 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
30	28, 29	sylibr 134	1 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (∃𝑤 𝑤 ∈ 𝑒 ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ℩cio 5213  ‘cfv 5254  (class class class)co 5918  Basecbs 12618  +_gcplusg 12695  0_gc0g 12867  Smgrpcsgrp 12984  Mndcmnd 12997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-cnex 7963  ax-resscn 7964  ax-1re 7966  ax-addrcl 7969

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-iota 5215  df-fun 5256  df-fn 5257  df-fv 5262  df-riota 5873  df-ov 5921  df-inn 8983  df-2 9041  df-ndx 12621  df-slot 12622  df-base 12624  df-plusg 12708  df-0g 12869  df-mnd 12998

This theorem is referenced by: (None)