Theorem sgrp0 12996

Description: Any set with an empty base set and any group operation is a semigroup. (Contributed by AV, 28-Aug-2021.)

Assertion

Ref	Expression
sgrp0	⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp)

Proof of Theorem sgrp0

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

Step	Hyp	Ref	Expression
1		mgm0 12955	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Mgm)
2		rzal 3545	. . 3 ⊢ ((Base‘𝑀) = ∅ → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
3	2	adantl 277	. 2 ⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧)))
4		eqid 2193	. . 3 ⊢ (Base‘𝑀) = (Base‘𝑀)
5		eqid 2193	. . 3 ⊢ (+g‘𝑀) = (+g‘𝑀)
6	4, 5	issgrp 12989	. 2 ⊢ (𝑀 ∈ Smgrp ↔ (𝑀 ∈ Mgm ∧ ∀𝑥 ∈ (Base‘𝑀)∀𝑦 ∈ (Base‘𝑀)∀𝑧 ∈ (Base‘𝑀)((𝑥(+g‘𝑀)𝑦)(+g‘𝑀)𝑧) = (𝑥(+g‘𝑀)(𝑦(+g‘𝑀)𝑧))))
7	1, 3, 6	sylanbrc 417	1 ⊢ ((𝑀 ∈ 𝑉 ∧ (Base‘𝑀) = ∅) → 𝑀 ∈ Smgrp)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∅c0 3447  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  +_gcplusg 12698  Mgmcmgm 12940  Smgrpcsgrp 12987

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-in1 615  ax-in2 616  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 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

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-ne 2365  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-mgm 12942  df-sgrp 12988

This theorem is referenced by: (None)