Theorem ismnd 13681

Description: The predicate "is a monoid". This is the defining theorem of a monoid by showing that a set is a monoid if and only if it is a set equipped with a closed, everywhere defined internal operation (so, a magma, see mndcl 13685), whose operation is associative (so, a semigroup, see also mndass 13686) and has a two-sided neutral element (see mndid 13687). (Contributed by Mario Carneiro, 6-Jan-2015.) (Revised by AV, 1-Feb-2020.)

Hypotheses

Ref	Expression
ismnd.b	⊢ 𝐵 = (Base‘𝐺)
ismnd.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
ismnd	⊢ (𝐺 ∈ Mnd ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Distinct variable groups:   𝐵,𝑎,𝑏,𝑐   𝐵,𝑒,𝑎   𝐺,𝑎,𝑏,𝑐   + ,𝑎,𝑒   + ,𝑏,𝑐

Allowed substitution hint:   𝐺(𝑒)

Proof of Theorem ismnd

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ismnd.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		ismnd.p	. . 3 ⊢ + = (+g‘𝐺)
3	1, 2	ismnddef 13680	. 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
4		rexm 3614	. . . . 5 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → ∃𝑒 𝑒 ∈ 𝐵)
5		eleq1w 2295	. . . . . 6 ⊢ (𝑒 = 𝑤 → (𝑒 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵))
6	5	cbvexv 1970	. . . . 5 ⊢ (∃𝑒 𝑒 ∈ 𝐵 ↔ ∃𝑤 𝑤 ∈ 𝐵)
7	4, 6	sylib 122	. . . 4 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → ∃𝑤 𝑤 ∈ 𝐵)
8	1	basmex 13362	. . . . 5 ⊢ (𝑤 ∈ 𝐵 → 𝐺 ∈ V)
9	8	exlimiv 1647	. . . 4 ⊢ (∃𝑤 𝑤 ∈ 𝐵 → 𝐺 ∈ V)
10	1, 2	issgrpv 13668	. . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ Smgrp ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))))
11	7, 9, 10	3syl 17	. . 3 ⊢ (∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎) → (𝐺 ∈ Smgrp ↔ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))))
12	11	pm5.32ri 455	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)) ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))
13	3, 12	bitri 184	1 ⊢ (𝐺 ∈ Mnd ↔ (∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝐵 ((𝑎 + 𝑏) ∈ 𝐵 ∧ ∀𝑐 ∈ 𝐵 ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐))) ∧ ∃𝑒 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑒 + 𝑎) = 𝑎 ∧ (𝑎 + 𝑒) = 𝑎)))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  ∀wral 2522  ∃wrex 2523  Vcvv 2815  ‘cfv 5359  (class class class)co 6060  Basecbs 13302  +_gcplusg 13380  Smgrpcsgrp 13665  Mndcmnd 13678

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4234  ax-pow 4293  ax-pr 4328  ax-un 4560  ax-cnex 8236  ax-resscn 8237  ax-1re 8239  ax-addrcl 8242

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3677  df-sn 3701  df-pr 3702  df-op 3704  df-uni 3921  df-int 3956  df-br 4116  df-opab 4178  df-mpt 4179  df-id 4420  df-xp 4762  df-rel 4763  df-cnv 4764  df-co 4765  df-dm 4766  df-rn 4767  df-res 4768  df-iota 5319  df-fun 5361  df-fn 5362  df-fv 5367  df-ov 6063  df-inn 9260  df-2 9318  df-ndx 13305  df-slot 13306  df-base 13308  df-plusg 13393  df-mgm 13625  df-sgrp 13666  df-mnd 13679

This theorem is referenced by:  mndid  13687  ismndd  13699  mndpropd  13702  mhmmnd  13868