Theorem sgrpidmndm 13716

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 548	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 ∈ 𝐵)
2		simpllr 540	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑤 ∈ 𝑒)
3	2	19.8ad 1644	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → ∃𝑤 𝑤 ∈ 𝑒)
4		simplr 533	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 = 0 )
5		sgrpidmnd.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐺)
6		eqid 2238	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
7		sgrpidmnd.0	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝐺)
8	5, 6, 7	grpidvalg 13676	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Smgrp → 0 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))
9	8	eqeq2d 2250	. . . . . . . . . . . 12 ⊢ (𝐺 ∈ Smgrp → (𝑒 = 0 ↔ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))))
10	9	ad4antr 498	. . . . . . . . . . 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 1208	. . . . . . . . 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 2299	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑒 → (𝑦 ∈ 𝐵 ↔ 𝑒 ∈ 𝐵))
15		oveq1 6086	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑒 → (𝑦(+g‘𝐺)𝑥) = (𝑒(+g‘𝐺)𝑥))
16	15	eqeq1d 2247	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑒 → ((𝑦(+g‘𝐺)𝑥) = 𝑥 ↔ (𝑒(+g‘𝐺)𝑥) = 𝑥))
17	16	ovanraleqv 6103	. . . . . . . . . . . 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 477	. . . . . . . . . . 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 5367	. . . . . . . . . 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 2597	. . . . . . . . . 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 390	. . . . . . . . 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 2623	. . . . . . 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 1872	. . . . 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 2652	. . 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 449	. 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 13714	. 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 1009   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ℩cio 5333  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  +_gcplusg 13414  0_gc0g 13593  Smgrpcsgrp 13699  Mndcmnd 13712

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-riota 6032  df-ov 6082  df-inn 9288  df-2 9346  df-ndx 13338  df-slot 13339  df-base 13341  df-plusg 13427  df-0g 13595  df-mnd 13713

This theorem is referenced by: (None)