Theorem sgrpidmndm 13564

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 544	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 ∈ 𝐵)
2		simpllr 536	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑤 ∈ 𝑒)
3	2	19.8ad 1640	. . . . . . . . . 10 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → ∃𝑤 𝑤 ∈ 𝑒)
4		simplr 529	. . . . . . . . . . 11 ⊢ (((((𝐺 ∈ Smgrp ∧ 𝑒 ∈ 𝐵) ∧ 𝑤 ∈ 𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥 ∈ 𝐵) → 𝑒 = 0 )
5		sgrpidmnd.b	. . . . . . . . . . . . . 14 ⊢ 𝐵 = (Base‘𝐺)
6		eqid 2231	. . . . . . . . . . . . . 14 ⊢ (+g‘𝐺) = (+g‘𝐺)
7		sgrpidmnd.0	. . . . . . . . . . . . . 14 ⊢ 0 = (0g‘𝐺)
8	5, 6, 7	grpidvalg 13517	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ Smgrp → 0 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))
9	8	eqeq2d 2243	. . . . . . . . . . . 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 1204	. . . . . . . . 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 2292	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑒 → (𝑦 ∈ 𝐵 ↔ 𝑒 ∈ 𝐵))
15		oveq1 6035	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑒 → (𝑦(+g‘𝐺)𝑥) = (𝑒(+g‘𝐺)𝑥))
16	15	eqeq1d 2240	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑒 → ((𝑦(+g‘𝐺)𝑥) = 𝑥 ↔ (𝑒(+g‘𝐺)𝑥) = 𝑥))
17	16	ovanraleqv 6052	. . . . . . . . . . . 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 5325	. . . . . . . . . 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 2580	. . . . . . . . . 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 2606	. . . . . . 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 1867	. . . . 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 2635	. . 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 13562	. 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 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  ℩cio 5291  ‘cfv 5333  (class class class)co 6028  Basecbs 13143  +_gcplusg 13221  0_gc0g 13400  Smgrpcsgrp 13545  Mndcmnd 13560

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-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-cnex 8166  ax-resscn 8167  ax-1re 8169  ax-addrcl 8172

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-fv 5341  df-riota 5981  df-ov 6031  df-inn 9187  df-2 9245  df-ndx 13146  df-slot 13147  df-base 13149  df-plusg 13234  df-0g 13402  df-mnd 13561

This theorem is referenced by: (None)