Theorem sgrpidmnd 18479

Description: A semigroup with an identity element which is not the empty set is a monoid. Of course there could be monoids with the empty set as identity element (see, for example, the monoid of the power set of a class under union, pwmnd 18664 and pwmndid 18663), 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
sgrpidmnd	⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)

Distinct variable groups:   𝐵,𝑒   𝑒,𝐺

Allowed substitution hint:   0 (𝑒)

Proof of Theorem sgrpidmnd

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

Step	Hyp	Ref	Expression
1		sgrpidmnd.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐺)
2		eqid 2736	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
3		sgrpidmnd.0	. . . . . . . . . 10 ⊢ 0 = (0g‘𝐺)
4	1, 2, 3	grpidval 18434	. . . . . . . . 9 ⊢ 0 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥)))
5	4	eqeq2i 2749	. . . . . . . 8 ⊢ (𝑒 = 0 ↔ 𝑒 = (℩𝑦(𝑦 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑦(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑦) = 𝑥))))
6		eleq1w 2819	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑒 → (𝑦 ∈ 𝐵 ↔ 𝑒 ∈ 𝐵))
7		oveq1 7336	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑒 → (𝑦(+g‘𝐺)𝑥) = (𝑒(+g‘𝐺)𝑥))
8	7	eqeq1d 2738	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑒 → ((𝑦(+g‘𝐺)𝑥) = 𝑥 ↔ (𝑒(+g‘𝐺)𝑥) = 𝑥))
9	8	ovanraleqv 7353	. . . . . . . . . . . . 13 ⊢ (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)))
10	6, 9	anbi12d 631	. . . . . . . . . . . 12 ⊢ (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))))
11	10	iotan0 6463	. . . . . . . . . . 11 ⊢ ((𝑒 ∈ 𝐵 ∧ 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)))
12		rsp 3226	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥) → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
13	11, 12	simpl2im 504	. . . . . . . . . 10 ⊢ ((𝑒 ∈ 𝐵 ∧ 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)))
14	13	3expb 1119	. . . . . . . . 9 ⊢ ((𝑒 ∈ 𝐵 ∧ (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)))
15	14	expcom 414	. . . . . . . 8 ⊢ ((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))))
16	5, 15	sylan2b 594	. . . . . . 7 ⊢ ((𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → (𝑒 ∈ 𝐵 → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))))
17	16	impcom 408	. . . . . 6 ⊢ ((𝑒 ∈ 𝐵 ∧ (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → (𝑥 ∈ 𝐵 → ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
18	17	ralrimiv 3138	. . . . 5 ⊢ ((𝑒 ∈ 𝐵 ∧ (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))
19	18	ex 413	. . . 4 ⊢ (𝑒 ∈ 𝐵 → ((𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
20	19	reximia 3080	. . 3 ⊢ (∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥))
21	20	anim2i 617	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
22	1, 2	ismnddef 18476	. 2 ⊢ (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒(+g‘𝐺)𝑥) = 𝑥 ∧ (𝑥(+g‘𝐺)𝑒) = 𝑥)))
23	21, 22	sylibr 233	1 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑒 ∈ 𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  ∅c0 4268  ℩cio 6423  ‘cfv 6473  (class class class)co 7329  Basecbs 17001  +_gcplusg 17051  0_gc0g 17239  Smgrpcsgrp 18463  Mndcmnd 18474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-sep 5240  ax-nul 5247  ax-pr 5369

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-sbc 3727  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4269  df-if 4473  df-sn 4573  df-pr 4575  df-op 4579  df-uni 4852  df-br 5090  df-opab 5152  df-mpt 5173  df-id 5512  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-iota 6425  df-fun 6475  df-fv 6481  df-ov 7332  df-0g 17241  df-mnd 18475

This theorem is referenced by: (None)