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Theorem sgrpidmnd 18561
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 18747 and pwmndid 18746), 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 = (0g𝐺)
Assertion
Ref Expression
sgrpidmnd ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (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.
StepHypRef Expression
1 sgrpidmnd.b . . . . . . . . . 10 𝐵 = (Base‘𝐺)
2 eqid 2736 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
3 sgrpidmnd.0 . . . . . . . . . 10 0 = (0g𝐺)
41, 2, 3grpidval 18516 . . . . . . . . 9 0 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))
54eqeq2i 2749 . . . . . . . 8 (𝑒 = 0𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥))))
6 eleq1w 2820 . . . . . . . . . . . . 13 (𝑦 = 𝑒 → (𝑦𝐵𝑒𝐵))
7 oveq1 7364 . . . . . . . . . . . . . . 15 (𝑦 = 𝑒 → (𝑦(+g𝐺)𝑥) = (𝑒(+g𝐺)𝑥))
87eqeq1d 2738 . . . . . . . . . . . . . 14 (𝑦 = 𝑒 → ((𝑦(+g𝐺)𝑥) = 𝑥 ↔ (𝑒(+g𝐺)𝑥) = 𝑥))
98ovanraleqv 7381 . . . . . . . . . . . . 13 (𝑦 = 𝑒 → (∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥) ↔ ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
106, 9anbi12d 631 . . . . . . . . . . . 12 (𝑦 = 𝑒 → ((𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)) ↔ (𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))))
1110iotan0 6486 . . . . . . . . . . 11 ((𝑒𝐵𝑒 ≠ ∅ ∧ 𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))) → (𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
12 rsp 3230 . . . . . . . . . . 11 (∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
1311, 12simpl2im 504 . . . . . . . . . 10 ((𝑒𝐵𝑒 ≠ ∅ ∧ 𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
14133expb 1120 . . . . . . . . 9 ((𝑒𝐵 ∧ (𝑒 ≠ ∅ ∧ 𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥))))) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
1514expcom 414 . . . . . . . 8 ((𝑒 ≠ ∅ ∧ 𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))) → (𝑒𝐵 → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))))
165, 15sylan2b 594 . . . . . . 7 ((𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → (𝑒𝐵 → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))))
1716impcom 408 . . . . . 6 ((𝑒𝐵 ∧ (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
1817ralrimiv 3142 . . . . 5 ((𝑒𝐵 ∧ (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))
1918ex 413 . . . 4 (𝑒𝐵 → ((𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
2019reximia 3084 . . 3 (∃𝑒𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → ∃𝑒𝐵𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))
2120anim2i 617 . 2 ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
221, 2ismnddef 18558 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
2321, 22sylibr 233 1 ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  c0 4282  cio 6446  cfv 6496  (class class class)co 7357  Basecbs 17083  +gcplusg 17133  0gc0g 17321  Smgrpcsgrp 18545  Mndcmnd 18556
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-0g 17323  df-mnd 18557
This theorem is referenced by: (None)
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