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Theorem sgrpidmnd 18752
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 18950 and pwmndid 18949), 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 2737 . . . . . . . . . 10 (+g𝐺) = (+g𝐺)
3 sgrpidmnd.0 . . . . . . . . . 10 0 = (0g𝐺)
41, 2, 3grpidval 18674 . . . . . . . . 9 0 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))
54eqeq2i 2750 . . . . . . . 8 (𝑒 = 0𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥))))
6 eleq1w 2824 . . . . . . . . . . . . 13 (𝑦 = 𝑒 → (𝑦𝐵𝑒𝐵))
7 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑦 = 𝑒 → (𝑦(+g𝐺)𝑥) = (𝑒(+g𝐺)𝑥))
87eqeq1d 2739 . . . . . . . . . . . . . 14 (𝑦 = 𝑒 → ((𝑦(+g𝐺)𝑥) = 𝑥 ↔ (𝑒(+g𝐺)𝑥) = 𝑥))
98ovanraleqv 7455 . . . . . . . . . . . . 13 (𝑦 = 𝑒 → (∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥) ↔ ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
106, 9anbi12d 632 . . . . . . . . . . . 12 (𝑦 = 𝑒 → ((𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)) ↔ (𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))))
1110iotan0 6551 . . . . . . . . . . 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 3247 . . . . . . . . . . 11 (∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
1311, 12simpl2im 503 . . . . . . . . . 10 ((𝑒𝐵𝑒 ≠ ∅ ∧ 𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
14133expb 1121 . . . . . . . . 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 413 . . . . . . . 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 407 . . . . . 6 ((𝑒𝐵 ∧ (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
1817ralrimiv 3145 . . . . 5 ((𝑒𝐵 ∧ (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))
1918ex 412 . . . 4 (𝑒𝐵 → ((𝑒 ≠ ∅ ∧ 𝑒 = 0 ) → ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
2019reximia 3081 . . 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 18749 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
2321, 22sylibr 234 1 ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (𝑒 ≠ ∅ ∧ 𝑒 = 0 )) → 𝐺 ∈ Mnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  c0 4333  cio 6512  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Smgrpcsgrp 18731  Mndcmnd 18747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-0g 17486  df-mnd 18748
This theorem is referenced by: (None)
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