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Theorem sgrpidmndm 12826
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 = (0g𝐺)
Assertion
Ref Expression
sgrpidmndm ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (∃𝑤 𝑤𝑒𝑒 = 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.
StepHypRef Expression
1 simp-4r 542 . . . . . . . . . 10 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → 𝑒𝐵)
2 simpllr 534 . . . . . . . . . . 11 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → 𝑤𝑒)
3219.8ad 1591 . . . . . . . . . 10 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → ∃𝑤 𝑤𝑒)
4 simplr 528 . . . . . . . . . . 11 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → 𝑒 = 0 )
5 sgrpidmnd.b . . . . . . . . . . . . . 14 𝐵 = (Base‘𝐺)
6 eqid 2177 . . . . . . . . . . . . . 14 (+g𝐺) = (+g𝐺)
7 sgrpidmnd.0 . . . . . . . . . . . . . 14 0 = (0g𝐺)
85, 6, 7grpidvalg 12797 . . . . . . . . . . . . 13 (𝐺 ∈ Smgrp → 0 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥))))
98eqeq2d 2189 . . . . . . . . . . . 12 (𝐺 ∈ Smgrp → (𝑒 = 0𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))))
109ad4antr 494 . . . . . . . . . . 11 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → (𝑒 = 0𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))))
114, 10mpbid 147 . . . . . . . . . 10 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → 𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥))))
121, 3, 113jca 1177 . . . . . . . . 9 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → (𝑒𝐵 ∧ ∃𝑤 𝑤𝑒𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))))
13 simpr 110 . . . . . . . . 9 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → 𝑥𝐵)
14 eleq1w 2238 . . . . . . . . . . . 12 (𝑦 = 𝑒 → (𝑦𝐵𝑒𝐵))
15 oveq1 5884 . . . . . . . . . . . . . 14 (𝑦 = 𝑒 → (𝑦(+g𝐺)𝑥) = (𝑒(+g𝐺)𝑥))
1615eqeq1d 2186 . . . . . . . . . . . . 13 (𝑦 = 𝑒 → ((𝑦(+g𝐺)𝑥) = 𝑥 ↔ (𝑒(+g𝐺)𝑥) = 𝑥))
1716ovanraleqv 5901 . . . . . . . . . . . 12 (𝑦 = 𝑒 → (∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥) ↔ ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
1814, 17anbi12d 473 . . . . . . . . . . 11 (𝑦 = 𝑒 → ((𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)) ↔ (𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))))
1918iotam 5210 . . . . . . . . . 10 ((𝑒𝐵 ∧ ∃𝑤 𝑤𝑒𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))) → (𝑒𝐵 ∧ ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
20 rsp 2524 . . . . . . . . . 10 (∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
2119, 20simpl2im 386 . . . . . . . . 9 ((𝑒𝐵 ∧ ∃𝑤 𝑤𝑒𝑒 = (℩𝑦(𝑦𝐵 ∧ ∀𝑥𝐵 ((𝑦(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑦) = 𝑥)))) → (𝑥𝐵 → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
2212, 13, 21sylc 62 . . . . . . . 8 (((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) ∧ 𝑥𝐵) → ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))
2322ralrimiva 2550 . . . . . . 7 ((((𝐺 ∈ Smgrp ∧ 𝑒𝐵) ∧ 𝑤𝑒) ∧ 𝑒 = 0 ) → ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))
2423exp31 364 . . . . . 6 ((𝐺 ∈ Smgrp ∧ 𝑒𝐵) → (𝑤𝑒 → (𝑒 = 0 → ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))))
2524exlimdv 1819 . . . . 5 ((𝐺 ∈ Smgrp ∧ 𝑒𝐵) → (∃𝑤 𝑤𝑒 → (𝑒 = 0 → ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥))))
2625impd 254 . . . 4 ((𝐺 ∈ Smgrp ∧ 𝑒𝐵) → ((∃𝑤 𝑤𝑒𝑒 = 0 ) → ∀𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
2726reximdva 2579 . . 3 (𝐺 ∈ Smgrp → (∃𝑒𝐵 (∃𝑤 𝑤𝑒𝑒 = 0 ) → ∃𝑒𝐵𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
2827imdistani 445 . 2 ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (∃𝑤 𝑤𝑒𝑒 = 0 )) → (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
295, 6ismnddef 12824 . 2 (𝐺 ∈ Mnd ↔ (𝐺 ∈ Smgrp ∧ ∃𝑒𝐵𝑥𝐵 ((𝑒(+g𝐺)𝑥) = 𝑥 ∧ (𝑥(+g𝐺)𝑒) = 𝑥)))
3028, 29sylibr 134 1 ((𝐺 ∈ Smgrp ∧ ∃𝑒𝐵 (∃𝑤 𝑤𝑒𝑒 = 0 )) → 𝐺 ∈ Mnd)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 978   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  cio 5178  cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  0gc0g 12710  Smgrpcsgrp 12812  Mndcmnd 12822
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-cnex 7904  ax-resscn 7905  ax-1re 7907  ax-addrcl 7910
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226  df-riota 5833  df-ov 5880  df-inn 8922  df-2 8980  df-ndx 12467  df-slot 12468  df-base 12470  df-plusg 12551  df-0g 12712  df-mnd 12823
This theorem is referenced by: (None)
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