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Theorem gsumvallem2 17994
Description: Lemma for properties of the set of identities of 𝐺. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
gsumvallem2.b 𝐵 = (Base‘𝐺)
gsumvallem2.z 0 = (0g𝐺)
gsumvallem2.p + = (+g𝐺)
gsumvallem2.o 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
Assertion
Ref Expression
gsumvallem2 (𝐺 ∈ Mnd → 𝑂 = { 0 })
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥, 0 ,𝑦
Allowed substitution hints:   𝑂(𝑥,𝑦)

Proof of Theorem gsumvallem2
StepHypRef Expression
1 gsumvallem2.b . . 3 𝐵 = (Base‘𝐺)
2 gsumvallem2.z . . 3 0 = (0g𝐺)
3 gsumvallem2.p . . 3 + = (+g𝐺)
4 gsumvallem2.o . . 3 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
51, 2, 3, 4mgmidsssn0 17878 . 2 (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
61, 2mndidcl 17922 . . . 4 (𝐺 ∈ Mnd → 0𝐵)
71, 3, 2mndlrid 17926 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
87ralrimiva 3152 . . . 4 (𝐺 ∈ Mnd → ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9 oveq1 7146 . . . . . . 7 (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
109eqeq1d 2803 . . . . . 6 (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
1110ovanraleqv 7163 . . . . 5 (𝑥 = 0 → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
1211, 4elrab2 3634 . . . 4 ( 0𝑂 ↔ ( 0𝐵 ∧ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
136, 8, 12sylanbrc 586 . . 3 (𝐺 ∈ Mnd → 0𝑂)
1413snssd 4705 . 2 (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
155, 14eqssd 3935 1 (𝐺 ∈ Mnd → 𝑂 = { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  {csn 4528  cfv 6328  (class class class)co 7139  Basecbs 16479  +gcplusg 16561  0gc0g 16709  Mndcmnd 17907
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-riota 7097  df-ov 7142  df-0g 16711  df-mgm 17848  df-sgrp 17897  df-mnd 17908
This theorem is referenced by:  gsumz  17996  gsumval3a  19020
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