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Theorem gsumvallem2 18848
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 18686 . 2 (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
61, 2mndidcl 18763 . . . 4 (𝐺 ∈ Mnd → 0𝐵)
71, 3, 2mndlrid 18767 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
87ralrimiva 3145 . . . 4 (𝐺 ∈ Mnd → ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9 oveq1 7439 . . . . . . 7 (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
109eqeq1d 2738 . . . . . 6 (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
1110ovanraleqv 7456 . . . . 5 (𝑥 = 0 → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
1211, 4elrab2 3694 . . . 4 ( 0𝑂 ↔ ( 0𝐵 ∧ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
136, 8, 12sylanbrc 583 . . 3 (𝐺 ∈ Mnd → 0𝑂)
1413snssd 4808 . 2 (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
155, 14eqssd 4000 1 (𝐺 ∈ Mnd → 𝑂 = { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1539  wcel 2107  wral 3060  {crab 3435  {csn 4625  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  0gc0g 17485  Mndcmnd 18748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-riota 7389  df-ov 7435  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749
This theorem is referenced by:  gsumz  18850  gsumval3a  19922
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