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Theorem gsumvallem2 18860
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 18698 . 2 (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
61, 2mndidcl 18775 . . . 4 (𝐺 ∈ Mnd → 0𝐵)
71, 3, 2mndlrid 18779 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
87ralrimiva 3144 . . . 4 (𝐺 ∈ Mnd → ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9 oveq1 7438 . . . . . . 7 (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
109eqeq1d 2737 . . . . . 6 (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
1110ovanraleqv 7455 . . . . 5 (𝑥 = 0 → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
1211, 4elrab2 3698 . . . 4 ( 0𝑂 ↔ ( 0𝐵 ∧ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
136, 8, 12sylanbrc 583 . . 3 (𝐺 ∈ Mnd → 0𝑂)
1413snssd 4814 . 2 (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
155, 14eqssd 4013 1 (𝐺 ∈ Mnd → 𝑂 = { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  {csn 4631  cfv 6563  (class class class)co 7431  Basecbs 17245  +gcplusg 17298  0gc0g 17486  Mndcmnd 18760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761
This theorem is referenced by:  gsumz  18862  gsumval3a  19936
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