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Theorem gsumvallem2 17419
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 17316 . 2 (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
61, 2mndidcl 17355 . . . 4 (𝐺 ∈ Mnd → 0𝐵)
71, 3, 2mndlrid 17357 . . . . 5 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
87ralrimiva 2995 . . . 4 (𝐺 ∈ Mnd → ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9 oveq1 6697 . . . . . . . 8 (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
109eqeq1d 2653 . . . . . . 7 (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
11 oveq2 6698 . . . . . . . 8 (𝑥 = 0 → (𝑦 + 𝑥) = (𝑦 + 0 ))
1211eqeq1d 2653 . . . . . . 7 (𝑥 = 0 → ((𝑦 + 𝑥) = 𝑦 ↔ (𝑦 + 0 ) = 𝑦))
1310, 12anbi12d 747 . . . . . 6 (𝑥 = 0 → (((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
1413ralbidv 3015 . . . . 5 (𝑥 = 0 → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
1514, 4elrab2 3399 . . . 4 ( 0𝑂 ↔ ( 0𝐵 ∧ ∀𝑦𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
166, 8, 15sylanbrc 699 . . 3 (𝐺 ∈ Mnd → 0𝑂)
1716snssd 4372 . 2 (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
185, 17eqssd 3653 1 (𝐺 ∈ Mnd → 𝑂 = { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  {csn 4210  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  0gc0g 16147  Mndcmnd 17341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-ov 6693  df-0g 16149  df-mgm 17289  df-sgrp 17331  df-mnd 17342
This theorem is referenced by:  gsumz  17421  gsumval3a  18350
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