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Theorem mgmidsssn0 17962
 Description: Property of the set of identities of 𝐺. Either 𝐺 has no identities, and 𝑂 = ∅, or it has one and this identity is unique and identified by the 0g function. (Contributed by Mario Carneiro, 7-Dec-2014.)
Hypotheses
Ref Expression
mgmidsssn0.b 𝐵 = (Base‘𝐺)
mgmidsssn0.z 0 = (0g𝐺)
mgmidsssn0.p + = (+g𝐺)
mgmidsssn0.o 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
Assertion
Ref Expression
mgmidsssn0 (𝐺𝑉𝑂 ⊆ { 0 })
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥,𝑉   𝑥, 0 ,𝑦
Allowed substitution hints:   𝑂(𝑥,𝑦)   𝑉(𝑦)

Proof of Theorem mgmidsssn0
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 mgmidsssn0.o . 2 𝑂 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
2 simpr 488 . . . . . . . 8 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
3 mgmidsssn0.b . . . . . . . . 9 𝐵 = (Base‘𝐺)
4 mgmidsssn0.z . . . . . . . . 9 0 = (0g𝐺)
5 mgmidsssn0.p . . . . . . . . 9 + = (+g𝐺)
6 oveq1 7163 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
76eqeq1d 2760 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ((𝑧 + 𝑦) = 𝑦 ↔ (𝑥 + 𝑦) = 𝑦))
87ovanraleqv 7180 . . . . . . . . . . 11 (𝑧 = 𝑥 → (∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦) ↔ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
98rspcev 3543 . . . . . . . . . 10 ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) → ∃𝑧𝐵𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
109adantl 485 . . . . . . . . 9 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → ∃𝑧𝐵𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
113, 4, 5, 10ismgmid 17955 . . . . . . . 8 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) ↔ 0 = 𝑥))
122, 11mpbid 235 . . . . . . 7 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 0 = 𝑥)
1312eqcomd 2764 . . . . . 6 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 = 0 )
14 velsn 4541 . . . . . 6 (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
1513, 14sylibr 237 . . . . 5 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 ∈ { 0 })
1615expr 460 . . . 4 ((𝐺𝑉𝑥𝐵) → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
1716ralrimiva 3113 . . 3 (𝐺𝑉 → ∀𝑥𝐵 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
18 rabss 3978 . . 3 ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 } ↔ ∀𝑥𝐵 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
1917, 18sylibr 237 . 2 (𝐺𝑉 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 })
201, 19eqsstrid 3942 1 (𝐺𝑉𝑂 ⊆ { 0 })
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  {crab 3074   ⊆ wss 3860  {csn 4525  ‘cfv 6340  (class class class)co 7156  Basecbs 16555  +gcplusg 16637  0gc0g 16785 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6299  df-fun 6342  df-fv 6348  df-riota 7114  df-ov 7159  df-0g 16787 This theorem is referenced by:  gsumress  17972  gsumval2  17976  gsumvallem2  18078
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