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Theorem mgmidsssn0 17581
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 478 . . . . . . . 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 6883 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
76eqeq1d 2799 . . . . . . . . . . . 12 (𝑧 = 𝑥 → ((𝑧 + 𝑦) = 𝑦 ↔ (𝑥 + 𝑦) = 𝑦))
87ovanraleqv 6900 . . . . . . . . . . 11 (𝑧 = 𝑥 → (∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦) ↔ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
98rspcev 3495 . . . . . . . . . 10 ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) → ∃𝑧𝐵𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
109adantl 474 . . . . . . . . 9 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → ∃𝑧𝐵𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
113, 4, 5, 10ismgmid 17576 . . . . . . . 8 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) ↔ 0 = 𝑥))
122, 11mpbid 224 . . . . . . 7 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 0 = 𝑥)
1312eqcomd 2803 . . . . . 6 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 = 0 )
14 velsn 4382 . . . . . 6 (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
1513, 14sylibr 226 . . . . 5 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 ∈ { 0 })
1615expr 449 . . . 4 ((𝐺𝑉𝑥𝐵) → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
1716ralrimiva 3145 . . 3 (𝐺𝑉 → ∀𝑥𝐵 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
18 rabss 3873 . . 3 ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 } ↔ ∀𝑥𝐵 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
1917, 18sylibr 226 . 2 (𝐺𝑉 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 })
201, 19syl5eqss 3843 1 (𝐺𝑉𝑂 ⊆ { 0 })
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3087  wrex 3088  {crab 3091  wss 3767  {csn 4366  cfv 6099  (class class class)co 6876  Basecbs 16181  +gcplusg 16264  0gc0g 16412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-reu 3094  df-rmo 3095  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107  df-riota 6837  df-ov 6879  df-0g 16414
This theorem is referenced by:  gsumress  17588  gsumval2  17592  gsumvallem2  17684
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