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Theorem mgmidsssn0 17092
 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 476 . . . . . . . 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 6556 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
76eqeq1d 2612 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝑧 + 𝑦) = 𝑦 ↔ (𝑥 + 𝑦) = 𝑦))
8 oveq2 6557 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑦 + 𝑧) = (𝑦 + 𝑥))
98eqeq1d 2612 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝑦 + 𝑧) = 𝑦 ↔ (𝑦 + 𝑥) = 𝑦))
107, 9anbi12d 743 . . . . . . . . . . . 12 (𝑧 = 𝑥 → (((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦) ↔ ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
1110ralbidv 2969 . . . . . . . . . . 11 (𝑧 = 𝑥 → (∀𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦) ↔ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
1211rspcev 3282 . . . . . . . . . 10 ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) → ∃𝑧𝐵𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
1312adantl 481 . . . . . . . . 9 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → ∃𝑧𝐵𝑦𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
143, 4, 5, 13ismgmid 17087 . . . . . . . 8 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → ((𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) ↔ 0 = 𝑥))
152, 14mpbid 221 . . . . . . 7 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 0 = 𝑥)
1615eqcomd 2616 . . . . . 6 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 = 0 )
17 velsn 4141 . . . . . 6 (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
1816, 17sylibr 223 . . . . 5 ((𝐺𝑉 ∧ (𝑥𝐵 ∧ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 ∈ { 0 })
1918expr 641 . . . 4 ((𝐺𝑉𝑥𝐵) → (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
2019ralrimiva 2949 . . 3 (𝐺𝑉 → ∀𝑥𝐵 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
21 rabss 3642 . . 3 ({𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 } ↔ ∀𝑥𝐵 (∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
2220, 21sylibr 223 . 2 (𝐺𝑉 → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 })
231, 22syl5eqss 3612 1 (𝐺𝑉𝑂 ⊆ { 0 })
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  {csn 4125  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  0gc0g 15923 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-0g 15925 This theorem is referenced by:  gsumress  17099  gsumval2  17103  gsumvallem2  17195
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