Theorem mgmidsssn0 18405

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 0_g function. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
mgmidsssn0.b	⊢ 𝐵 = (Base‘𝐺)
mgmidsssn0.z	⊢ 0 = (0g‘𝐺)
mgmidsssn0.p	⊢ + = (+g‘𝐺)
mgmidsssn0.o	⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}

Assertion

Ref	Expression
mgmidsssn0	⊢ (𝐺 ∈ 𝑉 → 𝑂 ⊆ { 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.

Step	Hyp	Ref	Expression
1		mgmidsssn0.o	. 2 ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
2		simpr 486	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
3		mgmidsssn0.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐺)
4		mgmidsssn0.z	. . . . . . . . 9 ⊢ 0 = (0g‘𝐺)
5		mgmidsssn0.p	. . . . . . . . 9 ⊢ + = (+g‘𝐺)
6		oveq1 7314	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
7	6	eqeq1d 2738	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((𝑧 + 𝑦) = 𝑦 ↔ (𝑥 + 𝑦) = 𝑦))
8	7	ovanraleqv 7331	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
9	8	rspcev 3566	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
10	9	adantl 483	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
11	3, 4, 5, 10	ismgmid 18398	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) ↔ 0 = 𝑥))
12	2, 11	mpbid 231	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 0 = 𝑥)
13	12	eqcomd 2742	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 = 0 )
14		velsn 4581	. . . . . 6 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
15	13, 14	sylibr 233	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 ∈ { 0 })
16	15	expr 458	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
17	16	ralrimiva 3140	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
18		rabss 4011	. . 3 ⊢ ({𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 } ↔ ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
19	17, 18	sylibr 233	. 2 ⊢ (𝐺 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 })
20	1, 19	eqsstrid 3974	1 ⊢ (𝐺 ∈ 𝑉 → 𝑂 ⊆ { 0 })

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539   ∈ wcel 2104  ∀wral 3062  ∃wrex 3071  {crab 3303   ⊆ wss 3892  {csn 4565  ‘cfv 6458  (class class class)co 7307  Basecbs 16961  +_gcplusg 17011  0_gc0g 17199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3304  df-reu 3305  df-rab 3306  df-v 3439  df-sbc 3722  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-iota 6410  df-fun 6460  df-fv 6466  df-riota 7264  df-ov 7310  df-0g 17201

This theorem is referenced by:  gsumress  18415  gsumval2  18419  gsumvallem2  18521