Theorem mgmidsssn0 18650

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 484	. . . . . . . 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 7412	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
7	6	eqeq1d 2737	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((𝑧 + 𝑦) = 𝑦 ↔ (𝑥 + 𝑦) = 𝑦))
8	7	ovanraleqv 7429	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
9	8	rspcev 3601	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
10	9	adantl 481	. . . . . . . . 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 18643	. . . . . . . 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 232	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 0 = 𝑥)
13	12	eqcomd 2741	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 = 0 )
14		velsn 4617	. . . . . 6 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
15	13, 14	sylibr 234	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 ∈ { 0 })
16	15	expr 456	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
17	16	ralrimiva 3132	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
18		rabss 4047	. . 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 234	. 2 ⊢ (𝐺 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 })
20	1, 19	eqsstrid 3997	1 ⊢ (𝐺 ∈ 𝑉 → 𝑂 ⊆ { 0 })

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  {crab 3415   ⊆ wss 3926  {csn 4601  ‘cfv 6531  (class class class)co 7405  Basecbs 17228  +_gcplusg 17271  0_gc0g 17453

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-iota 6484  df-fun 6533  df-fv 6539  df-riota 7362  df-ov 7408  df-0g 17455

This theorem is referenced by:  gsumress  18660  gsumval2  18664  gsumvallem2  18812