Theorem mgmidsssn0 13597

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 110	. . . . . . . 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 6057	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
7	6	eqeq1d 2241	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((𝑧 + 𝑦) = 𝑦 ↔ (𝑥 + 𝑦) = 𝑦))
8	7	ovanraleqv 6074	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
9	8	rspcev 2921	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)) → ∃𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦))
10	9	adantl 277	. . . . . . . . 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 13590	. . . . . . . 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 147	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 0 = 𝑥)
13	12	eqcomd 2238	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 = 0 )
14		velsn 3706	. . . . . 6 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 )
15	13, 14	sylibr 134	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 ∈ { 0 })
16	15	expr 375	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑥 ∈ 𝐵) → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
17	16	ralrimiva 2615	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
18		rabss 3315	. . 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 134	. 2 ⊢ (𝐺 ∈ 𝑉 → {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)} ⊆ { 0 })
20	1, 19	eqsstrid 3284	1 ⊢ (𝐺 ∈ 𝑉 → 𝑂 ⊆ { 0 })

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  {crab 2524   ⊆ wss 3211  {csn 3689  ‘cfv 5352  (class class class)co 6050  Basecbs 13212  +_gcplusg 13290  0_gc0g 13469

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-cnex 8218  ax-resscn 8219  ax-1re 8221  ax-addrcl 8224

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-iota 5312  df-fun 5354  df-fn 5355  df-fv 5360  df-riota 6003  df-ov 6053  df-inn 9238  df-ndx 13215  df-slot 13216  df-base 13218  df-0g 13471

This theorem is referenced by:  gsumress  13608  gsumvallem2  13706