Theorem mgmidsssn0 13763

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 6095	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → (𝑧 + 𝑦) = (𝑥 + 𝑦))
7	6	eqeq1d 2247	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑥 → ((𝑧 + 𝑦) = 𝑦 ↔ (𝑥 + 𝑦) = 𝑦))
8	7	ovanraleqv 6112	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐵 ((𝑧 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑧) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)))
9	8	rspcev 2929	. . . . . . . . . 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 13756	. . . . . . . 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 2244	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦))) → 𝑥 = 0 )
14		velsn 3726	. . . . . 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 2623	. . 3 ⊢ (𝐺 ∈ 𝑉 → ∀𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) → 𝑥 ∈ { 0 }))
18		rabss 3325	. . 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 3294	1 ⊢ (𝐺 ∈ 𝑉 → 𝑂 ⊆ { 0 })

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532   ⊆ wss 3220  {csn 3709  ‘cfv 5380  (class class class)co 6088  Basecbs 13410  +_gcplusg 13490  0_gc0g 13669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-cnex 8276  ax-resscn 8277  ax-1re 8279  ax-addrcl 8282

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-iota 5340  df-fun 5382  df-fn 5383  df-fv 5388  df-riota 6041  df-ov 6091  df-inn 9314  df-ndx 13413  df-slot 13414  df-base 13416  df-0g 13671

This theorem is used by:  gzsumress  13771  gsumvallem2  13859