Theorem gsumvallem2 13055

Description: Lemma for properties of the set of identities of 𝐺. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)

Hypotheses

Ref	Expression
gsumvallem2.b	⊢ 𝐵 = (Base‘𝐺)
gsumvallem2.z	⊢ 0 = (0g‘𝐺)
gsumvallem2.p	⊢ + = (+g‘𝐺)
gsumvallem2.o	⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}

Assertion

Ref	Expression
gsumvallem2	⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 })

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝑥, + ,𝑦   𝑥, 0 ,𝑦

Allowed substitution hints:   𝑂(𝑥,𝑦)

Proof of Theorem gsumvallem2

Step	Hyp	Ref	Expression
1		gsumvallem2.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		gsumvallem2.z	. . 3 ⊢ 0 = (0g‘𝐺)
3		gsumvallem2.p	. . 3 ⊢ + = (+g‘𝐺)
4		gsumvallem2.o	. . 3 ⊢ 𝑂 = {𝑥 ∈ 𝐵 ∣ ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦)}
5	1, 2, 3, 4	mgmidsssn0 12957	. 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
6	1, 2	mndidcl 13001	. . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
7	1, 3, 2	mndlrid 13005	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
8	7	ralrimiva 2567	. . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9		oveq1 5917	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
10	9	eqeq1d 2202	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
11	10	ovanraleqv 5934	. . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
12	11, 4	elrab2 2919	. . . 4 ⊢ ( 0 ∈ 𝑂 ↔ ( 0 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
13	6, 8, 12	sylanbrc 417	. . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑂)
14	13	snssd 3763	. 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
15	5, 14	eqssd 3196	1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 })

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476  {csn 3618  ‘cfv 5246  (class class class)co 5910  Basecbs 12608  +_gcplusg 12685  0_gc0g 12857  Mndcmnd 12987

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4462  ax-cnex 7953  ax-resscn 7954  ax-1re 7956  ax-addrcl 7959

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-iota 5207  df-fun 5248  df-fn 5249  df-fv 5254  df-riota 5865  df-ov 5913  df-inn 8973  df-2 9031  df-ndx 12611  df-slot 12612  df-base 12614  df-plusg 12698  df-0g 12859  df-mgm 12929  df-sgrp 12975  df-mnd 12988

This theorem is referenced by: (None)