Theorem gsumvallem2 13512

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 13403	. 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
6	1, 2	mndidcl 13449	. . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
7	1, 3, 2	mndlrid 13453	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
8	7	ralrimiva 2603	. . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9		oveq1 6001	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
10	9	eqeq1d 2238	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
11	10	ovanraleqv 6018	. . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
12	11, 4	elrab2 2962	. . . 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 3812	. 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
15	5, 14	eqssd 3241	1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 })

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  {crab 2512  {csn 3666  ‘cfv 5314  (class class class)co 5994  Basecbs 13018  +_gcplusg 13096  0_gc0g 13275  Mndcmnd 13435

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-cnex 8078  ax-resscn 8079  ax-1re 8081  ax-addrcl 8084

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4381  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-iota 5274  df-fun 5316  df-fn 5317  df-fv 5322  df-riota 5947  df-ov 5997  df-inn 9099  df-2 9157  df-ndx 13021  df-slot 13022  df-base 13024  df-plusg 13109  df-0g 13277  df-mgm 13375  df-sgrp 13421  df-mnd 13436

This theorem is referenced by: (None)