Theorem gsumvallem2 17758

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 17655	. 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
6	1, 2	mndidcl 17694	. . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
7	1, 3, 2	mndlrid 17696	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
8	7	ralrimiva 3148	. . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9		oveq1 6929	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
10	9	eqeq1d 2780	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
11	10	ovanraleqv 6946	. . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
12	11, 4	elrab2 3576	. . . 4 ⊢ ( 0 ∈ 𝑂 ↔ ( 0 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
13	6, 8, 12	sylanbrc 578	. . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑂)
14	13	snssd 4571	. 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
15	5, 14	eqssd 3838	1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 })

Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  ∀wral 3090  {crab 3094  {csn 4398  ‘cfv 6135  (class class class)co 6922  Basecbs 16255  +_gcplusg 16338  0_gc0g 16486  Mndcmnd 17680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-riota 6883  df-ov 6925  df-0g 16488  df-mgm 17628  df-sgrp 17670  df-mnd 17681

This theorem is referenced by:  gsumz  17760  gsumval3a  18690