Theorem gsumvallem2 18569

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 18453	. 2 ⊢ (𝐺 ∈ Mnd → 𝑂 ⊆ { 0 })
6	1, 2	mndidcl 18497	. . . 4 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝐵)
7	1, 3, 2	mndlrid 18501	. . . . 5 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
8	7	ralrimiva 3140	. . . 4 ⊢ (𝐺 ∈ Mnd → ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦))
9		oveq1 7348	. . . . . . 7 ⊢ (𝑥 = 0 → (𝑥 + 𝑦) = ( 0 + 𝑦))
10	9	eqeq1d 2739	. . . . . 6 ⊢ (𝑥 = 0 → ((𝑥 + 𝑦) = 𝑦 ↔ ( 0 + 𝑦) = 𝑦))
11	10	ovanraleqv 7365	. . . . 5 ⊢ (𝑥 = 0 → (∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) = 𝑦 ∧ (𝑦 + 𝑥) = 𝑦) ↔ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
12	11, 4	elrab2 3640	. . . 4 ⊢ ( 0 ∈ 𝑂 ↔ ( 0 ∈ 𝐵 ∧ ∀𝑦 ∈ 𝐵 (( 0 + 𝑦) = 𝑦 ∧ (𝑦 + 0 ) = 𝑦)))
13	6, 8, 12	sylanbrc 584	. . 3 ⊢ (𝐺 ∈ Mnd → 0 ∈ 𝑂)
14	13	snssd 4760	. 2 ⊢ (𝐺 ∈ Mnd → { 0 } ⊆ 𝑂)
15	5, 14	eqssd 3952	1 ⊢ (𝐺 ∈ Mnd → 𝑂 = { 0 })

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1541   ∈ wcel 2106  ∀wral 3062  {crab 3404  {csn 4577  ‘cfv 6483  (class class class)co 7341  Basecbs 17009  +_gcplusg 17059  0_gc0g 17247  Mndcmnd 18482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5247  ax-nul 5254  ax-pr 5376

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-mpt 5180  df-id 5522  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6435  df-fun 6485  df-fv 6491  df-riota 7297  df-ov 7344  df-0g 17249  df-mgm 18423  df-sgrp 18472  df-mnd 18483

This theorem is referenced by:  gsumz  18571  gsumval3a  19598