Theorem mgmlrid 18595

Description: The identity element of a magma, if it exists, is a left and right identity. (Contributed by Mario Carneiro, 27-Dec-2014.)

Hypotheses

Ref	Expression
ismgmid.b	⊢ 𝐵 = (Base‘𝐺)
ismgmid.o	⊢ 0 = (0g‘𝐺)
ismgmid.p	⊢ + = (+g‘𝐺)
mgmidcl.e	⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))

Assertion

Ref	Expression
mgmlrid	⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Distinct variable groups:   𝑥,𝑒, +   0 ,𝑒,𝑥   𝐵,𝑒,𝑥   𝑒,𝐺,𝑥   𝑥,𝑋

Allowed substitution hints:   𝜑(𝑥,𝑒)   𝑋(𝑒)

Proof of Theorem mgmlrid

Step	Hyp	Ref	Expression
1		eqid 2731	. . . 4 ⊢ 0 = 0
2		ismgmid.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3		ismgmid.o	. . . . 5 ⊢ 0 = (0g‘𝐺)
4		ismgmid.p	. . . . 5 ⊢ + = (+g‘𝐺)
5		mgmidcl.e	. . . . 5 ⊢ (𝜑 → ∃𝑒 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑒 + 𝑥) = 𝑥 ∧ (𝑥 + 𝑒) = 𝑥))
6	2, 3, 4, 5	ismgmid 18593	. . . 4 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ↔ 0 = 0 ))
7	1, 6	mpbiri 258	. . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
8	7	simprd 495	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
9		oveq2 7420	. . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 + 𝑥) = ( 0 + 𝑋))
10		id 22	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
11	9, 10	eqeq12d 2747	. . . 4 ⊢ (𝑥 = 𝑋 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 𝑋) = 𝑋))
12		oveq1 7419	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 + 0 ) = (𝑋 + 0 ))
13	12, 10	eqeq12d 2747	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 + 0 ) = 𝑥 ↔ (𝑋 + 0 ) = 𝑋))
14	11, 13	anbi12d 630	. . 3 ⊢ (𝑥 = 𝑋 → ((( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ↔ (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)))
15	14	rspccva 3611	. 2 ⊢ ((∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
16	8, 15	sylan 579	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069  ‘cfv 6543  (class class class)co 7412  Basecbs 17151  +_gcplusg 17204  0_gc0g 17392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-riota 7368  df-ov 7415  df-0g 17394

This theorem is referenced by:  mndlrid  18681