Theorem mgmlrid 18715

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 2765	. . . 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 18713	. . . 4 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ↔ 0 = 0 ))
7	1, 6	mpbiri 261	. . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
8	7	simprd 500	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
9		oveq2 7408	. . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 + 𝑥) = ( 0 + 𝑋))
10		id 23	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
11	9, 10	eqeq12d 2781	. . . 4 ⊢ (𝑥 = 𝑋 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 𝑋) = 𝑋))
12		oveq1 7407	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 + 0 ) = (𝑋 + 0 ))
13	12, 10	eqeq12d 2781	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 + 0 ) = 𝑥 ↔ (𝑋 + 0 ) = 𝑋))
14	11, 13	anbi12d 643	. . 3 ⊢ (𝑥 = 𝑋 → ((( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ↔ (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)))
15	14	rspccva 3583	. 2 ⊢ ((∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
16	8, 15	sylan 591	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃wrex 3089  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  +_gcplusg 17300  0_gc0g 17482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-riota 7357  df-ov 7403  df-0g 17484

This theorem is referenced by:  mndlrid  18801