Theorem mgmlrid 13682

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 2238	. . . 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 13680	. . . 4 ⊢ (𝜑 → (( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)) ↔ 0 = 0 ))
7	1, 6	mpbiri 168	. . 3 ⊢ (𝜑 → ( 0 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥)))
8	7	simprd 114	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥))
9		oveq2 6087	. . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 + 𝑥) = ( 0 + 𝑋))
10		id 19	. . . . 5 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
11	9, 10	eqeq12d 2253	. . . 4 ⊢ (𝑥 = 𝑋 → (( 0 + 𝑥) = 𝑥 ↔ ( 0 + 𝑋) = 𝑋))
12		oveq1 6086	. . . . 5 ⊢ (𝑥 = 𝑋 → (𝑥 + 0 ) = (𝑋 + 0 ))
13	12, 10	eqeq12d 2253	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑥 + 0 ) = 𝑥 ↔ (𝑋 + 0 ) = 𝑋))
14	11, 13	anbi12d 477	. . 3 ⊢ (𝑥 = 𝑋 → ((( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ↔ (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋)))
15	14	rspccva 2928	. 2 ⊢ ((∀𝑥 ∈ 𝐵 (( 0 + 𝑥) = 𝑥 ∧ (𝑥 + 0 ) = 𝑥) ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))
16	8, 15	sylan 283	1 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝐵) → (( 0 + 𝑋) = 𝑋 ∧ (𝑋 + 0 ) = 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  ‘cfv 5375  (class class class)co 6079  Basecbs 13335  +_gcplusg 13414  0_gc0g 13593

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-cnex 8264  ax-resscn 8265  ax-1re 8267  ax-addrcl 8270

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-mpt 4192  df-id 4436  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-iota 5335  df-fun 5377  df-fn 5378  df-fv 5383  df-riota 6032  df-ov 6082  df-inn 9288  df-ndx 13338  df-slot 13339  df-base 13341  df-0g 13595

This theorem is referenced by:  mndlrid  13730