Theorem grpidd 13259

Description: Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
grpidd.b	⊢ (𝜑 → 𝐵 = (Base‘𝐺))
grpidd.p	⊢ (𝜑 → + = (+g‘𝐺))
grpidd.z	⊢ (𝜑 → 0 ∈ 𝐵)
grpidd.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
grpidd.j	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)

Assertion

Ref	Expression
grpidd	⊢ (𝜑 → 0 = (0g‘𝐺))

Distinct variable groups:   𝑥,𝐺   𝜑,𝑥   𝑥, 0

Allowed substitution hints:   𝐵(𝑥)   + (𝑥)

Proof of Theorem grpidd

Step	Hyp	Ref	Expression
1		eqid 2206	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2206	. 2 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2206	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		grpidd.z	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
5		grpidd.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
6	4, 5	eleqtrd 2285	. 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
7	5	eleq2d 2276	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺)))
8	7	biimpar 297	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵)
9		grpidd.p	. . . . . 6 ⊢ (𝜑 → + = (+g‘𝐺))
10	9	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺))
11	10	oveqd 5968	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
12		grpidd.i	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
13	11, 12	eqtr3d 2241	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
14	8, 13	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺)𝑥) = 𝑥)
15	10	oveqd 5968	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
16		grpidd.j	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
17	15, 16	eqtr3d 2241	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
18	8, 17	syldan 282	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺) 0 ) = 𝑥)
19	1, 2, 3, 6, 14, 18	ismgmid2 13256	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  ‘cfv 5276  (class class class)co 5951  Basecbs 12876  +_gcplusg 12953  0_gc0g 13132

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-cnex 8023  ax-resscn 8024  ax-1re 8026  ax-addrcl 8029

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-br 4048  df-opab 4110  df-mpt 4111  df-id 4344  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-iota 5237  df-fun 5278  df-fn 5279  df-fv 5284  df-riota 5906  df-ov 5954  df-inn 9044  df-ndx 12879  df-slot 12880  df-base 12882  df-0g 13134

This theorem is referenced by:  ress0g  13319  imasmnd2  13328  mnd1id  13332  isgrpde  13398