Theorem grpidd 13489

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 2230	. 2 ⊢ (Base‘𝐺) = (Base‘𝐺)
2		eqid 2230	. 2 ⊢ (0g‘𝐺) = (0g‘𝐺)
3		eqid 2230	. 2 ⊢ (+g‘𝐺) = (+g‘𝐺)
4		grpidd.z	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
5		grpidd.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺))
6	4, 5	eleqtrd 2309	. 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺))
7	5	eleq2d 2300	. . . 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 6040	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥))
12		grpidd.i	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥)
13	11, 12	eqtr3d 2265	. . 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 6040	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 ))
16		grpidd.j	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥)
17	15, 16	eqtr3d 2265	. . 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 13486	1 ⊢ (𝜑 → 0 = (0g‘𝐺))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1397   ∈ wcel 2201  ‘cfv 5328  (class class class)co 6023  Basecbs 13105  +_gcplusg 13183  0_gc0g 13362

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-cnex 8128  ax-resscn 8129  ax-1re 8131  ax-addrcl 8134

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-id 4392  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-fv 5336  df-riota 5976  df-ov 6026  df-inn 9149  df-ndx 13108  df-slot 13109  df-base 13111  df-0g 13364

This theorem is referenced by:  ress0g  13549  imasmnd2  13558  mnd1id  13562  isgrpde  13628