Theorem mgmn0plusgplusf 18835

Description: The group addition function of a magma is the restriction of its group operation to its base set if the base set does not contain the empty set. (Contributed by AV, 16-Aug-2026.)

Hypotheses

Ref	Expression
mgmn0plusgf.b	⊢ 𝐵 = (Base‘𝐺)
mgmn0plusgf.p	⊢ + = (+g‘𝐺)
mgmn0plusgf.g	⊢ (𝜑 → 𝐺 ∈ Mgm)
mgmn0plusgf.0	⊢ (𝜑 → ∅ ∉ 𝐵)
mgmn0plusgplusf.p	⊢ ⨣ = (+𝑓‘𝐺)

Assertion

Ref	Expression
mgmn0plusgplusf	⊢ (𝜑 → ⨣ = ( + ↾ (𝐵 × 𝐵)))

Proof of Theorem mgmn0plusgplusf

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mgmn0plusgf.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Mgm)
2		mgmn0plusgf.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3		mgmn0plusgplusf.p	. . . . 5 ⊢ ⨣ = (+𝑓‘𝐺)
4	2, 3	mgmplusf 18833	. . . 4 ⊢ (𝐺 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → ⨣ :(𝐵 × 𝐵)⟶𝐵)
6	5	ffnd 6713	. 2 ⊢ (𝜑 → ⨣ Fn (𝐵 × 𝐵))
7		mgmn0plusgf.p	. . . 4 ⊢ + = (+g‘𝐺)
8		mgmn0plusgf.0	. . . 4 ⊢ (𝜑 → ∅ ∉ 𝐵)
9		eqid 2760	. . . 4 ⊢ ( + ↾ (𝐵 × 𝐵)) = ( + ↾ (𝐵 × 𝐵))
10	2, 7, 1, 8, 9	mgmn0plusgf 18834	. . 3 ⊢ (𝜑 → ( + ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶𝐵)
11	10	ffnd 6713	. 2 ⊢ (𝜑 → ( + ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
12		elxp 5673	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
13		df-ov 7425	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
14	2, 7, 3	plusfval 18830	. . . . . . . . . 10 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦))
15	14	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥 ⨣ 𝑦) = (𝑥 + 𝑦))
16		opelxpi 5687	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
17	16	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐵))
18	17	fvresd 6908	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩) = ( + ‘⟨𝑥, 𝑦⟩))
19	13, 15, 18	3eqtr4a 2821	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥 ⨣ 𝑦) = (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩))
20		fveq2 6888	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑧) = ( ⨣ ‘⟨𝑥, 𝑦⟩))
21		df-ov 7425	. . . . . . . . . . 11 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘⟨𝑥, 𝑦⟩)
22	20, 21	eqtr4di 2813	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑧) = (𝑥 ⨣ 𝑦))
23		fveq2 6888	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (( + ↾ (𝐵 × 𝐵))‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩))
24	22, 23	eqeq12d 2776	. . . . . . . . 9 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (( ⨣ ‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘𝑧) ↔ (𝑥 ⨣ 𝑦) = (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩)))
25	24	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (( ⨣ ‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘𝑧) ↔ (𝑥 ⨣ 𝑦) = (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩)))
26	19, 25	mpbird 260	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → ( ⨣ ‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘𝑧))
27	26	exp31 425	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑧 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘𝑧))))
28	27	impcomd 417	. . . . 5 ⊢ (𝜑 → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( ⨣ ‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘𝑧)))
29	28	exlimdvv 1966	. . . 4 ⊢ (𝜑 → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ( ⨣ ‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘𝑧)))
30	12, 29	biimtrid 245	. . 3 ⊢ (𝜑 → (𝑧 ∈ (𝐵 × 𝐵) → ( ⨣ ‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘𝑧)))
31	30	imp 412	. 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐵 × 𝐵)) → ( ⨣ ‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘𝑧))
32	6, 11, 31	eqfnfvd 7035	1 ⊢ (𝜑 → ⨣ = ( + ↾ (𝐵 × 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ∉ wnel 3061  ∅c0 4277  ⟨cop 4834   × cxp 5648   ↾ cres 5652  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  +_gcplusg 17435  +_𝑓cplusf 18820  Mgmcmgm 18821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-plusf 18822  df-mgm 18823

This theorem is used by: (None)