Theorem mgmn0plusgplusf 18732

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 18730	. . . 4 ⊢ (𝐺 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → ⨣ :(𝐵 × 𝐵)⟶𝐵)
6	5	ffnd 6710	. 2 ⊢ (𝜑 → ⨣ Fn (𝐵 × 𝐵))
7		mgmn0plusgf.p	. . . 4 ⊢ + = (+g‘𝐺)
8		mgmn0plusgf.0	. . . 4 ⊢ (𝜑 → ∅ ∉ 𝐵)
9		eqid 2765	. . . 4 ⊢ ( + ↾ (𝐵 × 𝐵)) = ( + ↾ (𝐵 × 𝐵))
10	2, 7, 1, 8, 9	mgmn0plusgf 18731	. . 3 ⊢ (𝜑 → ( + ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶𝐵)
11	10	ffnd 6710	. 2 ⊢ (𝜑 → ( + ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
12		elxp 5686	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
13		df-ov 7422	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
14	2, 7, 3	plusfval 18727	. . . . . . . . . 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 5700	. . . . . . . . . . 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 6905	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩) = ( + ‘⟨𝑥, 𝑦⟩))
19	13, 15, 18	3eqtr4a 2826	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥 ⨣ 𝑦) = (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩))
20		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑧) = ( ⨣ ‘⟨𝑥, 𝑦⟩))
21		df-ov 7422	. . . . . . . . . . 11 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘⟨𝑥, 𝑦⟩)
22	20, 21	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑧) = (𝑥 ⨣ 𝑦))
23		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (( + ↾ (𝐵 × 𝐵))‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩))
24	22, 23	eqeq12d 2781	. . . . . . . . 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 1967	. . . 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 7032	1 ⊢ (𝜑 → ⨣ = ( + ↾ (𝐵 × 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ∉ wnel 3066  ∅c0 4286  ⟨cop 4597   × cxp 5661   ↾ cres 5665  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  +_𝑓cplusf 18717  Mgmcmgm 18718

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-plusf 18719  df-mgm 18720

This theorem is used by: (None)