Theorem mgmn0plusgplusf 18878

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 18876	. . . 4 ⊢ (𝐺 ∈ Mgm → ⨣ :(𝐵 × 𝐵)⟶𝐵)
5	1, 4	syl 18	. . 3 ⊢ (𝜑 → ⨣ :(𝐵 × 𝐵)⟶𝐵)
6	5	ffnd 6740	. 2 ⊢ (𝜑 → ⨣ Fn (𝐵 × 𝐵))
7		mgmn0plusgf.p	. . . 4 ⊢ + = (+g‘𝐺)
8		mgmn0plusgf.0	. . . 4 ⊢ (𝜑 → ∅ ∉ 𝐵)
9		eqid 2763	. . . 4 ⊢ ( + ↾ (𝐵 × 𝐵)) = ( + ↾ (𝐵 × 𝐵))
10	2, 7, 1, 8, 9	mgmn0plusgf 18877	. . 3 ⊢ (𝜑 → ( + ↾ (𝐵 × 𝐵)):(𝐵 × 𝐵)⟶𝐵)
11	10	ffnd 6740	. 2 ⊢ (𝜑 → ( + ↾ (𝐵 × 𝐵)) Fn (𝐵 × 𝐵))
12		elxp 5681	. . . 4 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
13		df-ov 7454	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = ( + ‘⟨𝑥, 𝑦⟩)
14	2, 7, 3	plusfval 18873	. . . . . . . . . 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 5695	. . . . . . . . . . 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 6936	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩) = ( + ‘⟨𝑥, 𝑦⟩))
19	13, 15, 18	3eqtr4a 2824	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 = ⟨𝑥, 𝑦⟩) → (𝑥 ⨣ 𝑦) = (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩))
20		fveq2 6916	. . . . . . . . . . 11 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑧) = ( ⨣ ‘⟨𝑥, 𝑦⟩))
21		df-ov 7454	. . . . . . . . . . 11 ⊢ (𝑥 ⨣ 𝑦) = ( ⨣ ‘⟨𝑥, 𝑦⟩)
22	20, 21	eqtr4di 2816	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ( ⨣ ‘𝑧) = (𝑥 ⨣ 𝑦))
23		fveq2 6916	. . . . . . . . . 10 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (( + ↾ (𝐵 × 𝐵))‘𝑧) = (( + ↾ (𝐵 × 𝐵))‘⟨𝑥, 𝑦⟩))
24	22, 23	eqeq12d 2779	. . . . . . . . 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 1969	. . . 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 7063	1 ⊢ (𝜑 → ⨣ = ( + ↾ (𝐵 × 𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ∉ wnel 3065  ∅c0 4281  ⟨cop 4840   × cxp 5656   ↾ cres 5660  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  +_𝑓cplusf 18863  Mgmcmgm 18864

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-plusf 18865  df-mgm 18866

This theorem is used by: (None)