Theorem mgmn0plusgf 18834

Description: The restriction of the group operation of a magma to its base set is a function if the base set does not contain the empty set. Excluding the empty set from the base set is necessary because of the specific definition of an undefined operation value (see also ndmovcl 7608 and ndmovrcl 7609). (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)
mgmn0plusgf.r	⊢ 𝑃 = ( + ↾ (𝐵 × 𝐵))

Assertion

Ref	Expression
mgmn0plusgf	⊢ (𝜑 → 𝑃:(𝐵 × 𝐵)⟶𝐵)

Proof of Theorem mgmn0plusgf

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

Step	Hyp	Ref	Expression
1		mgmn0plusgf.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ Mgm)
2		mgmn0plusgf.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐺)
3		mgmn0plusgf.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
4	2, 3	mgmcl 18826	. . . . . . 7 ⊢ ((𝐺 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
5	1, 4	syl3an1 1181	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
6	5	3expb 1138	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ∈ 𝐵)
7		mgmn0plusgf.0	. . . . . . 7 ⊢ (𝜑 → ∅ ∉ 𝐵)
8		df-nel 3062	. . . . . . . 8 ⊢ (∅ ∉ 𝐵 ↔ ¬ ∅ ∈ 𝐵)
9		nelelne 3056	. . . . . . . 8 ⊢ (¬ ∅ ∈ 𝐵 → ((𝑥 + 𝑦) ∈ 𝐵 → (𝑥 + 𝑦) ≠ ∅))
10	8, 9	sylbi 220	. . . . . . 7 ⊢ (∅ ∉ 𝐵 → ((𝑥 + 𝑦) ∈ 𝐵 → (𝑥 + 𝑦) ≠ ∅))
11	7, 10	syl 18	. . . . . 6 ⊢ (𝜑 → ((𝑥 + 𝑦) ∈ 𝐵 → (𝑥 + 𝑦) ≠ ∅))
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + 𝑦) ∈ 𝐵 → (𝑥 + 𝑦) ≠ ∅))
13	6, 12	mpd 16	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) ≠ ∅)
14	13	ralrimivva 3205	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ≠ ∅)
15		ovn0ssdmfun 7591	. . 3 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ≠ ∅ → ((𝐵 × 𝐵) ⊆ dom + ∧ Fun ( + ↾ (𝐵 × 𝐵))))
16	14, 15	syl 18	. 2 ⊢ (𝜑 → ((𝐵 × 𝐵) ⊆ dom + ∧ Fun ( + ↾ (𝐵 × 𝐵))))
17		mgmn0plusgf.r	. . . . . 6 ⊢ 𝑃 = ( + ↾ (𝐵 × 𝐵))
18	17	eqcomi 2769	. . . . 5 ⊢ ( + ↾ (𝐵 × 𝐵)) = 𝑃
19	18	funeqi 6563	. . . 4 ⊢ (Fun ( + ↾ (𝐵 × 𝐵)) ↔ Fun 𝑃)
20		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → Fun 𝑃)
21	17	dmeqi 5886	. . . . . . . 8 ⊢ dom 𝑃 = dom ( + ↾ (𝐵 × 𝐵))
22		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) → (𝐵 × 𝐵) ⊆ dom + )
23	22	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → (𝐵 × 𝐵) ⊆ dom + )
24		ssdmres 6005	. . . . . . . . 9 ⊢ ((𝐵 × 𝐵) ⊆ dom + ↔ dom ( + ↾ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
25	23, 24	sylib 221	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → dom ( + ↾ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
26	21, 25	eqtrid 2807	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → dom 𝑃 = (𝐵 × 𝐵))
27		df-fn 6544	. . . . . . 7 ⊢ (𝑃 Fn (𝐵 × 𝐵) ↔ (Fun 𝑃 ∧ dom 𝑃 = (𝐵 × 𝐵)))
28	20, 26, 27	sylanbrc 595	. . . . . 6 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → 𝑃 Fn (𝐵 × 𝐵))
29	22, 24	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) → dom ( + ↾ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
30	21, 29	eqtrid 2807	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) → dom 𝑃 = (𝐵 × 𝐵))
31	30	anim1ci 628	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → (Fun 𝑃 ∧ dom 𝑃 = (𝐵 × 𝐵)))
32	31, 27	sylibr 237	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → 𝑃 Fn (𝐵 × 𝐵))
33		elxp 5673	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
34	17	oveqi 7435	. . . . . . . . . . . . . 14 ⊢ (𝑥𝑃𝑦) = (𝑥( + ↾ (𝐵 × 𝐵))𝑦)
35		simprrl 793	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → 𝑥 ∈ 𝐵)
36		simprrr 794	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → 𝑦 ∈ 𝐵)
37	35, 36	ovresd 7589	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥( + ↾ (𝐵 × 𝐵))𝑦) = (𝑥 + 𝑦))
38	34, 37	eqtrid 2807	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥𝑃𝑦) = (𝑥 + 𝑦))
39	6	ex 418	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵))
40	39	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵))
41	40	adantr 486	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵))
42	41	a1d 26	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)))
43	42	imp32 424	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥 + 𝑦) ∈ 𝐵)
44	38, 43	eqeltrd 2860	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥𝑃𝑦) ∈ 𝐵)
45		fveq2 6888	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑃‘𝑧) = (𝑃‘⟨𝑥, 𝑦⟩))
46		df-ov 7425	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑃𝑦) = (𝑃‘⟨𝑥, 𝑦⟩)
47	45, 46	eqtr4di 2813	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑃‘𝑧) = (𝑥𝑃𝑦))
48	47	eleq1d 2845	. . . . . . . . . . . . . 14 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → ((𝑃‘𝑧) ∈ 𝐵 ↔ (𝑥𝑃𝑦) ∈ 𝐵))
49	48	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑃‘𝑧) ∈ 𝐵 ↔ (𝑥𝑃𝑦) ∈ 𝐵))
50	49	adantl 487	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → ((𝑃‘𝑧) ∈ 𝐵 ↔ (𝑥𝑃𝑦) ∈ 𝐵))
51	44, 50	mpbird 260	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑃‘𝑧) ∈ 𝐵)
52	51	ex 418	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑃‘𝑧) ∈ 𝐵))
53	52	exlimdvv 1966	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑃‘𝑧) ∈ 𝐵))
54	33, 53	biimtrid 245	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → (𝑧 ∈ (𝐵 × 𝐵) → (𝑃‘𝑧) ∈ 𝐵))
55	54	ralrimiv 3153	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → ∀𝑧 ∈ (𝐵 × 𝐵)(𝑃‘𝑧) ∈ 𝐵)
56		fnfvrnss 7124	. . . . . . 7 ⊢ ((𝑃 Fn (𝐵 × 𝐵) ∧ ∀𝑧 ∈ (𝐵 × 𝐵)(𝑃‘𝑧) ∈ 𝐵) → ran 𝑃 ⊆ 𝐵)
57	32, 55, 56	syl2anc 596	. . . . . 6 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → ran 𝑃 ⊆ 𝐵)
58		df-f 6545	. . . . . 6 ⊢ (𝑃:(𝐵 × 𝐵)⟶𝐵 ↔ (𝑃 Fn (𝐵 × 𝐵) ∧ ran 𝑃 ⊆ 𝐵))
59	28, 57, 58	sylanbrc 595	. . . . 5 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → 𝑃:(𝐵 × 𝐵)⟶𝐵)
60	59	ex 418	. . . 4 ⊢ ((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) → (Fun 𝑃 → 𝑃:(𝐵 × 𝐵)⟶𝐵))
61	19, 60	biimtrid 245	. . 3 ⊢ ((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) → (Fun ( + ↾ (𝐵 × 𝐵)) → 𝑃:(𝐵 × 𝐵)⟶𝐵))
62	61	expimpd 459	. 2 ⊢ (𝜑 → (((𝐵 × 𝐵) ⊆ dom + ∧ Fun ( + ↾ (𝐵 × 𝐵))) → 𝑃:(𝐵 × 𝐵)⟶𝐵))
63	16, 62	mpd 16	1 ⊢ (𝜑 → 𝑃:(𝐵 × 𝐵)⟶𝐵)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955   ∉ wnel 3061  ∀wral 3076   ⊆ wss 3896  ∅c0 4277  ⟨cop 4834   × cxp 5648  dom cdm 5650  ran crn 5651   ↾ cres 5652  Fun wfun 6535   Fn wfn 6536  ⟶wf 6537  ‘cfv 6541  (class class class)co 7422  Basecbs 17394  +_gcplusg 17435  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-pr 5390

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-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-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-fv 6549  df-ov 7425  df-mgm 18823

This theorem is used by:  mgmn0plusgplusf  18835