Theorem mgmn0plusgf 18820

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 7604 and ndmovrcl 7605). (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 18812	. . . . . . 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 3063	. . . . . . . 8 ⊢ (∅ ∉ 𝐵 ↔ ¬ ∅ ∈ 𝐵)
9		nelelne 3057	. . . . . . . 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 3206	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ≠ ∅)
15		ovn0ssdmfun 7587	. . 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 2770	. . . . 5 ⊢ ( + ↾ (𝐵 × 𝐵)) = 𝑃
19	18	funeqi 6558	. . . 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 6004	. . . . . . . . 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 2808	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → dom 𝑃 = (𝐵 × 𝐵))
27		df-fn 6540	. . . . . . 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 2808	. . . . . . . . 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 5674	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
34	17	oveqi 7431	. . . . . . . . . . . . . 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 7585	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥( + ↾ (𝐵 × 𝐵))𝑦) = (𝑥 + 𝑦))
38	34, 37	eqtrid 2808	. . . . . . . . . . . . 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 2861	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥𝑃𝑦) ∈ 𝐵)
45		fveq2 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑃‘𝑧) = (𝑃‘⟨𝑥, 𝑦⟩))
46		df-ov 7421	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑃𝑦) = (𝑃‘⟨𝑥, 𝑦⟩)
47	45, 46	eqtr4di 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑃‘𝑧) = (𝑥𝑃𝑦))
48	47	eleq1d 2846	. . . . . . . . . . . . . 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 1967	. . . . . . . . 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 3154	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → ∀𝑧 ∈ (𝐵 × 𝐵)(𝑃‘𝑧) ∈ 𝐵)
56		fnfvrnss 7119	. . . . . . 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 6541	. . . . . 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 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956   ∉ wnel 3062  ∀wral 3077   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590   × cxp 5649  dom cdm 5651  ran crn 5652   ↾ cres 5653  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  ‘cfv 6537  (class class class)co 7418  Basecbs 17380  +_gcplusg 17421  Mgmcmgm 18807

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7421  df-mgm 18809

This theorem is used by:  mgmn0plusgplusf  18821