Theorem mgmn0plusgf 18741

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 7599 and ndmovrcl 7600). (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 18733	. . . . . . 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 7582	. . 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 6554	. . . 4 ⊢ (Fun ( + ↾ (𝐵 × 𝐵)) ↔ Fun 𝑃)
20		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → Fun 𝑃)
21	17	dmeqi 5888	. . . . . . . 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 6006	. . . . . . . . 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 6536	. . . . . . 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 5678	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
34	17	oveqi 7426	. . . . . . . . . . . . . 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 7580	. . . . . . . . . . . . . 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 6878	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑃‘𝑧) = (𝑃‘⟨𝑥, 𝑦⟩))
46		df-ov 7416	. . . . . . . . . . . . . . . 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 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 3153	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → ∀𝑧 ∈ (𝐵 × 𝐵)(𝑃‘𝑧) ∈ 𝐵)
56		fnfvrnss 7114	. . . . . . 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 6537	. . . . . 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 2955   ∉ wnel 3061  ∀wral 3076   ⊆ wss 3899  ∅c0 4279  ⟨cop 4590   × cxp 5653  dom cdm 5655  ran crn 5656   ↾ cres 5657  Fun wfun 6527   Fn wfn 6528  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  Basecbs 17301  +_gcplusg 17342  Mgmcmgm 18728

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 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 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 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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-mgm 18730

This theorem is used by:  mgmn0plusgplusf  18742