Theorem mgmn0plusgf 18877

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 7638 and ndmovrcl 7639). (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 18869	. . . . . . 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 3066	. . . . . . . 8 ⊢ (∅ ∉ 𝐵 ↔ ¬ ∅ ∈ 𝐵)
9		nelelne 3060	. . . . . . . 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 3209	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) ≠ ∅)
15		ovn0ssdmfun 7621	. . 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 2772	. . . . 5 ⊢ ( + ↾ (𝐵 × 𝐵)) = 𝑃
19	18	funeqi 6588	. . . 4 ⊢ (Fun ( + ↾ (𝐵 × 𝐵)) ↔ Fun 𝑃)
20		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → Fun 𝑃)
21	17	dmeqi 5898	. . . . . . . 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 6021	. . . . . . . . 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 2810	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → dom 𝑃 = (𝐵 × 𝐵))
27		df-fn 6569	. . . . . . 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 2810	. . . . . . . . 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 5681	. . . . . . . . 9 ⊢ (𝑧 ∈ (𝐵 × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)))
34	17	oveqi 7464	. . . . . . . . . . . . . 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 7619	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥( + ↾ (𝐵 × 𝐵))𝑦) = (𝑥 + 𝑦))
38	34, 37	eqtrid 2810	. . . . . . . . . . . . 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 2864	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) ∧ (𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))) → (𝑥𝑃𝑦) ∈ 𝐵)
45		fveq2 6916	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑃‘𝑧) = (𝑃‘⟨𝑥, 𝑦⟩))
46		df-ov 7454	. . . . . . . . . . . . . . . 16 ⊢ (𝑥𝑃𝑦) = (𝑃‘⟨𝑥, 𝑦⟩)
47	45, 46	eqtr4di 2816	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑃‘𝑧) = (𝑥𝑃𝑦))
48	47	eleq1d 2849	. . . . . . . . . . . . . 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 1969	. . . . . . . . 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 3157	. . . . . . 7 ⊢ (((𝜑 ∧ (𝐵 × 𝐵) ⊆ dom + ) ∧ Fun 𝑃) → ∀𝑧 ∈ (𝐵 × 𝐵)(𝑃‘𝑧) ∈ 𝐵)
56		fnfvrnss 7153	. . . . . . 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 6570	. . . . . 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 2147   ≠ wne 2959   ∉ wnel 3065  ∀wral 3080   ⊆ wss 3900  ∅c0 4281  ⟨cop 4840   × cxp 5656  dom cdm 5658  ran crn 5659   ↾ cres 5660  Fun wfun 6560   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451  Basecbs 17431  +_gcplusg 17472  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-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 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-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-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fv 6574  df-ov 7454  df-mgm 18866

This theorem is used by:  mgmn0plusgplusf  18878