Theorem degenmgm2 19053

Description: A degenerate magma: although the operation is not defined for all pairs of elements of the base set, and its domain is not (a subset of) the base set, and the operation is not a function (see degenmgm2nfun 19052), the structure 𝑀 is still a magma according to our definition. (Contributed by AV, 21-Aug-2026.)

Hypothesis

Ref	Expression
degenmgm2.m	⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}⟩}

Assertion

Ref	Expression
degenmgm2	⊢ 𝑀 ∈ Mgm

Proof of Theorem degenmgm2

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

Step	Hyp	Ref	Expression
1		0ex 5264	. . 3 ⊢ ∅ ∈ V
2	1	prid1 4723	. 2 ⊢ ∅ ∈ {∅, 1o}
3	2, 2	pm3.2i 476	. . . . 5 ⊢ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o})
4		1oelpr 8466	. . . . . 6 ⊢ 1o ∈ {∅, 1o}
5	2, 4	pm3.2i 476	. . . . 5 ⊢ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})
6	3, 5	pm3.2i 476	. . . 4 ⊢ ((∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}) ∧ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o}))
7		1oex 8465	. . . . . 6 ⊢ 1o ∈ V
8		oveq1 7420	. . . . . . . . 9 ⊢ (𝑥 = ∅ → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) = (∅{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦))
9		df-ov 7416	. . . . . . . . 9 ⊢ (∅{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩)
10	8, 9	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑥 = ∅ → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩))
11	10	eleq1d 2845	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) ∈ {∅, 1o} ↔ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o}))
12	11	ralbidv 3185	. . . . . 6 ⊢ (𝑥 = ∅ → (∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) ∈ {∅, 1o} ↔ ∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o}))
13		oveq1 7420	. . . . . . . . 9 ⊢ (𝑥 = 1o → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) = (1o{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦))
14		df-ov 7416	. . . . . . . . 9 ⊢ (1o{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩)
15	13, 14	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑥 = 1o → (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩))
16	15	eleq1d 2845	. . . . . . 7 ⊢ (𝑥 = 1o → ((𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) ∈ {∅, 1o} ↔ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o}))
17	16	ralbidv 3185	. . . . . 6 ⊢ (𝑥 = 1o → (∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) ∈ {∅, 1o} ↔ ∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o}))
18	1, 7, 12, 17	ralpr 4661	. . . . 5 ⊢ (∀𝑥 ∈ {∅, 1o}∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) ∈ {∅, 1o} ↔ (∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ∧ ∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o}))
19		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ⟨∅, 𝑦⟩ = ⟨∅, ∅⟩)
20	19	fveq2d 6882	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, ∅⟩))
21		1n0 8474	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
22	21	necomi 3009	. . . . . . . . . . . 12 ⊢ ∅ ≠ 1o
23	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 1o)
24	1, 1	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 1o))
25	23, 24	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩
26	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 2o)
27	1, 1	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 2o))
28	26, 27	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩
29		2oex 8467	. . . . . . . . . . 11 ⊢ 2o ∈ V
30		opex 5439	. . . . . . . . . . 11 ⊢ ⟨∅, ∅⟩ ∈ V
31	7, 7, 29, 30	fvtp0 7199	. . . . . . . . . 10 ⊢ ((⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩ ∧ ⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩ ∧ ⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩) → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, ∅⟩) = ∅)
32	25, 28, 28, 31	mp3an 1490	. . . . . . . . 9 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, ∅⟩) = ∅
33	20, 32	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) = ∅)
34	33	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = ∅ → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
35		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨∅, 𝑦⟩ = ⟨∅, 1o⟩)
36	35	fveq2d 6882	. . . . . . . . 9 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 1o⟩))
37	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ 1o ≠ 1o)
38	1, 7	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨∅, 1o⟩ ≠ ⟨1o, 1o⟩ ↔ (∅ ≠ 1o ∨ 1o ≠ 1o))
39	37, 38	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 1o⟩
40	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ 1o ≠ 2o)
41	1, 7	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ 1o ≠ 2o))
42	40, 41	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩
43		1one2o 8634	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
44	43	olci 880	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ 1o ≠ 2o)
45	44, 41	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩
46		opex 5439	. . . . . . . . . . 11 ⊢ ⟨∅, 1o⟩ ∈ V
47	7, 7, 29, 46	fvtp0 7199	. . . . . . . . . 10 ⊢ ((⟨∅, 1o⟩ ≠ ⟨1o, 1o⟩ ∧ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩ ∧ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩) → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 1o⟩) = ∅)
48	39, 42, 45, 47	mp3an 1490	. . . . . . . . 9 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 1o⟩) = ∅
49	36, 48	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) = ∅)
50	49	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = 1o → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
51	1, 7, 34, 50	ralpr 4661	. . . . . 6 ⊢ (∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}))
52		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ⟨1o, 𝑦⟩ = ⟨1o, ∅⟩)
53	52	fveq2d 6882	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, ∅⟩))
54	22	olci 880	. . . . . . . . . . 11 ⊢ (1o ≠ 1o ∨ ∅ ≠ 1o)
55	7, 1	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨1o, ∅⟩ ≠ ⟨1o, 1o⟩ ↔ (1o ≠ 1o ∨ ∅ ≠ 1o))
56	54, 55	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, ∅⟩ ≠ ⟨1o, 1o⟩
57		2on0 8470	. . . . . . . . . . . . 13 ⊢ 2o ≠ ∅
58	57	olci 880	. . . . . . . . . . . 12 ⊢ (1o ≠ 1o ∨ 2o ≠ ∅)
59	7, 29	opthne 5458	. . . . . . . . . . . 12 ⊢ (⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩ ↔ (1o ≠ 1o ∨ 2o ≠ ∅))
60	58, 59	mpbir 234	. . . . . . . . . . 11 ⊢ ⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩
61	60	necomi 3009	. . . . . . . . . 10 ⊢ ⟨1o, ∅⟩ ≠ ⟨1o, 2o⟩
62		opex 5439	. . . . . . . . . . 11 ⊢ ⟨1o, ∅⟩ ∈ V
63	7, 7, 29, 62	fvtp0 7199	. . . . . . . . . 10 ⊢ ((⟨1o, ∅⟩ ≠ ⟨1o, 1o⟩ ∧ ⟨1o, ∅⟩ ≠ ⟨1o, 2o⟩ ∧ ⟨1o, ∅⟩ ≠ ⟨1o, 2o⟩) → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, ∅⟩) = ∅)
64	56, 61, 61, 63	mp3an 1490	. . . . . . . . 9 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, ∅⟩) = ∅
65	53, 64	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) = ∅)
66	65	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = ∅ → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
67		opeq2 4834	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨1o, 𝑦⟩ = ⟨1o, 1o⟩)
68	67	fveq2d 6882	. . . . . . . . 9 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 1o⟩))
69	43	olci 880	. . . . . . . . . . 11 ⊢ (1o ≠ 1o ∨ 1o ≠ 2o)
70	7, 7	opthne 5458	. . . . . . . . . . 11 ⊢ (⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (1o ≠ 1o ∨ 1o ≠ 2o))
71	69, 70	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩
72		opex 5439	. . . . . . . . . . 11 ⊢ ⟨1o, 1o⟩ ∈ V
73	72, 7	fvtp1 7193	. . . . . . . . . 10 ⊢ ((⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩ ∧ ⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩) → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 1o⟩) = 1o)
74	71, 71, 73	mp2an 705	. . . . . . . . 9 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 1o⟩) = 1o
75	68, 74	eqtrdi 2811	. . . . . . . 8 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) = 1o)
76	75	eleq1d 2845	. . . . . . 7 ⊢ (𝑦 = 1o → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o} ↔ 1o ∈ {∅, 1o}))
77	1, 7, 66, 76	ralpr 4661	. . . . . 6 ⊢ (∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o} ↔ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o}))
78	51, 77	anbi12i 640	. . . . 5 ⊢ ((∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ∧ ∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o}) ↔ ((∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}) ∧ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})))
79	18, 78	bitri 278	. . . 4 ⊢ (∀𝑥 ∈ {∅, 1o}∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) ∈ {∅, 1o} ↔ ((∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}) ∧ (∅ ∈ {∅, 1o} ∧ 1o ∈ {∅, 1o})))
80	6, 79	mpbir 234	. . 3 ⊢ ∀𝑥 ∈ {∅, 1o}∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) ∈ {∅, 1o}
81		prex 5403	. . . . 5 ⊢ {∅, 1o} ∈ V
82		degenmgm2.m	. . . . . 6 ⊢ 𝑀 = {⟨(Base‘ndx), {∅, 1o}⟩, ⟨(+g‘ndx), {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}⟩}
83	82	grpbase 17374	. . . . 5 ⊢ ({∅, 1o} ∈ V → {∅, 1o} = (Base‘𝑀))
84	81, 83	ax-mp 5	. . . 4 ⊢ {∅, 1o} = (Base‘𝑀)
85		tpex 7747	. . . . 5 ⊢ {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩} ∈ V
86	82	grpplusg 17375	. . . . 5 ⊢ ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩} ∈ V → {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩} = (+g‘𝑀))
87	85, 86	ax-mp 5	. . . 4 ⊢ {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩} = (+g‘𝑀)
88	84, 87	ismgmn0 18732	. . 3 ⊢ (∅ ∈ {∅, 1o} → (𝑀 ∈ Mgm ↔ ∀𝑥 ∈ {∅, 1o}∀𝑦 ∈ {∅, 1o} (𝑥{⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}𝑦) ∈ {∅, 1o}))
89	80, 88	mpbiri 261	. 2 ⊢ (∅ ∈ {∅, 1o} → 𝑀 ∈ Mgm)
90	2, 89	ax-mp 5	1 ⊢ 𝑀 ∈ Mgm

This proof depends on syntax axioms:   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  Vcvv 3450  ∅c0 4279  {cpr 4586  {ctp 4588  ⟨cop 4590  ‘cfv 6533  (class class class)co 7413  1_oc1o 8448  2_oc2o 8449  ndxcnx 17285  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-pow 5330  ax-pr 5398  ax-un 7736  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-reu 3366  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-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-nn 12258  df-2 12327  df-n0 12529  df-z 12616  df-uz 12888  df-fz 13562  df-struct 17239  df-slot 17274  df-ndx 17286  df-base 17302  df-plusg 17355  df-mgm 18730

This theorem is used by: (None)