Theorem degenmgm2 19040

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 19039), 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 5272	. . 3 ⊢ ∅ ∈ V
2	1	prid1 4730	. 2 ⊢ ∅ ∈ {∅, 1o}
3	2, 2	pm3.2i 476	. . . . 5 ⊢ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o})
4		1oelpr 8470	. . . . . 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 8469	. . . . . 6 ⊢ 1o ∈ V
8		oveq1 7426	. . . . . . . . 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 7422	. . . . . . . . 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 2816	. . . . . . . 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 2850	. . . . . . 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 3190	. . . . . 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 7426	. . . . . . . . 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 7422	. . . . . . . . 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 2816	. . . . . . . 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 2850	. . . . . . 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 3190	. . . . . 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 4668	. . . . 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 4841	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ⟨∅, 𝑦⟩ = ⟨∅, ∅⟩)
20	19	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, ∅⟩))
21		1n0 8478	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
22	21	necomi 3014	. . . . . . . . . . . 12 ⊢ ∅ ≠ 1o
23	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 1o)
24	1, 1	opthne 5466	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 1o))
25	23, 24	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩
26	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 2o)
27	1, 1	opthne 5466	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 2o))
28	26, 27	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩
29		2oex 8471	. . . . . . . . . . 11 ⊢ 2o ∈ V
30		opex 5447	. . . . . . . . . . 11 ⊢ ⟨∅, ∅⟩ ∈ V
31	7, 7, 29, 30	fvtp0 7205	. . . . . . . . . 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 2816	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) = ∅)
34	33	eleq1d 2850	. . . . . . 7 ⊢ (𝑦 = ∅ → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
35		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨∅, 𝑦⟩ = ⟨∅, 1o⟩)
36	35	fveq2d 6889	. . . . . . . . 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 5466	. . . . . . . . . . 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 5466	. . . . . . . . . . 11 ⊢ (⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ 1o ≠ 2o))
42	40, 41	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩
43		1one2o 8638	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
44	43	olci 880	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ 1o ≠ 2o)
45	44, 41	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩
46		opex 5447	. . . . . . . . . . 11 ⊢ ⟨∅, 1o⟩ ∈ V
47	7, 7, 29, 46	fvtp0 7205	. . . . . . . . . 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 2816	. . . . . . . 8 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) = ∅)
50	49	eleq1d 2850	. . . . . . 7 ⊢ (𝑦 = 1o → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
51	1, 7, 34, 50	ralpr 4668	. . . . . 6 ⊢ (∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}))
52		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ⟨1o, 𝑦⟩ = ⟨1o, ∅⟩)
53	52	fveq2d 6889	. . . . . . . . 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 5466	. . . . . . . . . . 11 ⊢ (⟨1o, ∅⟩ ≠ ⟨1o, 1o⟩ ↔ (1o ≠ 1o ∨ ∅ ≠ 1o))
56	54, 55	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, ∅⟩ ≠ ⟨1o, 1o⟩
57		2on0 8474	. . . . . . . . . . . . 13 ⊢ 2o ≠ ∅
58	57	olci 880	. . . . . . . . . . . 12 ⊢ (1o ≠ 1o ∨ 2o ≠ ∅)
59	7, 29	opthne 5466	. . . . . . . . . . . 12 ⊢ (⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩ ↔ (1o ≠ 1o ∨ 2o ≠ ∅))
60	58, 59	mpbir 234	. . . . . . . . . . 11 ⊢ ⟨1o, 2o⟩ ≠ ⟨1o, ∅⟩
61	60	necomi 3014	. . . . . . . . . 10 ⊢ ⟨1o, ∅⟩ ≠ ⟨1o, 2o⟩
62		opex 5447	. . . . . . . . . . 11 ⊢ ⟨1o, ∅⟩ ∈ V
63	7, 7, 29, 62	fvtp0 7205	. . . . . . . . . 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 2816	. . . . . . . 8 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) = ∅)
66	65	eleq1d 2850	. . . . . . 7 ⊢ (𝑦 = ∅ → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o} ↔ ∅ ∈ {∅, 1o}))
67		opeq2 4841	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨1o, 𝑦⟩ = ⟨1o, 1o⟩)
68	67	fveq2d 6889	. . . . . . . . 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 5466	. . . . . . . . . . 11 ⊢ (⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (1o ≠ 1o ∨ 1o ≠ 2o))
71	69, 70	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩
72		opex 5447	. . . . . . . . . . 11 ⊢ ⟨1o, 1o⟩ ∈ V
73	72, 7	fvtp1 7199	. . . . . . . . . 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 2816	. . . . . . . 8 ⊢ (𝑦 = 1o → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) = 1o)
76	75	eleq1d 2850	. . . . . . 7 ⊢ (𝑦 = 1o → (({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨1o, 𝑦⟩) ∈ {∅, 1o} ↔ 1o ∈ {∅, 1o}))
77	1, 7, 66, 76	ralpr 4668	. . . . . 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 5411	. . . . 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 17364	. . . . 5 ⊢ ({∅, 1o} ∈ V → {∅, 1o} = (Base‘𝑀))
84	81, 83	ax-mp 5	. . . 4 ⊢ {∅, 1o} = (Base‘𝑀)
85		tpex 7753	. . . . 5 ⊢ {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩} ∈ V
86	82	grpplusg 17365	. . . . 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 18722	. . 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 2146   ≠ wne 2960  ∀wral 3081  Vcvv 3457  ∅c0 4286  {cpr 4593  {ctp 4595  ⟨cop 4597  ‘cfv 6540  (class class class)co 7419  1_oc1o 8452  2_oc2o 8453  ndxcnx 17275  Basecbs 17291  +_gcplusg 17332  Mgmcmgm 18718

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11171  ax-resscn 11172  ax-1cn 11173  ax-icn 11174  ax-addcl 11175  ax-addrcl 11176  ax-mulcl 11177  ax-mulrcl 11178  ax-mulcom 11179  ax-addass 11180  ax-mulass 11181  ax-distr 11182  ax-i2m1 11183  ax-1ne0 11184  ax-1rid 11185  ax-rnegex 11186  ax-rrecex 11187  ax-cnre 11188  ax-pre-lttri 11189  ax-pre-lttrn 11190  ax-pre-ltadd 11191  ax-pre-mulgt0 11192

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-pnf 11260  df-mnf 11261  df-xr 11262  df-ltxr 11263  df-le 11264  df-sub 11458  df-neg 11459  df-nn 12249  df-2 12318  df-n0 12520  df-z 12607  df-uz 12879  df-fz 13552  df-struct 17229  df-slot 17264  df-ndx 17276  df-base 17292  df-plusg 17345  df-mgm 18720

This theorem is used by: (None)