Theorem degenmgm2 19147

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 19146), 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 5260	. . 3 ⊢ ∅ ∈ V
2	1	prid1 4719	. 2 ⊢ ∅ ∈ {∅, 1o}
3	2, 2	pm3.2i 476	. . . . 5 ⊢ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o})
4		1oelpr 8492	. . . . . 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 8491	. . . . . 6 ⊢ 1o ∈ V
8		oveq1 7429	. . . . . . . . 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 7425	. . . . . . . . 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 7429	. . . . . . . . 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 7425	. . . . . . . . 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 4657	. . . . 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 4840	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ⟨∅, 𝑦⟩ = ⟨∅, ∅⟩)
20	19	fveq2d 6892	. . . . . . . . 9 ⊢ (𝑦 = ∅ → ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) = ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, ∅⟩))
21		1n0 8500	. . . . . . . . . . . . 13 ⊢ 1o ≠ ∅
22	21	necomi 3009	. . . . . . . . . . . 12 ⊢ ∅ ≠ 1o
23	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 1o)
24	1, 1	opthne 5450	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 1o))
25	23, 24	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 1o⟩
26	22	orci 879	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ ∅ ≠ 2o)
27	1, 1	opthne 5450	. . . . . . . . . . 11 ⊢ (⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ ∅ ≠ 2o))
28	26, 27	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, ∅⟩ ≠ ⟨1o, 2o⟩
29		2oex 8493	. . . . . . . . . . 11 ⊢ 2o ∈ V
30		opex 5431	. . . . . . . . . . 11 ⊢ ⟨∅, ∅⟩ ∈ V
31	7, 7, 29, 30	fvtp0 7208	. . . . . . . . . 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 4840	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨∅, 𝑦⟩ = ⟨∅, 1o⟩)
36	35	fveq2d 6892	. . . . . . . . 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 5450	. . . . . . . . . . 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 5450	. . . . . . . . . . 11 ⊢ (⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (∅ ≠ 1o ∨ 1o ≠ 2o))
42	40, 41	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩
43		1one2o 8660	. . . . . . . . . . . 12 ⊢ 1o ≠ 2o
44	43	olci 880	. . . . . . . . . . 11 ⊢ (∅ ≠ 1o ∨ 1o ≠ 2o)
45	44, 41	mpbir 234	. . . . . . . . . 10 ⊢ ⟨∅, 1o⟩ ≠ ⟨1o, 2o⟩
46		opex 5431	. . . . . . . . . . 11 ⊢ ⟨∅, 1o⟩ ∈ V
47	7, 7, 29, 46	fvtp0 7208	. . . . . . . . . 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 4657	. . . . . 6 ⊢ (∀𝑦 ∈ {∅, 1o} ({⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩}‘⟨∅, 𝑦⟩) ∈ {∅, 1o} ↔ (∅ ∈ {∅, 1o} ∧ ∅ ∈ {∅, 1o}))
52		opeq2 4840	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ⟨1o, 𝑦⟩ = ⟨1o, ∅⟩)
53	52	fveq2d 6892	. . . . . . . . 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 5450	. . . . . . . . . . 11 ⊢ (⟨1o, ∅⟩ ≠ ⟨1o, 1o⟩ ↔ (1o ≠ 1o ∨ ∅ ≠ 1o))
56	54, 55	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, ∅⟩ ≠ ⟨1o, 1o⟩
57		2on0 8496	. . . . . . . . . . . . 13 ⊢ 2o ≠ ∅
58	57	olci 880	. . . . . . . . . . . 12 ⊢ (1o ≠ 1o ∨ 2o ≠ ∅)
59	7, 29	opthne 5450	. . . . . . . . . . . 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 5431	. . . . . . . . . . 11 ⊢ ⟨1o, ∅⟩ ∈ V
63	7, 7, 29, 62	fvtp0 7208	. . . . . . . . . 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 4840	. . . . . . . . . 10 ⊢ (𝑦 = 1o → ⟨1o, 𝑦⟩ = ⟨1o, 1o⟩)
68	67	fveq2d 6892	. . . . . . . . 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 5450	. . . . . . . . . . 11 ⊢ (⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩ ↔ (1o ≠ 1o ∨ 1o ≠ 2o))
71	69, 70	mpbir 234	. . . . . . . . . 10 ⊢ ⟨1o, 1o⟩ ≠ ⟨1o, 2o⟩
72		opex 5431	. . . . . . . . . . 11 ⊢ ⟨1o, 1o⟩ ∈ V
73	72, 7	fvtp1 7202	. . . . . . . . . 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 4657	. . . . . 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 5395	. . . . 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 17467	. . . . 5 ⊢ ({∅, 1o} ∈ V → {∅, 1o} = (Base‘𝑀))
84	81, 83	ax-mp 5	. . . 4 ⊢ {∅, 1o} = (Base‘𝑀)
85		tpex 7764	. . . . 5 ⊢ {⟨⟨1o, 1o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 1o⟩, ⟨⟨1o, 2o⟩, 2o⟩} ∈ V
86	82	grpplusg 17468	. . . . 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 18825	. . 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 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  Vcvv 3450  ∅c0 4277  {cpr 4586  {ctp 4588  ⟨cop 4834  ‘cfv 6541  (class class class)co 7422  1_oc1o 8474  2_oc2o 8475  ndxcnx 17378  Basecbs 17394  +_gcplusg 17435  Mgmcmgm 18821

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753  ax-cnex 11261  ax-resscn 11262  ax-1cn 11263  ax-icn 11264  ax-addcl 11265  ax-addrcl 11266  ax-mulcl 11267  ax-mulrcl 11268  ax-mulcom 11269  ax-addass 11270  ax-mulass 11271  ax-distr 11272  ax-i2m1 11273  ax-1ne0 11274  ax-1rid 11275  ax-rnegex 11276  ax-rrecex 11277  ax-cnre 11278  ax-pre-lttri 11279  ax-pre-lttrn 11280  ax-pre-ltadd 11281  ax-pre-mulgt0 11282

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  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 3739  df-csb 3848  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-riota 7379  df-ov 7425  df-oprab 7426  df-mpo 7427  df-om 7880  df-1st 8003  df-2nd 8004  df-frecs 8304  df-wrecs 8335  df-recs 8384  df-rdg 8423  df-1o 8481  df-2o 8482  df-er 8722  df-en 8979  df-dom 8980  df-sdom 8981  df-fin 8982  df-pnf 11350  df-mnf 11351  df-xr 11352  df-ltxr 11353  df-le 11354  df-sub 11548  df-neg 11549  df-nn 12341  df-2 12410  df-n0 12612  df-z 12699  df-uz 12971  df-fz 13645  df-struct 17332  df-slot 17367  df-ndx 17379  df-base 17395  df-plusg 17448  df-mgm 18823

This theorem is used by: (None)