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Theorem ring1 13028

Description: The (smallest) structure representing a zero ring. (Contributed by AV, 28-Apr-2019.)

**Hypothesis**
Ref	Expression
ring1.m	⊢ 𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(._r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}

**Assertion**
Ref	Expression
ring1	⊢ (𝑍 ∈ 𝑉 → 𝑀 ∈ Ring)

Proof of Theorem ring1 Dummy variables 𝑎 𝑏 𝑐 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		snexg 4179	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑉 → {𝑍} ∈ V)
2		opexg 4222	. . . . . . . . . . 11 ⊢ ((𝑍 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → ⟨𝑍, 𝑍⟩ ∈ V)
3	2	anidms 397	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑉 → ⟨𝑍, 𝑍⟩ ∈ V)
4		opexg 4222	. . . . . . . . . 10 ⊢ ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍 ∈ 𝑉) → ⟨⟨𝑍, 𝑍⟩, 𝑍⟩ ∈ V)
5	3, 4	mpancom 422	. . . . . . . . 9 ⊢ (𝑍 ∈ 𝑉 → ⟨⟨𝑍, 𝑍⟩, 𝑍⟩ ∈ V)
6		snexg 4179	. . . . . . . . 9 ⊢ (⟨⟨𝑍, 𝑍⟩, 𝑍⟩ ∈ V → {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V)
7	5, 6	syl 14	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑉 → {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V)
8		ring1.m	. . . . . . . . 9 ⊢ 𝑀 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(._r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}
9	8	rngbaseg 12545	. . . . . . . 8 ⊢ (({𝑍} ∈ V ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V) → {𝑍} = (Base‘𝑀))
10	1, 7, 7, 9	syl3anc 1238	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → {𝑍} = (Base‘𝑀))
11	10	opeq2d 3781	. . . . . 6 ⊢ (𝑍 ∈ 𝑉 → ⟨(Base‘ndx), {𝑍}⟩ = ⟨(Base‘ndx), (Base‘𝑀)⟩)
12	8	rngplusgg 12546	. . . . . . . 8 ⊢ (({𝑍} ∈ V ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V) → {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} = (+_g‘𝑀))
13	1, 7, 7, 12	syl3anc 1238	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} = (+_g‘𝑀))
14	13	opeq2d 3781	. . . . . 6 ⊢ (𝑍 ∈ 𝑉 → ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ = ⟨(+_g‘ndx), (+_g‘𝑀)⟩)
15	11, 14	preq12d 3674	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩} = {⟨(Base‘ndx), (Base‘𝑀)⟩, ⟨(+_g‘ndx), (+_g‘𝑀)⟩})
16		eqid 2175	. . . . . 6 ⊢ {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩} = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}
17	16	grp1 12835	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩} ∈ Grp)
18	15, 17	eqeltrrd 2253	. . . 4 ⊢ (𝑍 ∈ 𝑉 → {⟨(Base‘ndx), (Base‘𝑀)⟩, ⟨(+_g‘ndx), (+_g‘𝑀)⟩} ∈ Grp)
19		basendxnn 12482	. . . . . . . 8 ⊢ (Base‘ndx) ∈ ℕ
20		opexg 4222	. . . . . . . 8 ⊢ (((Base‘ndx) ∈ ℕ ∧ {𝑍} ∈ V) → ⟨(Base‘ndx), {𝑍}⟩ ∈ V)
21	19, 1, 20	sylancr 414	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → ⟨(Base‘ndx), {𝑍}⟩ ∈ V)
22		plusgslid 12524	. . . . . . . . 9 ⊢ (+_g = Slot (+_g‘ndx) ∧ (+_g‘ndx) ∈ ℕ)
23	22	simpri 113	. . . . . . . 8 ⊢ (+_g‘ndx) ∈ ℕ
24		opexg 4222	. . . . . . . 8 ⊢ (((+_g‘ndx) ∈ ℕ ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V) → ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ V)
25	23, 7, 24	sylancr 414	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ V)
26		mulrslid 12541	. . . . . . . . 9 ⊢ (._r = Slot (._r‘ndx) ∧ (._r‘ndx) ∈ ℕ)
27	26	simpri 113	. . . . . . . 8 ⊢ (._r‘ndx) ∈ ℕ
28		opexg 4222	. . . . . . . 8 ⊢ (((._r‘ndx) ∈ ℕ ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V) → ⟨(._r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ V)
29	27, 7, 28	sylancr 414	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → ⟨(._r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ V)
30		tpexg 4438	. . . . . . 7 ⊢ ((⟨(Base‘ndx), {𝑍}⟩ ∈ V ∧ ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ V ∧ ⟨(._r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩ ∈ V) → {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(._r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩} ∈ V)
31	21, 25, 29, 30	syl3anc 1238	. . . . . 6 ⊢ (𝑍 ∈ 𝑉 → {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(._r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩} ∈ V)
32	8, 31	eqeltrid 2262	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∈ V)
33		eqid 2175	. . . . . 6 ⊢ (Base‘𝑀) = (Base‘𝑀)
34		eqid 2175	. . . . . 6 ⊢ (+_g‘𝑀) = (+_g‘𝑀)
35		eqid 2175	. . . . . 6 ⊢ {⟨(Base‘ndx), (Base‘𝑀)⟩, ⟨(+_g‘ndx), (+_g‘𝑀)⟩} = {⟨(Base‘ndx), (Base‘𝑀)⟩, ⟨(+_g‘ndx), (+_g‘𝑀)⟩}
36	33, 34, 35	grppropstrg 12755	. . . . 5 ⊢ (𝑀 ∈ V → (𝑀 ∈ Grp ↔ {⟨(Base‘ndx), (Base‘𝑀)⟩, ⟨(+_g‘ndx), (+_g‘𝑀)⟩} ∈ Grp))
37	32, 36	syl 14	. . . 4 ⊢ (𝑍 ∈ 𝑉 → (𝑀 ∈ Grp ↔ {⟨(Base‘ndx), (Base‘𝑀)⟩, ⟨(+_g‘ndx), (+_g‘𝑀)⟩} ∈ Grp))
38	18, 37	mpbird 167	. . 3 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∈ Grp)
39	16	mnd1 12708	. . . 4 ⊢ (𝑍 ∈ 𝑉 → {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩} ∈ Mnd)
40		eqidd 2176	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (Base‘(mulGrp‘𝑀)) = (Base‘(mulGrp‘𝑀)))
41	16	grpbaseg 12537	. . . . . . 7 ⊢ (({𝑍} ∈ V ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V) → {𝑍} = (Base‘{⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}))
42	1, 7, 41	syl2anc 411	. . . . . 6 ⊢ (𝑍 ∈ 𝑉 → {𝑍} = (Base‘{⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}))
43		eqid 2175	. . . . . . . 8 ⊢ (mulGrp‘𝑀) = (mulGrp‘𝑀)
44	43, 33	mgpbasg 12930	. . . . . . 7 ⊢ (𝑀 ∈ V → (Base‘𝑀) = (Base‘(mulGrp‘𝑀)))
45	32, 44	syl 14	. . . . . 6 ⊢ (𝑍 ∈ 𝑉 → (Base‘𝑀) = (Base‘(mulGrp‘𝑀)))
46	10, 42, 45	3eqtr3rd 2217	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (Base‘(mulGrp‘𝑀)) = (Base‘{⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}))
47	8	rngmulrg 12547	. . . . . . . 8 ⊢ (({𝑍} ∈ V ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V) → {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} = (._r‘𝑀))
48	1, 7, 7, 47	syl3anc 1238	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} = (._r‘𝑀))
49	16	grpplusgg 12538	. . . . . . . 8 ⊢ (({𝑍} ∈ V ∧ {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} ∈ V) → {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} = (+_g‘{⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}))
50	1, 7, 49	syl2anc 411	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → {⟨⟨𝑍, 𝑍⟩, 𝑍⟩} = (+_g‘{⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}))
51		eqid 2175	. . . . . . . . 9 ⊢ (._r‘𝑀) = (._r‘𝑀)
52	43, 51	mgpplusgg 12929	. . . . . . . 8 ⊢ (𝑀 ∈ V → (._r‘𝑀) = (+_g‘(mulGrp‘𝑀)))
53	32, 52	syl 14	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → (._r‘𝑀) = (+_g‘(mulGrp‘𝑀)))
54	48, 50, 53	3eqtr3rd 2217	. . . . . 6 ⊢ (𝑍 ∈ 𝑉 → (+_g‘(mulGrp‘𝑀)) = (+_g‘{⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}))
55	54	oveqdr 5893	. . . . 5 ⊢ ((𝑍 ∈ 𝑉 ∧ (𝑥 ∈ (Base‘(mulGrp‘𝑀)) ∧ 𝑦 ∈ (Base‘(mulGrp‘𝑀)))) → (𝑥(+_g‘(mulGrp‘𝑀))𝑦) = (𝑥(+_g‘{⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩})𝑦))
56	40, 46, 55	mndpropd 12705	. . . 4 ⊢ (𝑍 ∈ 𝑉 → ((mulGrp‘𝑀) ∈ Mnd ↔ {⟨(Base‘ndx), {𝑍}⟩, ⟨(+_g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩} ∈ Mnd))
57	39, 56	mpbird 167	. . 3 ⊢ (𝑍 ∈ 𝑉 → (mulGrp‘𝑀) ∈ Mnd)
58		df-ov 5868	. . . . . . 7 ⊢ (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍) = ({⟨⟨𝑍, 𝑍⟩, 𝑍⟩}‘⟨𝑍, 𝑍⟩)
59		fvsng 5704	. . . . . . . 8 ⊢ ((⟨𝑍, 𝑍⟩ ∈ V ∧ 𝑍 ∈ 𝑉) → ({⟨⟨𝑍, 𝑍⟩, 𝑍⟩}‘⟨𝑍, 𝑍⟩) = 𝑍)
60	3, 59	mpancom 422	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → ({⟨⟨𝑍, 𝑍⟩, 𝑍⟩}‘⟨𝑍, 𝑍⟩) = 𝑍)
61	58, 60	eqtrid 2220	. . . . . 6 ⊢ (𝑍 ∈ 𝑉 → (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍) = 𝑍)
62	61	oveq2d 5881	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍))
63	61, 61	oveq12d 5883	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍))
64	62, 63	eqtr4d 2211	. . . 4 ⊢ (𝑍 ∈ 𝑉 → (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)))
65	61	oveq1d 5880	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍))
66	65, 63	eqtr4d 2211	. . . 4 ⊢ (𝑍 ∈ 𝑉 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)))
67		oveq1 5872	. . . . . . . . 9 ⊢ (𝑎 = 𝑍 → (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))
68		oveq1 5872	. . . . . . . . . 10 ⊢ (𝑎 = 𝑍 → (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏))
69		oveq1 5872	. . . . . . . . . 10 ⊢ (𝑎 = 𝑍 → (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))
70	68, 69	oveq12d 5883	. . . . . . . . 9 ⊢ (𝑎 = 𝑍 → ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))
71	67, 70	eqeq12d 2190	. . . . . . . 8 ⊢ (𝑎 = 𝑍 → ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ↔ (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))))
72	68	oveq1d 5880	. . . . . . . . 9 ⊢ (𝑎 = 𝑍 → ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))
73	69	oveq1d 5880	. . . . . . . . 9 ⊢ (𝑎 = 𝑍 → ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))
74	72, 73	eqeq12d 2190	. . . . . . . 8 ⊢ (𝑎 = 𝑍 → (((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ↔ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))))
75	71, 74	anbi12d 473	. . . . . . 7 ⊢ (𝑎 = 𝑍 → (((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))))
76	75	2ralbidv 2499	. . . . . 6 ⊢ (𝑎 = 𝑍 → (∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))))
77	76	ralsng 3629	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (∀𝑎 ∈ {𝑍}∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))))
78		oveq1 5872	. . . . . . . . . 10 ⊢ (𝑏 = 𝑍 → (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))
79	78	oveq2d 5881	. . . . . . . . 9 ⊢ (𝑏 = 𝑍 → (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))
80		oveq2 5873	. . . . . . . . . 10 ⊢ (𝑏 = 𝑍 → (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍))
81	80	oveq1d 5880	. . . . . . . . 9 ⊢ (𝑏 = 𝑍 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))
82	79, 81	eqeq12d 2190	. . . . . . . 8 ⊢ (𝑏 = 𝑍 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ↔ (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))))
83	80	oveq1d 5880	. . . . . . . . 9 ⊢ (𝑏 = 𝑍 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))
84	78	oveq2d 5881	. . . . . . . . 9 ⊢ (𝑏 = 𝑍 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))
85	83, 84	eqeq12d 2190	. . . . . . . 8 ⊢ (𝑏 = 𝑍 → (((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ↔ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))))
86	82, 85	anbi12d 473	. . . . . . 7 ⊢ (𝑏 = 𝑍 → (((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))))
87	86	ralbidv 2475	. . . . . 6 ⊢ (𝑏 = 𝑍 → (∀𝑐 ∈ {𝑍} ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ∀𝑐 ∈ {𝑍} ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))))
88	87	ralsng 3629	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ∀𝑐 ∈ {𝑍} ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))))
89		oveq2 5873	. . . . . . . . 9 ⊢ (𝑐 = 𝑍 → (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍))
90	89	oveq2d 5881	. . . . . . . 8 ⊢ (𝑐 = 𝑍 → (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)))
91	89	oveq2d 5881	. . . . . . . 8 ⊢ (𝑐 = 𝑍 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)))
92	90, 91	eqeq12d 2190	. . . . . . 7 ⊢ (𝑐 = 𝑍 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ↔ (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍))))
93		oveq2 5873	. . . . . . . 8 ⊢ (𝑐 = 𝑍 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍))
94	89, 89	oveq12d 5883	. . . . . . . 8 ⊢ (𝑐 = 𝑍 → ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)))
95	93, 94	eqeq12d 2190	. . . . . . 7 ⊢ (𝑐 = 𝑍 → (((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ↔ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍))))
96	92, 95	anbi12d 473	. . . . . 6 ⊢ (𝑐 = 𝑍 → (((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)))))
97	96	ralsng 3629	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (∀𝑐 ∈ {𝑍} ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)))))
98	77, 88, 97	3bitrd 214	. . . 4 ⊢ (𝑍 ∈ 𝑉 → (∀𝑎 ∈ {𝑍}∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)) ∧ ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍) = ((𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑍{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑍)))))
99	64, 66, 98	mpbir2and 944	. . 3 ⊢ (𝑍 ∈ 𝑉 → ∀𝑎 ∈ {𝑍}∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))))
100	38, 57, 99	3jca 1177	. 2 ⊢ (𝑍 ∈ 𝑉 → (𝑀 ∈ Grp ∧ (mulGrp‘𝑀) ∈ Mnd ∧ ∀𝑎 ∈ {𝑍}∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))))
101		eqidd 2176	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑉 → 𝑎 = 𝑎)
102	13	oveqd 5882	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑉 → (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = (𝑏(+_g‘𝑀)𝑐))
103	48, 101, 102	oveq123d 5886	. . . . . . . . 9 ⊢ (𝑍 ∈ 𝑉 → (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = (𝑎(._r‘𝑀)(𝑏(+_g‘𝑀)𝑐)))
104	48	oveqd 5882	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑉 → (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏) = (𝑎(._r‘𝑀)𝑏))
105	48	oveqd 5882	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑉 → (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = (𝑎(._r‘𝑀)𝑐))
106	13, 104, 105	oveq123d 5886	. . . . . . . . 9 ⊢ (𝑍 ∈ 𝑉 → ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎(._r‘𝑀)𝑏)(+_g‘𝑀)(𝑎(._r‘𝑀)𝑐)))
107	103, 106	eqeq12d 2190	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑉 → ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ↔ (𝑎(._r‘𝑀)(𝑏(+_g‘𝑀)𝑐)) = ((𝑎(._r‘𝑀)𝑏)(+_g‘𝑀)(𝑎(._r‘𝑀)𝑐))))
108	13	oveqd 5882	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑉 → (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏) = (𝑎(+_g‘𝑀)𝑏))
109		eqidd 2176	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑉 → 𝑐 = 𝑐)
110	48, 108, 109	oveq123d 5886	. . . . . . . . 9 ⊢ (𝑍 ∈ 𝑉 → ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎(+_g‘𝑀)𝑏)(._r‘𝑀)𝑐))
111	48	oveqd 5882	. . . . . . . . . 10 ⊢ (𝑍 ∈ 𝑉 → (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = (𝑏(._r‘𝑀)𝑐))
112	13, 105, 111	oveq123d 5886	. . . . . . . . 9 ⊢ (𝑍 ∈ 𝑉 → ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎(._r‘𝑀)𝑐)(+_g‘𝑀)(𝑏(._r‘𝑀)𝑐)))
113	110, 112	eqeq12d 2190	. . . . . . . 8 ⊢ (𝑍 ∈ 𝑉 → (((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ↔ ((𝑎(+_g‘𝑀)𝑏)(._r‘𝑀)𝑐) = ((𝑎(._r‘𝑀)𝑐)(+_g‘𝑀)(𝑏(._r‘𝑀)𝑐))))
114	107, 113	anbi12d 473	. . . . . . 7 ⊢ (𝑍 ∈ 𝑉 → (((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ((𝑎(._r‘𝑀)(𝑏(+_g‘𝑀)𝑐)) = ((𝑎(._r‘𝑀)𝑏)(+_g‘𝑀)(𝑎(._r‘𝑀)𝑐)) ∧ ((𝑎(+_g‘𝑀)𝑏)(._r‘𝑀)𝑐) = ((𝑎(._r‘𝑀)𝑐)(+_g‘𝑀)(𝑏(._r‘𝑀)𝑐)))))
115	10, 114	raleqbidv 2682	. . . . . 6 ⊢ (𝑍 ∈ 𝑉 → (∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ∀𝑐 ∈ (Base‘𝑀)((𝑎(._r‘𝑀)(𝑏(+_g‘𝑀)𝑐)) = ((𝑎(._r‘𝑀)𝑏)(+_g‘𝑀)(𝑎(._r‘𝑀)𝑐)) ∧ ((𝑎(+_g‘𝑀)𝑏)(._r‘𝑀)𝑐) = ((𝑎(._r‘𝑀)𝑐)(+_g‘𝑀)(𝑏(._r‘𝑀)𝑐)))))
116	10, 115	raleqbidv 2682	. . . . 5 ⊢ (𝑍 ∈ 𝑉 → (∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)((𝑎(._r‘𝑀)(𝑏(+_g‘𝑀)𝑐)) = ((𝑎(._r‘𝑀)𝑏)(+_g‘𝑀)(𝑎(._r‘𝑀)𝑐)) ∧ ((𝑎(+_g‘𝑀)𝑏)(._r‘𝑀)𝑐) = ((𝑎(._r‘𝑀)𝑐)(+_g‘𝑀)(𝑏(._r‘𝑀)𝑐)))))
117	10, 116	raleqbidv 2682	. . . 4 ⊢ (𝑍 ∈ 𝑉 → (∀𝑎 ∈ {𝑍}∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐))) ↔ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)((𝑎(._r‘𝑀)(𝑏(+_g‘𝑀)𝑐)) = ((𝑎(._r‘𝑀)𝑏)(+_g‘𝑀)(𝑎(._r‘𝑀)𝑐)) ∧ ((𝑎(+_g‘𝑀)𝑏)(._r‘𝑀)𝑐) = ((𝑎(._r‘𝑀)𝑐)(+_g‘𝑀)(𝑏(._r‘𝑀)𝑐)))))
118	117	3anbi3d 1318	. . 3 ⊢ (𝑍 ∈ 𝑉 → ((𝑀 ∈ Grp ∧ (mulGrp‘𝑀) ∈ Mnd ∧ ∀𝑎 ∈ {𝑍}∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))) ↔ (𝑀 ∈ Grp ∧ (mulGrp‘𝑀) ∈ Mnd ∧ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)((𝑎(._r‘𝑀)(𝑏(+_g‘𝑀)𝑐)) = ((𝑎(._r‘𝑀)𝑏)(+_g‘𝑀)(𝑎(._r‘𝑀)𝑐)) ∧ ((𝑎(+_g‘𝑀)𝑏)(._r‘𝑀)𝑐) = ((𝑎(._r‘𝑀)𝑐)(+_g‘𝑀)(𝑏(._r‘𝑀)𝑐))))))
119	33, 43, 34, 51	isring 12976	. . 3 ⊢ (𝑀 ∈ Ring ↔ (𝑀 ∈ Grp ∧ (mulGrp‘𝑀) ∈ Mnd ∧ ∀𝑎 ∈ (Base‘𝑀)∀𝑏 ∈ (Base‘𝑀)∀𝑐 ∈ (Base‘𝑀)((𝑎(._r‘𝑀)(𝑏(+_g‘𝑀)𝑐)) = ((𝑎(._r‘𝑀)𝑏)(+_g‘𝑀)(𝑎(._r‘𝑀)𝑐)) ∧ ((𝑎(+_g‘𝑀)𝑏)(._r‘𝑀)𝑐) = ((𝑎(._r‘𝑀)𝑐)(+_g‘𝑀)(𝑏(._r‘𝑀)𝑐)))))
120	118, 119	bitr4di 198	. 2 ⊢ (𝑍 ∈ 𝑉 → ((𝑀 ∈ Grp ∧ (mulGrp‘𝑀) ∈ Mnd ∧ ∀𝑎 ∈ {𝑍}∀𝑏 ∈ {𝑍}∀𝑐 ∈ {𝑍} ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)) ∧ ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑏){⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐) = ((𝑎{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐){⟨⟨𝑍, 𝑍⟩, 𝑍⟩} (𝑏{⟨⟨𝑍, 𝑍⟩, 𝑍⟩}𝑐)))) ↔ 𝑀 ∈ Ring))
121	100, 120	mpbid 147	1 ⊢ (𝑍 ∈ 𝑉 → 𝑀 ∈ Ring)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2146 ∀wral 2453 Vcvv 2735 {csn 3589 {cpr 3590 {ctp 3591 ⟨cop 3592 ‘cfv 5208 (class class class)co 5865 ℕcn 8890 ndxcnx 12424 Slot cslot 12426 Basecbs 12427 +_gcplusg 12491 ._rcmulr 12492 Mndcmnd 12681 Grpcgrp 12737 mulGrpcmgp 12925 Ringcrg 12972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-sep 4116 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-addcom 7886 ax-addass 7888 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-0id 7894 ax-rnegex 7895 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902

This theorem depends on definitions: df-bi 117 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-pw 3574 df-sn 3595 df-pr 3596 df-tp 3597 df-op 3598 df-uni 3806 df-int 3841 df-br 3999 df-opab 4060 df-mpt 4061 df-id 4287 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-inn 8891 df-2 8949 df-3 8950 df-n0 9148 df-z 9225 df-uz 9500 df-fz 9978 df-struct 12429 df-ndx 12430 df-slot 12431 df-base 12433 df-sets 12434 df-plusg 12504 df-mulr 12505 df-0g 12627 df-mgm 12639 df-sgrp 12672 df-mnd 12682 df-grp 12740 df-mgp 12926 df-ring 12974

This theorem is referenced by: ringn0 13029

W3C validator