Theorem mdetunilem8 22747

Description: Lemma for mdetuni 22750. (Contributed by SO, 15-Jul-2018.)

Hypotheses

Ref	Expression
mdetuni.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
mdetuni.b	⊢ 𝐵 = (Base‘𝐴)
mdetuni.k	⊢ 𝐾 = (Base‘𝑅)
mdetuni.0g	⊢ 0 = (0g‘𝑅)
mdetuni.1r	⊢ 1 = (1r‘𝑅)
mdetuni.pg	⊢ + = (+g‘𝑅)
mdetuni.tg	⊢ · = (.r‘𝑅)
mdetuni.n	⊢ (𝜑 → 𝑁 ∈ Fin)
mdetuni.r	⊢ (𝜑 → 𝑅 ∈ Ring)
mdetuni.ff	⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
mdetuni.al	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
mdetuni.li	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
mdetuni.sc	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))
mdetunilem8.id	⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )

Assertion

Ref	Expression
mdetunilem8	⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )

Distinct variable groups:   𝜑,𝑥,𝑦,𝑧,𝑤,𝑎,𝑏   𝑥,𝐵,𝑦,𝑧,𝑤,𝑎,𝑏   𝑥,𝐾,𝑦,𝑧,𝑤,𝑎,𝑏   𝑥,𝑁,𝑦,𝑧,𝑤,𝑎,𝑏   𝑥,𝐷,𝑦,𝑧,𝑤,𝑎,𝑏   𝑥, · ,𝑦,𝑧,𝑤   + ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤   0 ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤   1 ,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤   𝑥,𝑅,𝑦,𝑧,𝑤   𝐴,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤   𝑥,𝐸,𝑦,𝑧,𝑤,𝑎,𝑏

Allowed substitution hints:   𝑅(𝑎,𝑏)   · (𝑎,𝑏)

Proof of Theorem mdetunilem8

Dummy variables 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝜑)
2		mdetuni.n	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ Fin)
3		enrefg 8983	. . . . . . . . 9 ⊢ (𝑁 ∈ Fin → 𝑁 ≈ 𝑁)
4	2, 3	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ≈ 𝑁)
5		f1finf1o 9235	. . . . . . . 8 ⊢ ((𝑁 ≈ 𝑁 ∧ 𝑁 ∈ Fin) → (𝐸:𝑁–1-1→𝑁 ↔ 𝐸:𝑁–1-1-onto→𝑁))
6	4, 2, 5	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐸:𝑁–1-1→𝑁 ↔ 𝐸:𝑁–1-1-onto→𝑁))
7	6	biimpa 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝐸:𝑁–1-1-onto→𝑁)
8		mdetuni.r	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ Ring)
9		mdetuni.a	. . . . . . . . . 10 ⊢ 𝐴 = (𝑁 Mat 𝑅)
10	9	matring 22571	. . . . . . . . 9 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
11	2, 8, 10	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ Ring)
12		mdetuni.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐴)
13		eqid 2769	. . . . . . . . 9 ⊢ (1r‘𝐴) = (1r‘𝐴)
14	12, 13	ringidcl 20350	. . . . . . . 8 ⊢ (𝐴 ∈ Ring → (1r‘𝐴) ∈ 𝐵)
15	11, 14	syl 18	. . . . . . 7 ⊢ (𝜑 → (1r‘𝐴) ∈ 𝐵)
16	15	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (1r‘𝐴) ∈ 𝐵)
17		mdetuni.k	. . . . . . 7 ⊢ 𝐾 = (Base‘𝑅)
18		mdetuni.0g	. . . . . . 7 ⊢ 0 = (0g‘𝑅)
19		mdetuni.1r	. . . . . . 7 ⊢ 1 = (1r‘𝑅)
20		mdetuni.pg	. . . . . . 7 ⊢ + = (+g‘𝑅)
21		mdetuni.tg	. . . . . . 7 ⊢ · = (.r‘𝑅)
22		mdetuni.ff	. . . . . . 7 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
23		mdetuni.al	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
24		mdetuni.li	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((𝑦 ↾ ({𝑤} × 𝑁)) ∘f + (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑦 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = ((𝐷‘𝑦) + (𝐷‘𝑧))))
25		mdetuni.sc	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐾 ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))))
26	9, 12, 17, 18, 19, 20, 21, 2, 8, 22, 23, 24, 25	mdetunilem7 22746	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1-onto→𝑁 ∧ (1r‘𝐴) ∈ 𝐵) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)(1r‘𝐴)𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴))))
27	1, 7, 16, 26	syl3anc 1396	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)(1r‘𝐴)𝑏))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴))))
28	2	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝑁 ∈ Fin)
29	28	3ad2ant1 1149	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑁 ∈ Fin)
30	8	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝑅 ∈ Ring)
31	30	3ad2ant1 1149	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑅 ∈ Ring)
32		simp1r 1215	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐸:𝑁–1-1→𝑁)
33		f1f 6777	. . . . . . . . . 10 ⊢ (𝐸:𝑁–1-1→𝑁 → 𝐸:𝑁⟶𝑁)
34	32, 33	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐸:𝑁⟶𝑁)
35		simp2 1153	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑎 ∈ 𝑁)
36	34, 35	ffvelcdmd 7083	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → (𝐸‘𝑎) ∈ 𝑁)
37		simp3 1154	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝑏 ∈ 𝑁)
38	9, 19, 18, 29, 31, 36, 37, 13	mat1ov 22576	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → ((𝐸‘𝑎)(1r‘𝐴)𝑏) = if((𝐸‘𝑎) = 𝑏, 1 , 0 ))
39	38	mpoeq3dva 7490	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)(1r‘𝐴)𝑏)) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 )))
40	39	fveq2d 6888	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ ((𝐸‘𝑎)(1r‘𝐴)𝑏))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))
41		mdetunilem8.id	. . . . . . . 8 ⊢ (𝜑 → (𝐷‘(1r‘𝐴)) = 0 )
42	41	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(1r‘𝐴)) = 0 )
43	42	oveq2d 7429	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴))) = ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · 0 ))
44		zrhpsgnmhm 21705	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
45	8, 2, 44	syl2anc 595	. . . . . . . . . 10 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)))
46		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘(SymGrp‘𝑁)) = (Base‘(SymGrp‘𝑁))
47		eqid 2769	. . . . . . . . . . . 12 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅)
48	47, 17	mgpbas 20223	. . . . . . . . . . 11 ⊢ 𝐾 = (Base‘(mulGrp‘𝑅))
49	46, 48	mhmf 18849	. . . . . . . . . 10 ⊢ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)) ∈ ((SymGrp‘𝑁) MndHom (mulGrp‘𝑅)) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
50	45, 49	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
51	50	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁)):(Base‘(SymGrp‘𝑁))⟶𝐾)
52		eqid 2769	. . . . . . . . . . 11 ⊢ (SymGrp‘𝑁) = (SymGrp‘𝑁)
53	52, 46	elsymgbas 19446	. . . . . . . . . 10 ⊢ (𝑁 ∈ Fin → (𝐸 ∈ (Base‘(SymGrp‘𝑁)) ↔ 𝐸:𝑁–1-1-onto→𝑁))
54	28, 53	syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐸 ∈ (Base‘(SymGrp‘𝑁)) ↔ 𝐸:𝑁–1-1-onto→𝑁))
55	7, 54	mpbird 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → 𝐸 ∈ (Base‘(SymGrp‘𝑁)))
56	51, 55	ffvelcdmd 7083	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) ∈ 𝐾)
57	17, 21, 18	ringrz 20379	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ (((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) ∈ 𝐾) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · 0 ) = 0 )
58	30, 56, 57	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · 0 ) = 0 )
59	43, 58	eqtrd 2804	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → ((((ℤRHom‘𝑅) ∘ (pmSgn‘𝑁))‘𝐸) · (𝐷‘(1r‘𝐴))) = 0 )
60	27, 40, 59	3eqtr3d 2812	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁–1-1→𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )
61	60	ex 417	. . 3 ⊢ (𝜑 → (𝐸:𝑁–1-1→𝑁 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ))
62	61	adantr 485	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐸:𝑁–1-1→𝑁 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ))
63		dff13 7255	. . . . . 6 ⊢ (𝐸:𝑁–1-1→𝑁 ↔ (𝐸:𝑁⟶𝑁 ∧ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)))
64		ibar 537	. . . . . . 7 ⊢ (𝐸:𝑁⟶𝑁 → (∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ (𝐸:𝑁⟶𝑁 ∧ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))))
65	64	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ (𝐸:𝑁⟶𝑁 ∧ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))))
66	63, 65	bitr4id 293	. . . . 5 ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐸:𝑁–1-1→𝑁 ↔ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)))
67	66	notbid 321	. . . 4 ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 ↔ ¬ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑)))
68		rexnal 3123	. . . . 5 ⊢ (∃𝑐 ∈ 𝑁 ¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ¬ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))
69		rexnal 3123	. . . . . . 7 ⊢ (∃𝑑 ∈ 𝑁 ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))
70		df-ne 2965	. . . . . . . . . 10 ⊢ (𝑐 ≠ 𝑑 ↔ ¬ 𝑐 = 𝑑)
71	70	anbi2i 634	. . . . . . . . 9 ⊢ (((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑) ↔ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ ¬ 𝑐 = 𝑑))
72		annim 408	. . . . . . . . 9 ⊢ (((𝐸‘𝑐) = (𝐸‘𝑑) ∧ ¬ 𝑐 = 𝑑) ↔ ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑))
73	71, 72	bitr2i 279	. . . . . . . 8 ⊢ (¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))
74	73	rexbii 3118	. . . . . . 7 ⊢ (∃𝑑 ∈ 𝑁 ¬ ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))
75	69, 74	bitr3i 280	. . . . . 6 ⊢ (¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))
76	75	rexbii 3118	. . . . 5 ⊢ (∃𝑐 ∈ 𝑁 ¬ ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))
77	68, 76	bitr3i 280	. . . 4 ⊢ (¬ ∀𝑐 ∈ 𝑁 ∀𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) → 𝑐 = 𝑑) ↔ ∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))
78	67, 77	bitrdi 290	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 ↔ ∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑)))
79		simprrl 792	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐸‘𝑐) = (𝐸‘𝑑))
80		fveqeq2 6893	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑐 → ((𝐸‘𝑎) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏))
81	80	ifbid 4516	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if((𝐸‘𝑐) = 𝑏, 1 , 0 ))
82		iftrue 4498	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑐 → if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = if((𝐸‘𝑐) = 𝑏, 1 , 0 ))
83	81, 82	eqtr4d 2807	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑐 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))
84		fveqeq2 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑎 = 𝑑 → ((𝐸‘𝑎) = 𝑏 ↔ (𝐸‘𝑑) = 𝑏))
85	84	ifbid 4516	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if((𝐸‘𝑑) = 𝑏, 1 , 0 ))
86		iftrue 4498	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑑 → if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = if((𝐸‘𝑑) = 𝑏, 1 , 0 ))
87	85, 86	eqtr4d 2807	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))
88		iffalse 4501	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑎 = 𝑑 → if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = if((𝐸‘𝑎) = 𝑏, 1 , 0 ))
89	88	eqcomd 2775	. . . . . . . . . . . . 13 ⊢ (¬ 𝑎 = 𝑑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))
90	87, 89	pm2.61i 184	. . . . . . . . . . . 12 ⊢ if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))
91		iffalse 4501	. . . . . . . . . . . 12 ⊢ (¬ 𝑎 = 𝑐 → if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))
92	90, 91	eqtr4id 2823	. . . . . . . . . . 11 ⊢ (¬ 𝑎 = 𝑐 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))
93	83, 92	pm2.61i 184	. . . . . . . . . 10 ⊢ if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))
94		eqeq1 2773	. . . . . . . . . . . . . 14 ⊢ ((𝐸‘𝑑) = (𝐸‘𝑐) → ((𝐸‘𝑑) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏))
95	94	eqcoms 2777	. . . . . . . . . . . . 13 ⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → ((𝐸‘𝑑) = 𝑏 ↔ (𝐸‘𝑐) = 𝑏))
96	95	ifbid 4516	. . . . . . . . . . . 12 ⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if((𝐸‘𝑑) = 𝑏, 1 , 0 ) = if((𝐸‘𝑐) = 𝑏, 1 , 0 ))
97	96	ifeq1d 4512	. . . . . . . . . . 11 ⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))
98	97	ifeq2d 4513	. . . . . . . . . 10 ⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑑) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))
99	93, 98	eqtrid 2816	. . . . . . . . 9 ⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) = if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))
100	99	mpoeq3dv 7492	. . . . . . . 8 ⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 )) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 )))))
101	100	fveq2d 6888	. . . . . . 7 ⊢ ((𝐸‘𝑐) = (𝐸‘𝑑) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))))
102	79, 101	syl 18	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))))
103		simpll 778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝜑)
104		simprll 790	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑐 ∈ 𝑁)
105		simprlr 791	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑑 ∈ 𝑁)
106		simprrr 793	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → 𝑐 ≠ 𝑑)
107	104, 105, 106	3jca 1144	. . . . . . 7 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁 ∧ 𝑐 ≠ 𝑑))
108	17, 19	ringidcl 20350	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 1 ∈ 𝐾)
109	8, 108	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 1 ∈ 𝐾)
110	17, 18	ring0cl 20352	. . . . . . . . . 10 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐾)
111	8, 110	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 0 ∈ 𝐾)
112	109, 111	ifcld 4539	. . . . . . . 8 ⊢ (𝜑 → if((𝐸‘𝑐) = 𝑏, 1 , 0 ) ∈ 𝐾)
113	112	ad3antrrr 742	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) ∧ 𝑏 ∈ 𝑁) → if((𝐸‘𝑐) = 𝑏, 1 , 0 ) ∈ 𝐾)
114		simp1ll 1253	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝜑)
115	109, 111	ifcld 4539	. . . . . . . 8 ⊢ (𝜑 → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) ∈ 𝐾)
116	114, 115	syl 18	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if((𝐸‘𝑎) = 𝑏, 1 , 0 ) ∈ 𝐾)
117	9, 12, 17, 18, 19, 20, 21, 2, 8, 22, 23, 24, 25, 103, 107, 113, 116	mdetunilem2 22741	. . . . . 6 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝑐, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if(𝑎 = 𝑑, if((𝐸‘𝑐) = 𝑏, 1 , 0 ), if((𝐸‘𝑎) = 𝑏, 1 , 0 ))))) = 0 )
118	102, 117	eqtrd 2804	. . . . 5 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ ((𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁) ∧ ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑))) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )
119	118	expr 461	. . . 4 ⊢ (((𝜑 ∧ 𝐸:𝑁⟶𝑁) ∧ (𝑐 ∈ 𝑁 ∧ 𝑑 ∈ 𝑁)) → (((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ))
120	119	rexlimdvva 3228	. . 3 ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (∃𝑐 ∈ 𝑁 ∃𝑑 ∈ 𝑁 ((𝐸‘𝑐) = (𝐸‘𝑑) ∧ 𝑐 ≠ 𝑑) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ))
121	78, 120	sylbid 243	. 2 ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (¬ 𝐸:𝑁–1-1→𝑁 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 ))
122	62, 121	pm2.61d 181	1 ⊢ ((𝜑 ∧ 𝐸:𝑁⟶𝑁) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if((𝐸‘𝑎) = 𝑏, 1 , 0 ))) = 0 )

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910  ifcif 4492  {csn 4594   class class class wbr 5113   × cxp 5662   ↾ cres 5666   ∘ ccom 5668  ⟶wf 6535  –1-1→wf1 6536  –1-1-onto→wf1o 6538  ‘cfv 6539  (class class class)co 7413   ∈ cmpo 7415   ∘_f cof 7675   ≈ cen 8942  Fincfn 8945  Basecbs 17271  +_gcplusg 17312  ._rcmulr 17313  0_gc0g 17494   MndHom cmhm 18841  SymGrpcsymg 19441  pmSgncpsgn 19561  mulGrpcmgp 20218  1_rcur 20265  Ringcrg 20317  ℤRHomczrh 21620   Mat cmat 22535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179  ax-addf 11181  ax-mulf 11182

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-xor 1539  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-ot 4603  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7677  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-tpos 8224  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9324  df-sup 9404  df-oi 9474  df-card 9927  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-div 11874  df-nn 12236  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12507  df-xnn0 12580  df-z 12594  df-dec 12714  df-uz 12865  df-rp 13019  df-fz 13538  df-fzo 13685  df-seq 14040  df-exp 14100  df-hash 14369  df-word 14553  df-lsw 14602  df-concat 14610  df-s1 14636  df-substr 14681  df-pfx 14711  df-splice 14789  df-reverse 14798  df-s2 14887  df-struct 17209  df-sets 17226  df-slot 17244  df-ndx 17256  df-base 17272  df-ress 17293  df-plusg 17325  df-mulr 17326  df-starv 17327  df-sca 17328  df-vsca 17329  df-ip 17330  df-tset 17331  df-ple 17332  df-ds 17334  df-unif 17335  df-hom 17336  df-cco 17337  df-0g 17496  df-gsum 17497  df-prds 17502  df-pws 17504  df-mre 17640  df-mrc 17641  df-acs 17643  df-mgm 18700  df-sgrp 18779  df-mnd 18795  df-mhm 18843  df-submnd 18844  df-efmnd 18930  df-grp 19005  df-minusg 19006  df-sbg 19007  df-mulg 19136  df-subg 19191  df-ghm 19286  df-gim 19331  df-cntz 19389  df-oppg 19418  df-symg 19442  df-pmtr 19514  df-psgn 19563  df-evpm 19564  df-cmn 19854  df-abl 19855  df-mgp 20219  df-rng 20233  df-ur 20266  df-ring 20319  df-cring 20320  df-oppr 20421  df-dvdsr 20441  df-unit 20442  df-invr 20472  df-dvr 20485  df-rhm 20556  df-subrng 20633  df-subrg 20657  df-drng 20817  df-lmod 20963  df-lss 21033  df-sra 21274  df-rgmod 21275  df-cnfld 21494  df-zring 21568  df-zrh 21624  df-dsmm 21853  df-frlm 21868  df-mamu 22519  df-mat 22536

This theorem is referenced by:  mdetunilem9  22748