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Theorem mdetunilem2 21670

Description: Lemma for mdetuni 21679. (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 = (0_g‘𝑅)
mdetuni.1r	⊢ 1 = (1_r‘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))))
mdetunilem2.ph	⊢ (𝜓 → 𝜑)
mdetunilem2.eg	⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐸 ≠ 𝐺))
mdetunilem2.f	⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾)
mdetunilem2.h	⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)

**Assertion**
Ref	Expression
mdetunilem2	⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))) = 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 𝑥,𝐹,𝑦,𝑧,𝑤 𝑥,𝐺,𝑦,𝑧,𝑤 𝑥,𝐻,𝑦,𝑧,𝑤 𝜓,𝑎,𝑏,𝑥,𝑦,𝑧,𝑤 𝐸,𝑎,𝑏 𝐺,𝑎,𝑏 𝐹,𝑎 Allowed substitution hints: 𝑅(𝑎,𝑏) · (𝑎,𝑏) 𝐹(𝑏) 𝐻(𝑎,𝑏)

**Proof of Theorem mdetunilem2**
Step	Hyp	Ref	Expression
1		mdetunilem2.ph	. 2 ⊢ (𝜓 → 𝜑)
2		mdetuni.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		mdetuni.k	. . 3 ⊢ 𝐾 = (Base‘𝑅)
4		mdetuni.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
5		mdetuni.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ Fin)
6	1, 5	syl 17	. . 3 ⊢ (𝜓 → 𝑁 ∈ Fin)
7		mdetuni.r	. . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring)
8	1, 7	syl 17	. . 3 ⊢ (𝜓 → 𝑅 ∈ Ring)
9		mdetunilem2.f	. . . . 5 ⊢ ((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾)
10	9	3adant2 1129	. . . 4 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾)
11		mdetunilem2.h	. . . . 5 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)
12	10, 11	ifcld 4502	. . . 4 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐺, 𝐹, 𝐻) ∈ 𝐾)
13	10, 12	ifcld 4502	. . 3 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) ∈ 𝐾)
14	2, 3, 4, 6, 8, 13	matbas2d 21480	. 2 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻))) ∈ 𝐵)
15		eqidd 2739	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻))) = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻))))
16		iftrue 4462	. . . . . . 7 ⊢ (𝑎 = 𝐸 → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = 𝐹)
17		csbeq1a 3842	. . . . . . 7 ⊢ (𝑏 = 𝑤 → 𝐹 = ⦋𝑤 / 𝑏⦌𝐹)
18	16, 17	sylan9eq 2799	. . . . . 6 ⊢ ((𝑎 = 𝐸 ∧ 𝑏 = 𝑤) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = ⦋𝑤 / 𝑏⦌𝐹)
19	18	adantl 481	. . . . 5 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐸 ∧ 𝑏 = 𝑤)) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = ⦋𝑤 / 𝑏⦌𝐹)
20		eqidd 2739	. . . . 5 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐸) → 𝑁 = 𝑁)
21		mdetunilem2.eg	. . . . . . 7 ⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐸 ≠ 𝐺))
22	21	simp1d 1140	. . . . . 6 ⊢ (𝜓 → 𝐸 ∈ 𝑁)
23	22	adantr 480	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → 𝐸 ∈ 𝑁)
24		simpr 484	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → 𝑤 ∈ 𝑁)
25		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑏(𝜓 ∧ 𝑤 ∈ 𝑁)
26		nfcsb1v 3853	. . . . . . . 8 ⊢ Ⅎ𝑏⦋𝑤 / 𝑏⦌𝐹
27	26	nfel1 2922	. . . . . . 7 ⊢ Ⅎ𝑏⦋𝑤 / 𝑏⦌𝐹 ∈ 𝐾
28	25, 27	nfim 1900	. . . . . 6 ⊢ Ⅎ𝑏((𝜓 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑏⦌𝐹 ∈ 𝐾)
29		eleq1w 2821	. . . . . . . 8 ⊢ (𝑏 = 𝑤 → (𝑏 ∈ 𝑁 ↔ 𝑤 ∈ 𝑁))
30	29	anbi2d 628	. . . . . . 7 ⊢ (𝑏 = 𝑤 → ((𝜓 ∧ 𝑏 ∈ 𝑁) ↔ (𝜓 ∧ 𝑤 ∈ 𝑁)))
31	17	eleq1d 2823	. . . . . . 7 ⊢ (𝑏 = 𝑤 → (𝐹 ∈ 𝐾 ↔ ⦋𝑤 / 𝑏⦌𝐹 ∈ 𝐾))
32	30, 31	imbi12d 344	. . . . . 6 ⊢ (𝑏 = 𝑤 → (((𝜓 ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾) ↔ ((𝜓 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑏⦌𝐹 ∈ 𝐾)))
33	28, 32, 9	chvarfv 2236	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑏⦌𝐹 ∈ 𝐾)
34		nfv 1918	. . . . 5 ⊢ Ⅎ𝑎(𝜓 ∧ 𝑤 ∈ 𝑁)
35		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑏𝐸
36		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑎𝑤
37		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑎⦋𝑤 / 𝑏⦌𝐹
38	15, 19, 20, 23, 24, 33, 34, 25, 35, 36, 37, 26	ovmpodxf 7401	. . . 4 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → (𝐸(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))𝑤) = ⦋𝑤 / 𝑏⦌𝐹)
39	21	simp3d 1142	. . . . . . . . . . . . 13 ⊢ (𝜓 → 𝐸 ≠ 𝐺)
40	39	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → 𝐸 ≠ 𝐺)
41		neeq2 3006	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐺 → (𝐸 ≠ 𝑎 ↔ 𝐸 ≠ 𝐺))
42	40, 41	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → (𝑎 = 𝐺 → 𝐸 ≠ 𝑎))
43	42	imp 406	. . . . . . . . . 10 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐺) → 𝐸 ≠ 𝑎)
44	43	necomd 2998	. . . . . . . . 9 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐺) → 𝑎 ≠ 𝐸)
45	44	neneqd 2947	. . . . . . . 8 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐺) → ¬ 𝑎 = 𝐸)
46	45	adantrr 713	. . . . . . 7 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐺 ∧ 𝑏 = 𝑤)) → ¬ 𝑎 = 𝐸)
47	46	iffalsed 4467	. . . . . 6 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐺 ∧ 𝑏 = 𝑤)) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = if(𝑎 = 𝐺, 𝐹, 𝐻))
48		iftrue 4462	. . . . . . . 8 ⊢ (𝑎 = 𝐺 → if(𝑎 = 𝐺, 𝐹, 𝐻) = 𝐹)
49	48, 17	sylan9eq 2799	. . . . . . 7 ⊢ ((𝑎 = 𝐺 ∧ 𝑏 = 𝑤) → if(𝑎 = 𝐺, 𝐹, 𝐻) = ⦋𝑤 / 𝑏⦌𝐹)
50	49	adantl 481	. . . . . 6 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐺 ∧ 𝑏 = 𝑤)) → if(𝑎 = 𝐺, 𝐹, 𝐻) = ⦋𝑤 / 𝑏⦌𝐹)
51	47, 50	eqtrd 2778	. . . . 5 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐺 ∧ 𝑏 = 𝑤)) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = ⦋𝑤 / 𝑏⦌𝐹)
52		eqidd 2739	. . . . 5 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐺) → 𝑁 = 𝑁)
53	21	simp2d 1141	. . . . . 6 ⊢ (𝜓 → 𝐺 ∈ 𝑁)
54	53	adantr 480	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → 𝐺 ∈ 𝑁)
55		nfcv 2906	. . . . 5 ⊢ Ⅎ𝑏𝐺
56	15, 51, 52, 54, 24, 33, 34, 25, 55, 36, 37, 26	ovmpodxf 7401	. . . 4 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → (𝐺(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))𝑤) = ⦋𝑤 / 𝑏⦌𝐹)
57	38, 56	eqtr4d 2781	. . 3 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → (𝐸(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))𝑤) = (𝐺(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))𝑤))
58	57	ralrimiva 3107	. 2 ⊢ (𝜓 → ∀𝑤 ∈ 𝑁 (𝐸(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))𝑤) = (𝐺(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))𝑤))
59		mdetuni.0g	. . 3 ⊢ 0 = (0_g‘𝑅)
60		mdetuni.1r	. . 3 ⊢ 1 = (1_r‘𝑅)
61		mdetuni.pg	. . 3 ⊢ + = (+_g‘𝑅)
62		mdetuni.tg	. . 3 ⊢ · = (._r‘𝑅)
63		mdetuni.ff	. . 3 ⊢ (𝜑 → 𝐷:𝐵⟶𝐾)
64		mdetuni.al	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
65		mdetuni.li	. . 3 ⊢ (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))))
66		mdetuni.sc	. . 3 ⊢ (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))))
67	2, 4, 3, 59, 60, 61, 62, 5, 7, 63, 64, 65, 66	mdetunilem1 21669	. 2 ⊢ (((𝜑 ∧ (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻))) ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐸(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))𝑤) = (𝐺(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))𝑤)) ∧ (𝐸 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐸 ≠ 𝐺)) → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))) = 0 )
68	1, 14, 58, 21, 67	syl31anc 1371	1 ⊢ (𝜓 → (𝐷‘(𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)))) = 0 )

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ⦋csb 3828 ∖ cdif 3880 ifcif 4456 {csn 4558 × cxp 5578 ↾ cres 5582 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ∈ cmpo 7257 ∘_f cof 7509 Fincfn 8691 Basecbs 16840 +_gcplusg 16888 ._rcmulr 16889 0_gc0g 17067 1_rcur 19652 Ringcrg 19698 Mat cmat 21464

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-supp 7949 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-fsupp 9059 df-sup 9131 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-ip 16906 df-tset 16907 df-ple 16908 df-ds 16910 df-hom 16912 df-cco 16913 df-0g 17069 df-prds 17075 df-pws 17077 df-sra 20349 df-rgmod 20350 df-dsmm 20849 df-frlm 20864 df-mat 21465

This theorem is referenced by: mdetunilem6 21674 mdetunilem8 21676

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