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Theorem mdetunilem2 21962

Description: Lemma for mdetuni 21971. (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 1131	. . . 4 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐹 ∈ 𝐾)
11		mdetunilem2.h	. . . . 5 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → 𝐻 ∈ 𝐾)
12	10, 11	ifcld 4532	. . . 4 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐺, 𝐹, 𝐻) ∈ 𝐾)
13	10, 12	ifcld 4532	. . 3 ⊢ ((𝜓 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑁) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) ∈ 𝐾)
14	2, 3, 4, 6, 8, 13	matbas2d 21772	. 2 ⊢ (𝜓 → (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑁 ↦ if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻))) ∈ 𝐵)
15		eqidd 2737	. . . . 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 4492	. . . . . . 7 ⊢ (𝑎 = 𝐸 → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = 𝐹)
17		csbeq1a 3869	. . . . . . 7 ⊢ (𝑏 = 𝑤 → 𝐹 = ⦋𝑤 / 𝑏⦌𝐹)
18	16, 17	sylan9eq 2796	. . . . . 6 ⊢ ((𝑎 = 𝐸 ∧ 𝑏 = 𝑤) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = ⦋𝑤 / 𝑏⦌𝐹)
19	18	adantl 482	. . . . 5 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐸 ∧ 𝑏 = 𝑤)) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = ⦋𝑤 / 𝑏⦌𝐹)
20		eqidd 2737	. . . . 5 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐸) → 𝑁 = 𝑁)
21		mdetunilem2.eg	. . . . . . 7 ⊢ (𝜓 → (𝐸 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐸 ≠ 𝐺))
22	21	simp1d 1142	. . . . . 6 ⊢ (𝜓 → 𝐸 ∈ 𝑁)
23	22	adantr 481	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → 𝐸 ∈ 𝑁)
24		simpr 485	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → 𝑤 ∈ 𝑁)
25		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑏(𝜓 ∧ 𝑤 ∈ 𝑁)
26		nfcsb1v 3880	. . . . . . . 8 ⊢ Ⅎ𝑏⦋𝑤 / 𝑏⦌𝐹
27	26	nfel1 2923	. . . . . . 7 ⊢ Ⅎ𝑏⦋𝑤 / 𝑏⦌𝐹 ∈ 𝐾
28	25, 27	nfim 1899	. . . . . 6 ⊢ Ⅎ𝑏((𝜓 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑏⦌𝐹 ∈ 𝐾)
29		eleq1w 2820	. . . . . . . 8 ⊢ (𝑏 = 𝑤 → (𝑏 ∈ 𝑁 ↔ 𝑤 ∈ 𝑁))
30	29	anbi2d 629	. . . . . . 7 ⊢ (𝑏 = 𝑤 → ((𝜓 ∧ 𝑏 ∈ 𝑁) ↔ (𝜓 ∧ 𝑤 ∈ 𝑁)))
31	17	eleq1d 2822	. . . . . . 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 2233	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → ⦋𝑤 / 𝑏⦌𝐹 ∈ 𝐾)
34		nfv 1917	. . . . 5 ⊢ Ⅎ𝑎(𝜓 ∧ 𝑤 ∈ 𝑁)
35		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑏𝐸
36		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑎𝑤
37		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑎⦋𝑤 / 𝑏⦌𝐹
38	15, 19, 20, 23, 24, 33, 34, 25, 35, 36, 37, 26	ovmpodxf 7505	. . . 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 1144	. . . . . . . . . . . . 13 ⊢ (𝜓 → 𝐸 ≠ 𝐺)
40	39	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → 𝐸 ≠ 𝐺)
41		neeq2 3007	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐺 → (𝐸 ≠ 𝑎 ↔ 𝐸 ≠ 𝐺))
42	40, 41	syl5ibrcom 246	. . . . . . . . . . 11 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → (𝑎 = 𝐺 → 𝐸 ≠ 𝑎))
43	42	imp 407	. . . . . . . . . 10 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐺) → 𝐸 ≠ 𝑎)
44	43	necomd 2999	. . . . . . . . 9 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐺) → 𝑎 ≠ 𝐸)
45	44	neneqd 2948	. . . . . . . 8 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐺) → ¬ 𝑎 = 𝐸)
46	45	adantrr 715	. . . . . . 7 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐺 ∧ 𝑏 = 𝑤)) → ¬ 𝑎 = 𝐸)
47	46	iffalsed 4497	. . . . . 6 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐺 ∧ 𝑏 = 𝑤)) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = if(𝑎 = 𝐺, 𝐹, 𝐻))
48		iftrue 4492	. . . . . . . 8 ⊢ (𝑎 = 𝐺 → if(𝑎 = 𝐺, 𝐹, 𝐻) = 𝐹)
49	48, 17	sylan9eq 2796	. . . . . . 7 ⊢ ((𝑎 = 𝐺 ∧ 𝑏 = 𝑤) → if(𝑎 = 𝐺, 𝐹, 𝐻) = ⦋𝑤 / 𝑏⦌𝐹)
50	49	adantl 482	. . . . . 6 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐺 ∧ 𝑏 = 𝑤)) → if(𝑎 = 𝐺, 𝐹, 𝐻) = ⦋𝑤 / 𝑏⦌𝐹)
51	47, 50	eqtrd 2776	. . . . 5 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ (𝑎 = 𝐺 ∧ 𝑏 = 𝑤)) → if(𝑎 = 𝐸, 𝐹, if(𝑎 = 𝐺, 𝐹, 𝐻)) = ⦋𝑤 / 𝑏⦌𝐹)
52		eqidd 2737	. . . . 5 ⊢ (((𝜓 ∧ 𝑤 ∈ 𝑁) ∧ 𝑎 = 𝐺) → 𝑁 = 𝑁)
53	21	simp2d 1143	. . . . . 6 ⊢ (𝜓 → 𝐺 ∈ 𝑁)
54	53	adantr 481	. . . . 5 ⊢ ((𝜓 ∧ 𝑤 ∈ 𝑁) → 𝐺 ∈ 𝑁)
55		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑏𝐺
56	15, 51, 52, 54, 24, 33, 34, 25, 55, 36, 37, 26	ovmpodxf 7505	. . . 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 2779	. . 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 3143	. 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 21961	. 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 1373	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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ⦋csb 3855 ∖ cdif 3907 ifcif 4486 {csn 4586 × cxp 5631 ↾ cres 5635 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ∈ cmpo 7359 ∘_f cof 7615 Fincfn 8883 Basecbs 17083 +_gcplusg 17133 ._rcmulr 17134 0_gc0g 17321 1_rcur 19913 Ringcrg 19964 Mat cmat 21754

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-sup 9378 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-0g 17323 df-prds 17329 df-pws 17331 df-sra 20633 df-rgmod 20634 df-dsmm 21138 df-frlm 21153 df-mat 21755

This theorem is referenced by: mdetunilem6 21966 mdetunilem8 21968

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