Theorem mdetunilem1 22954

Description: Lemma for mdetuni 22964. (Contributed by SO, 14-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))))

Assertion

Ref	Expression
mdetunilem1	⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → (𝐷‘𝐸) = 0 )

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

Proof of Theorem mdetunilem1

Step	Hyp	Ref	Expression
1		simpr3 1215	. 2 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → 𝐹 ≠ 𝐺)
2		simpl3 1212	. 2 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))
3		neeq2 3022	. . . . 5 ⊢ (𝑧 = 𝐺 → (𝐹 ≠ 𝑧 ↔ 𝐹 ≠ 𝐺))
4		oveq1 7447	. . . . . . 7 ⊢ (𝑧 = 𝐺 → (𝑧𝐸𝑤) = (𝐺𝐸𝑤))
5	4	eqeq2d 2774	. . . . . 6 ⊢ (𝑧 = 𝐺 → ((𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
6	5	ralbidv 3189	. . . . 5 ⊢ (𝑧 = 𝐺 → (∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
7	3, 6	anbi12d 644	. . . 4 ⊢ (𝑧 = 𝐺 → ((𝐹 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) ↔ (𝐹 ≠ 𝐺 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))))
8	7	imbi1d 344	. . 3 ⊢ (𝑧 = 𝐺 → (((𝐹 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 ) ↔ ((𝐹 ≠ 𝐺 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷‘𝐸) = 0 )))
9		simpl2 1211	. . . 4 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → 𝐸 ∈ 𝐵)
10		simpr1 1213	. . . 4 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → 𝐹 ∈ 𝑁)
11		simpl1 1210	. . . . 5 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → 𝜑)
12		mdetuni.al	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
13	11, 12	syl 18	. . . 4 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ))
14		oveq 7446	. . . . . . . . . 10 ⊢ (𝑥 = 𝐸 → (𝑦𝑥𝑤) = (𝑦𝐸𝑤))
15		oveq 7446	. . . . . . . . . 10 ⊢ (𝑥 = 𝐸 → (𝑧𝑥𝑤) = (𝑧𝐸𝑤))
16	14, 15	eqeq12d 2779	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → ((𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
17	16	ralbidv 3189	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
18	17	anbi2d 642	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) ↔ (𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤))))
19		fveqeq2 6914	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((𝐷‘𝑥) = 0 ↔ (𝐷‘𝐸) = 0 ))
20	18, 19	imbi12d 347	. . . . . 6 ⊢ (𝑥 = 𝐸 → (((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ) ↔ ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 )))
21	20	ralbidv 3189	. . . . 5 ⊢ (𝑥 = 𝐸 → (∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 ) ↔ ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 )))
22		neeq1 3021	. . . . . . . 8 ⊢ (𝑦 = 𝐹 → (𝑦 ≠ 𝑧 ↔ 𝐹 ≠ 𝑧))
23		oveq1 7447	. . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → (𝑦𝐸𝑤) = (𝐹𝐸𝑤))
24	23	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑦 = 𝐹 → ((𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
25	24	ralbidv 3189	. . . . . . . 8 ⊢ (𝑦 = 𝐹 → (∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
26	22, 25	anbi12d 644	. . . . . . 7 ⊢ (𝑦 = 𝐹 → ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) ↔ (𝐹 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤))))
27	26	imbi1d 344	. . . . . 6 ⊢ (𝑦 = 𝐹 → (((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 ) ↔ ((𝐹 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 )))
28	27	ralbidv 3189	. . . . 5 ⊢ (𝑦 = 𝐹 → (∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 ) ↔ ∀𝑧 ∈ 𝑁 ((𝐹 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 )))
29	21, 28	rspc2va 3591	. . . 4 ⊢ (((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝑁) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝑁 ∀𝑧 ∈ 𝑁 ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) → (𝐷‘𝑥) = 0 )) → ∀𝑧 ∈ 𝑁 ((𝐹 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 ))
30	9, 10, 13, 29	syl21anc 851	. . 3 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → ∀𝑧 ∈ 𝑁 ((𝐹 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)) → (𝐷‘𝐸) = 0 ))
31		simpr2 1214	. . 3 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → 𝐺 ∈ 𝑁)
32	8, 30, 31	rspcdva 3581	. 2 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → ((𝐹 ≠ 𝐺 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) → (𝐷‘𝐸) = 0 ))
33	1, 2, 32	mp2and 712	1 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → (𝐷‘𝐸) = 0 )

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ∖ cdif 4048  {csn 4588   × cxp 5653   ↾ cres 5657  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440   ∘_f cof 7712  Fincfn 8998  Basecbs 17413  +_gcplusg 17454  ._rcmulr 17455  0_gc0g 17636  1_rcur 20434  Ringcrg 20486   Mat cmat 22749

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-ov 7443

This theorem is used by:  mdetunilem2  22955  mdetuni0  22963