Theorem mdetunilem1 22780

Description: Lemma for mdetuni 22790. (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 1214	. 2 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → 𝐹 ≠ 𝐺)
2		simpl3 1211	. 2 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤))
3		neeq2 3020	. . . . 5 ⊢ (𝑧 = 𝐺 → (𝐹 ≠ 𝑧 ↔ 𝐹 ≠ 𝐺))
4		oveq1 7419	. . . . . . 7 ⊢ (𝑧 = 𝐺 → (𝑧𝐸𝑤) = (𝐺𝐸𝑤))
5	4	eqeq2d 2773	. . . . . 6 ⊢ (𝑧 = 𝐺 → ((𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
6	5	ralbidv 3187	. . . . 5 ⊢ (𝑧 = 𝐺 → (∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)))
7	3, 6	anbi12d 643	. . . 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 1210	. . . 4 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → 𝐸 ∈ 𝐵)
10		simpr1 1212	. . . 4 ⊢ (((𝜑 ∧ 𝐸 ∈ 𝐵 ∧ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝐺𝐸𝑤)) ∧ (𝐹 ∈ 𝑁 ∧ 𝐺 ∈ 𝑁 ∧ 𝐹 ≠ 𝐺)) → 𝐹 ∈ 𝑁)
11		simpl1 1209	. . . . 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 7418	. . . . . . . . . 10 ⊢ (𝑥 = 𝐸 → (𝑦𝑥𝑤) = (𝑦𝐸𝑤))
15		oveq 7418	. . . . . . . . . 10 ⊢ (𝑥 = 𝐸 → (𝑧𝑥𝑤) = (𝑧𝐸𝑤))
16	14, 15	eqeq12d 2778	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → ((𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
17	16	ralbidv 3187	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤) ↔ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤)))
18	17	anbi2d 641	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝑥𝑤) = (𝑧𝑥𝑤)) ↔ (𝑦 ≠ 𝑧 ∧ ∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤))))
19		fveqeq2 6890	. . . . . . 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 3187	. . . . 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 3019	. . . . . . . 8 ⊢ (𝑦 = 𝐹 → (𝑦 ≠ 𝑧 ↔ 𝐹 ≠ 𝑧))
23		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → (𝑦𝐸𝑤) = (𝐹𝐸𝑤))
24	23	eqeq1d 2764	. . . . . . . . 9 ⊢ (𝑦 = 𝐹 → ((𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
25	24	ralbidv 3187	. . . . . . . 8 ⊢ (𝑦 = 𝐹 → (∀𝑤 ∈ 𝑁 (𝑦𝐸𝑤) = (𝑧𝐸𝑤) ↔ ∀𝑤 ∈ 𝑁 (𝐹𝐸𝑤) = (𝑧𝐸𝑤)))
26	22, 25	anbi12d 643	. . . . . . 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 3187	. . . . 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 3592	. . . 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 850	. . 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 1213	. . 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 711	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 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142   ≠ wne 2957  ∀wral 3078   ∖ cdif 3901  {csn 4588   × cxp 5658   ↾ cres 5662  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   ∘_f cof 7674  Fincfn 8941  Basecbs 17275  +_gcplusg 17316  ._rcmulr 17317  0_gc0g 17498  1_rcur 20269  Ringcrg 20321   Mat cmat 22575

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544  df-ov 7415

This theorem is used by:  mdetunilem2  22781  mdetuni0  22789