Theorem mdetunilem4 22637

Description: Lemma for mdetuni 22644. (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))))

Assertion

Ref	Expression
mdetunilem4	⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ (𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺)))

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   𝑥,𝐻,𝑦,𝑧,𝑤

Proof of Theorem mdetunilem4

Step	Hyp	Ref	Expression
1		simp32 1209	. 2 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ (𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))))
2		simp33 1210	. 2 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ (𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))
3		simp1 1135	. . 3 ⊢ ((𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁))) → 𝐻 ∈ 𝑁)
4		simp23 1207	. . . 4 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → 𝐺 ∈ 𝐵)
5		simp3 1137	. . . 4 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → 𝐻 ∈ 𝑁)
6		simp21 1205	. . . . 5 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → 𝐸 ∈ 𝐵)
7		simp22 1206	. . . . 5 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → 𝐹 ∈ 𝐾)
8		mdetuni.sc	. . . . . 6 ⊢ (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))))
9	8	3ad2ant1 1132	. . . . 5 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → ∀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))))
10		reseq1 5994	. . . . . . . . . 10 ⊢ (𝑥 = 𝐸 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝐸 ↾ ({𝑤} × 𝑁)))
11	10	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → ((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁)))))
12		reseq1 5994	. . . . . . . . . 10 ⊢ (𝑥 = 𝐸 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
13	12	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
14	11, 13	anbi12d 632	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → (((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
15		fveqeq2 6916	. . . . . . . 8 ⊢ (𝑥 = 𝐸 → ((𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘𝐸) = (𝑦 · (𝐷‘𝑧))))
16	14, 15	imbi12d 344	. . . . . . 7 ⊢ (𝑥 = 𝐸 → ((((𝑥 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝑥) = (𝑦 · (𝐷‘𝑧))) ↔ (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝑦 · (𝐷‘𝑧)))))
17	16	2ralbidv 3219	. . . . . 6 ⊢ (𝑥 = 𝐸 → (∀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))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝑦 · (𝐷‘𝑧)))))
18		sneq 4641	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐹 → {𝑦} = {𝐹})
19	18	xpeq2d 5719	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐹 → (({𝑤} × 𝑁) × {𝑦}) = (({𝑤} × 𝑁) × {𝐹}))
20	19	oveq1d 7446	. . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))))
21	20	eqeq2d 2746	. . . . . . . . 9 ⊢ (𝑦 = 𝐹 → ((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁)))))
22	21	anbi1d 631	. . . . . . . 8 ⊢ (𝑦 = 𝐹 → (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
23		oveq1 7438	. . . . . . . . 9 ⊢ (𝑦 = 𝐹 → (𝑦 · (𝐷‘𝑧)) = (𝐹 · (𝐷‘𝑧)))
24	23	eqeq2d 2746	. . . . . . . 8 ⊢ (𝑦 = 𝐹 → ((𝐷‘𝐸) = (𝑦 · (𝐷‘𝑧)) ↔ (𝐷‘𝐸) = (𝐹 · (𝐷‘𝑧))))
25	22, 24	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = 𝐹 → ((((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝑦 · (𝐷‘𝑧))) ↔ (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝑧)))))
26	25	2ralbidv 3219	. . . . . 6 ⊢ (𝑦 = 𝐹 → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝑦}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝑦 · (𝐷‘𝑧))) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝑧)))))
27	17, 26	rspc2va 3634	. . . . 5 ⊢ (((𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾) ∧ ∀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)))) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝑧))))
28	6, 7, 9, 27	syl21anc 838	. . . 4 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝑧))))
29		reseq1 5994	. . . . . . . . 9 ⊢ (𝑧 = 𝐺 → (𝑧 ↾ ({𝑤} × 𝑁)) = (𝐺 ↾ ({𝑤} × 𝑁)))
30	29	oveq2d 7447	. . . . . . . 8 ⊢ (𝑧 = 𝐺 → ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝑤} × 𝑁))))
31	30	eqeq2d 2746	. . . . . . 7 ⊢ (𝑧 = 𝐺 → ((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ↔ (𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝑤} × 𝑁)))))
32		reseq1 5994	. . . . . . . 8 ⊢ (𝑧 = 𝐺 → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
33	32	eqeq2d 2746	. . . . . . 7 ⊢ (𝑧 = 𝐺 → ((𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
34	31, 33	anbi12d 632	. . . . . 6 ⊢ (𝑧 = 𝐺 → (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))))
35		fveq2 6907	. . . . . . . 8 ⊢ (𝑧 = 𝐺 → (𝐷‘𝑧) = (𝐷‘𝐺))
36	35	oveq2d 7447	. . . . . . 7 ⊢ (𝑧 = 𝐺 → (𝐹 · (𝐷‘𝑧)) = (𝐹 · (𝐷‘𝐺)))
37	36	eqeq2d 2746	. . . . . 6 ⊢ (𝑧 = 𝐺 → ((𝐷‘𝐸) = (𝐹 · (𝐷‘𝑧)) ↔ (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺))))
38	34, 37	imbi12d 344	. . . . 5 ⊢ (𝑧 = 𝐺 → ((((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝑧))) ↔ (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺)))))
39		sneq 4641	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → {𝑤} = {𝐻})
40	39	xpeq1d 5718	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → ({𝑤} × 𝑁) = ({𝐻} × 𝑁))
41	40	reseq2d 6000	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → (𝐸 ↾ ({𝑤} × 𝑁)) = (𝐸 ↾ ({𝐻} × 𝑁)))
42	40	xpeq1d 5718	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (({𝑤} × 𝑁) × {𝐹}) = (({𝐻} × 𝑁) × {𝐹}))
43	40	reseq2d 6000	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (𝐺 ↾ ({𝑤} × 𝑁)) = (𝐺 ↾ ({𝐻} × 𝑁)))
44	42, 43	oveq12d 7449	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝑤} × 𝑁))) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))))
45	41, 44	eqeq12d 2751	. . . . . . 7 ⊢ (𝑤 = 𝐻 → ((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝑤} × 𝑁))) ↔ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁)))))
46	39	difeq2d 4136	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {𝐻}))
47	46	xpeq1d 5718	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {𝐻}) × 𝑁))
48	47	reseq2d 6000	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))
49	47	reseq2d 6000	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))
50	48, 49	eqeq12d 2751	. . . . . . 7 ⊢ (𝑤 = 𝐻 → ((𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁))))
51	45, 50	anbi12d 632	. . . . . 6 ⊢ (𝑤 = 𝐻 → (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) ↔ ((𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))))
52	51	imbi1d 341	. . . . 5 ⊢ (𝑤 = 𝐻 → ((((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺))) ↔ (((𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺)))))
53	38, 52	rspc2va 3634	. . . 4 ⊢ (((𝐺 ∈ 𝐵 ∧ 𝐻 ∈ 𝑁) ∧ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝑁 (((𝐸 ↾ ({𝑤} × 𝑁)) = ((({𝑤} × 𝑁) × {𝐹}) ∘f · (𝑧 ↾ ({𝑤} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝑧)))) → (((𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺))))
54	4, 5, 28, 53	syl21anc 838	. . 3 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → (((𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺))))
55	3, 54	syl3an3 1164	. 2 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ (𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (((𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺))))
56	1, 2, 55	mp2and 699	1 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ (𝐻 ∈ 𝑁 ∧ (𝐸 ↾ ({𝐻} × 𝑁)) = ((({𝐻} × 𝑁) × {𝐹}) ∘f · (𝐺 ↾ ({𝐻} × 𝑁))) ∧ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))) → (𝐷‘𝐸) = (𝐹 · (𝐷‘𝐺)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059   ∖ cdif 3960  {csn 4631   × cxp 5687   ↾ cres 5691  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ∘_f cof 7695  Fincfn 8984  Basecbs 17245  +_gcplusg 17298  ._rcmulr 17299  0_gc0g 17486  1_rcur 20199  Ringcrg 20251   Mat cmat 22427

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701  df-iota 6516  df-fv 6571  df-ov 7434

This theorem is referenced by:  mdetuni0  22643