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Theorem mdetunilem4 22491

Description: Lemma for mdetuni 22498. (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))))

**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 1208	. 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 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 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))
3		simp1 1134	. . 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 1206	. . . 4 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → 𝐺 ∈ 𝐵)
5		simp3 1136	. . . 4 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → 𝐻 ∈ 𝑁)
6		simp21 1204	. . . . 5 ⊢ ((𝜑 ∧ (𝐸 ∈ 𝐵 ∧ 𝐹 ∈ 𝐾 ∧ 𝐺 ∈ 𝐵) ∧ 𝐻 ∈ 𝑁) → 𝐸 ∈ 𝐵)
7		simp22 1205	. . . . 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 1131	. . . . 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 5973	. . . . . . . . . 10 ⊢ (𝑥 = 𝐸 → (𝑥 ↾ ({𝑤} × 𝑁)) = (𝐸 ↾ ({𝑤} × 𝑁)))
11	10	eqeq1d 2729	. . . . . . . . 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 5973	. . . . . . . . . 10 ⊢ (𝑥 = 𝐸 → (𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
13	12	eqeq1d 2729	. . . . . . . . 9 ⊢ (𝑥 = 𝐸 → ((𝑥 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
14	11, 13	anbi12d 630	. . . . . . . 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 6900	. . . . . . . 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 3213	. . . . . 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 4634	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐹 → {𝑦} = {𝐹})
19	18	xpeq2d 5702	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐹 → (({𝑤} × 𝑁) × {𝑦}) = (({𝑤} × 𝑁) × {𝐹}))
20	19	oveq1d 7429	. . . . . . . . . 10 ⊢ (𝑦 = 𝐹 → ((({𝑤} × 𝑁) × {𝑦}) ∘_f · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {𝐹}) ∘_f · (𝑧 ↾ ({𝑤} × 𝑁))))
21	20	eqeq2d 2738	. . . . . . . . 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 629	. . . . . . . 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 7421	. . . . . . . . 9 ⊢ (𝑦 = 𝐹 → (𝑦 · (𝐷‘𝑧)) = (𝐹 · (𝐷‘𝑧)))
24	23	eqeq2d 2738	. . . . . . . 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 3213	. . . . . 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 3619	. . . . 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 837	. . . 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 5973	. . . . . . . . 9 ⊢ (𝑧 = 𝐺 → (𝑧 ↾ ({𝑤} × 𝑁)) = (𝐺 ↾ ({𝑤} × 𝑁)))
30	29	oveq2d 7430	. . . . . . . 8 ⊢ (𝑧 = 𝐺 → ((({𝑤} × 𝑁) × {𝐹}) ∘_f · (𝑧 ↾ ({𝑤} × 𝑁))) = ((({𝑤} × 𝑁) × {𝐹}) ∘_f · (𝐺 ↾ ({𝑤} × 𝑁))))
31	30	eqeq2d 2738	. . . . . . 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 5973	. . . . . . . 8 ⊢ (𝑧 = 𝐺 → (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)))
33	32	eqeq2d 2738	. . . . . . 7 ⊢ (𝑧 = 𝐺 → ((𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝑧 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁))))
34	31, 33	anbi12d 630	. . . . . 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 6891	. . . . . . . 8 ⊢ (𝑧 = 𝐺 → (𝐷‘𝑧) = (𝐷‘𝐺))
36	35	oveq2d 7430	. . . . . . 7 ⊢ (𝑧 = 𝐺 → (𝐹 · (𝐷‘𝑧)) = (𝐹 · (𝐷‘𝐺)))
37	36	eqeq2d 2738	. . . . . 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 4634	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → {𝑤} = {𝐻})
40	39	xpeq1d 5701	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → ({𝑤} × 𝑁) = ({𝐻} × 𝑁))
41	40	reseq2d 5979	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → (𝐸 ↾ ({𝑤} × 𝑁)) = (𝐸 ↾ ({𝐻} × 𝑁)))
42	40	xpeq1d 5701	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (({𝑤} × 𝑁) × {𝐹}) = (({𝐻} × 𝑁) × {𝐹}))
43	40	reseq2d 5979	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (𝐺 ↾ ({𝑤} × 𝑁)) = (𝐺 ↾ ({𝐻} × 𝑁)))
44	42, 43	oveq12d 7432	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → ((({𝑤} × 𝑁) × {𝐹}) ∘_f · (𝐺 ↾ ({𝑤} × 𝑁))) = ((({𝐻} × 𝑁) × {𝐹}) ∘_f · (𝐺 ↾ ({𝐻} × 𝑁))))
45	41, 44	eqeq12d 2743	. . . . . . 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 4118	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → (𝑁 ∖ {𝑤}) = (𝑁 ∖ {𝐻}))
47	46	xpeq1d 5701	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → ((𝑁 ∖ {𝑤}) × 𝑁) = ((𝑁 ∖ {𝐻}) × 𝑁))
48	47	reseq2d 5979	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → (𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))
49	47	reseq2d 5979	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)))
50	48, 49	eqeq12d 2743	. . . . . . 7 ⊢ (𝑤 = 𝐻 → ((𝐸 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝑤}) × 𝑁)) ↔ (𝐸 ↾ ((𝑁 ∖ {𝐻}) × 𝑁)) = (𝐺 ↾ ((𝑁 ∖ {𝐻}) × 𝑁))))
51	45, 50	anbi12d 630	. . . . . 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 3619	. . . 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 837	. . 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 1163	. 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 698	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)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∖ cdif 3941 {csn 4624 × cxp 5670 ↾ cres 5674 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ∘_f cof 7675 Fincfn 8953 Basecbs 17165 +_gcplusg 17218 ._rcmulr 17219 0_gc0g 17406 1_rcur 20105 Ringcrg 20157 Mat cmat 22281

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2698

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-xp 5678 df-res 5684 df-iota 6494 df-fv 6550 df-ov 7417

This theorem is referenced by: mdetuni0 22497

W3C validator