Theorem dmatelnd 22219

Description: An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)

Hypotheses

Ref	Expression
dmatid.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
dmatid.b	⊢ 𝐵 = (Base‘𝐴)
dmatid.0	⊢ 0 = (0g‘𝑅)
dmatid.d	⊢ 𝐷 = (𝑁 DMat 𝑅)

Assertion

Ref	Expression
dmatelnd	⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽)) → (𝐼𝑋𝐽) = 0 )

Proof of Theorem dmatelnd

Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmatid.a	. . . . 5 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		dmatid.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
3		dmatid.0	. . . . 5 ⊢ 0 = (0g‘𝑅)
4		dmatid.d	. . . . 5 ⊢ 𝐷 = (𝑁 DMat 𝑅)
5	1, 2, 3, 4	dmatel 22216	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ))))
6		neeq1 3002	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖 ≠ 𝑗 ↔ 𝐼 ≠ 𝑗))
7		oveq1 7419	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
8	7	eqeq1d 2733	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ((𝑖𝑋𝑗) = 0 ↔ (𝐼𝑋𝑗) = 0 ))
9	6, 8	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) ↔ (𝐼 ≠ 𝑗 → (𝐼𝑋𝑗) = 0 )))
10		neeq2 3003	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼 ≠ 𝑗 ↔ 𝐼 ≠ 𝐽))
11		oveq2 7420	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
12	11	eqeq1d 2733	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ((𝐼𝑋𝑗) = 0 ↔ (𝐼𝑋𝐽) = 0 ))
13	10, 12	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼 ≠ 𝑗 → (𝐼𝑋𝑗) = 0 ) ↔ (𝐼 ≠ 𝐽 → (𝐼𝑋𝐽) = 0 )))
14	9, 13	rspc2v 3622	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼 ≠ 𝐽 → (𝐼𝑋𝐽) = 0 )))
15	14	com23 86	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼 ≠ 𝐽 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 )))
16	15	3impia 1116	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 ))
17	16	com12 32	. . . . . 6 ⊢ (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 ))
18	17	2a1i 12	. . . . 5 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐵 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 ))))
19	18	impd 410	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑋 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 )) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 )))
20	5, 19	sylbid 239	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 )))
21	20	3impia 1116	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 ))
22	21	imp 406	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽)) → (𝐼𝑋𝐽) = 0 )

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ‘cfv 6543  (class class class)co 7412  Fincfn 8942  Basecbs 17149  0_gc0g 17390  Ringcrg 20128   Mat cmat 22128   DMat cdmat 22211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-dmat 22213

This theorem is referenced by:  dmatmul  22220  dmatsubcl  22221