Theorem dmatelnd 22518

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 22515	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ))))
6		neeq1 3001	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖 ≠ 𝑗 ↔ 𝐼 ≠ 𝑗))
7		oveq1 7438	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
8	7	eqeq1d 2737	. . . . . . . . . . 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 3002	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼 ≠ 𝑗 ↔ 𝐼 ≠ 𝐽))
11		oveq2 7439	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
12	11	eqeq1d 2737	. . . . . . . . . . 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 3633	. . . . . . . . 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 240	. . 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 1537   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ‘cfv 6563  (class class class)co 7431  Fincfn 8984  Basecbs 17245  0_gc0g 17486  Ringcrg 20251   Mat cmat 22427   DMat cdmat 22510

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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

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-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-dmat 22512

This theorem is referenced by:  dmatmul  22519  dmatsubcl  22520