Theorem dmatelnd 22614

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 22611	. . . 4 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 ↔ (𝑋 ∈ 𝐵 ∧ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ))))
6		neeq1 3022	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → (𝑖 ≠ 𝑗 ↔ 𝐼 ≠ 𝑗))
7		oveq1 7407	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
8	7	eqeq1d 2767	. . . . . . . . . . 11 ⊢ (𝑖 = 𝐼 → ((𝑖𝑋𝑗) = 0 ↔ (𝐼𝑋𝑗) = 0 ))
9	6, 8	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑖 = 𝐼 → ((𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) ↔ (𝐼 ≠ 𝑗 → (𝐼𝑋𝑗) = 0 )))
10		neeq2 3023	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → (𝐼 ≠ 𝑗 ↔ 𝐼 ≠ 𝐽))
11		oveq2 7408	. . . . . . . . . . . 12 ⊢ (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
12	11	eqeq1d 2767	. . . . . . . . . . 11 ⊢ (𝑗 = 𝐽 → ((𝐼𝑋𝑗) = 0 ↔ (𝐼𝑋𝐽) = 0 ))
13	10, 12	imbi12d 347	. . . . . . . . . 10 ⊢ (𝑗 = 𝐽 → ((𝐼 ≠ 𝑗 → (𝐼𝑋𝑗) = 0 ) ↔ (𝐼 ≠ 𝐽 → (𝐼𝑋𝐽) = 0 )))
14	9, 13	rspc2v 3595	. . . . . . . . 9 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼 ≠ 𝐽 → (𝐼𝑋𝐽) = 0 )))
15	14	com23 87	. . . . . . . 8 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁) → (𝐼 ≠ 𝐽 → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 )))
16	15	3impia 1133	. . . . . . 7 ⊢ ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 ))
17	16	com12 33	. . . . . 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 415	. . . 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 243	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋 ∈ 𝐷 → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 )))
21	20	3impia 1133	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) → ((𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽) → (𝐼𝑋𝐽) = 0 ))
22	21	imp 411	1 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐷) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑁 ∧ 𝐼 ≠ 𝐽)) → (𝐼𝑋𝐽) = 0 )

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960  ∀wral 3079  ‘cfv 6525  (class class class)co 7400  Fincfn 8931  Basecbs 17259  0_gc0g 17482  Ringcrg 20306   Mat cmat 22525   DMat cdmat 22606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-dmat 22608

This theorem is referenced by:  dmatmul  22615  dmatsubcl  22616