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Theorem dmatelnd 20221
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 = (0g𝑅)
dmatid.d 𝐷 = (𝑁 DMat 𝑅)
Assertion
Ref Expression
dmatelnd (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝐼𝑁𝐽𝑁𝐼𝐽)) → (𝐼𝑋𝐽) = 0 )

Proof of Theorem dmatelnd
Dummy variables 𝑖 𝑗 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmatid.a . . . . 5 𝐴 = (𝑁 Mat 𝑅)
2 dmatid.b . . . . 5 𝐵 = (Base‘𝐴)
3 dmatid.0 . . . . 5 0 = (0g𝑅)
4 dmatid.d . . . . 5 𝐷 = (𝑁 DMat 𝑅)
51, 2, 3, 4dmatel 20218 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 ↔ (𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ))))
6 neeq1 2852 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑖𝑗𝐼𝑗))
7 oveq1 6611 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
87eqeq1d 2623 . . . . . . . . . . 11 (𝑖 = 𝐼 → ((𝑖𝑋𝑗) = 0 ↔ (𝐼𝑋𝑗) = 0 ))
96, 8imbi12d 334 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) ↔ (𝐼𝑗 → (𝐼𝑋𝑗) = 0 )))
10 neeq2 2853 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝐼𝑗𝐼𝐽))
11 oveq2 6612 . . . . . . . . . . . 12 (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
1211eqeq1d 2623 . . . . . . . . . . 11 (𝑗 = 𝐽 → ((𝐼𝑋𝑗) = 0 ↔ (𝐼𝑋𝐽) = 0 ))
1310, 12imbi12d 334 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝐼𝑗 → (𝐼𝑋𝑗) = 0 ) ↔ (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
149, 13rspc2v 3306 . . . . . . . . 9 ((𝐼𝑁𝐽𝑁) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
1514com23 86 . . . . . . . 8 ((𝐼𝑁𝐽𝑁) → (𝐼𝐽 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 )))
16153impia 1258 . . . . . . 7 ((𝐼𝑁𝐽𝑁𝐼𝐽) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 ))
1716com12 32 . . . . . 6 (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 ))
18172a1i 12 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐵 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 ))))
1918impd 447 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 )) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 )))
205, 19sylbid 230 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 )))
21203impia 1258 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 ))
2221imp 445 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝐼𝑁𝐽𝑁𝐼𝐽)) → (𝐼𝑋𝐽) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  cfv 5847  (class class class)co 6604  Fincfn 7899  Basecbs 15781  0gc0g 16021  Ringcrg 18468   Mat cmat 20132   DMat cdmat 20213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-dmat 20215
This theorem is referenced by:  dmatmul  20222  dmatsubcl  20223
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