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Theorem dmatelnd 22503
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 22500 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 ↔ (𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ))))
6 neeq1 3002 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑖𝑗𝐼𝑗))
7 oveq1 7439 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
87eqeq1d 2738 . . . . . . . . . . 11 (𝑖 = 𝐼 → ((𝑖𝑋𝑗) = 0 ↔ (𝐼𝑋𝑗) = 0 ))
96, 8imbi12d 344 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) ↔ (𝐼𝑗 → (𝐼𝑋𝑗) = 0 )))
10 neeq2 3003 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝐼𝑗𝐼𝐽))
11 oveq2 7440 . . . . . . . . . . . 12 (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
1211eqeq1d 2738 . . . . . . . . . . 11 (𝑗 = 𝐽 → ((𝐼𝑋𝑗) = 0 ↔ (𝐼𝑋𝐽) = 0 ))
1310, 12imbi12d 344 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝐼𝑗 → (𝐼𝑋𝑗) = 0 ) ↔ (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
149, 13rspc2v 3632 . . . . . . . . 9 ((𝐼𝑁𝐽𝑁) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
1514com23 86 . . . . . . . 8 ((𝐼𝑁𝐽𝑁) → (𝐼𝐽 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 )))
16153impia 1117 . . . . . . 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 410 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 )) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 )))
205, 19sylbid 240 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 )))
21203impia 1117 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 ))
2221imp 406 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝐼𝑁𝐽𝑁𝐼𝐽)) → (𝐼𝑋𝐽) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  cfv 6560  (class class class)co 7432  Fincfn 8986  Basecbs 17248  0gc0g 17485  Ringcrg 20231   Mat cmat 22412   DMat cdmat 22495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-dmat 22497
This theorem is referenced by:  dmatmul  22504  dmatsubcl  22505
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