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Theorem dmatelnd 20625
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 20622 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 ↔ (𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ))))
6 neeq1 3031 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑖𝑗𝐼𝑗))
7 oveq1 6883 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
87eqeq1d 2799 . . . . . . . . . . 11 (𝑖 = 𝐼 → ((𝑖𝑋𝑗) = 0 ↔ (𝐼𝑋𝑗) = 0 ))
96, 8imbi12d 336 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) ↔ (𝐼𝑗 → (𝐼𝑋𝑗) = 0 )))
10 neeq2 3032 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝐼𝑗𝐼𝐽))
11 oveq2 6884 . . . . . . . . . . . 12 (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
1211eqeq1d 2799 . . . . . . . . . . 11 (𝑗 = 𝐽 → ((𝐼𝑋𝑗) = 0 ↔ (𝐼𝑋𝐽) = 0 ))
1310, 12imbi12d 336 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝐼𝑗 → (𝐼𝑋𝑗) = 0 ) ↔ (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
149, 13rspc2v 3508 . . . . . . . . 9 ((𝐼𝑁𝐽𝑁) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
1514com23 86 . . . . . . . 8 ((𝐼𝑁𝐽𝑁) → (𝐼𝐽 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 )))
16153impia 1146 . . . . . . 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 399 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 )) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 )))
205, 19sylbid 232 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 )))
21203impia 1146 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 ))
2221imp 396 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝐼𝑁𝐽𝑁𝐼𝐽)) → (𝐼𝑋𝐽) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2969  wral 3087  cfv 6099  (class class class)co 6876  Fincfn 8193  Basecbs 16181  0gc0g 16412  Ringcrg 18860   Mat cmat 20535   DMat cdmat 20617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-iota 6062  df-fun 6101  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-dmat 20619
This theorem is referenced by:  dmatmul  20626  dmatsubcl  20627
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