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Theorem dmatelnd 20350
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 20347 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 ↔ (𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ))))
6 neeq1 2885 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑖𝑗𝐼𝑗))
7 oveq1 6697 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
87eqeq1d 2653 . . . . . . . . . . 11 (𝑖 = 𝐼 → ((𝑖𝑋𝑗) = 0 ↔ (𝐼𝑋𝑗) = 0 ))
96, 8imbi12d 333 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) ↔ (𝐼𝑗 → (𝐼𝑋𝑗) = 0 )))
10 neeq2 2886 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝐼𝑗𝐼𝐽))
11 oveq2 6698 . . . . . . . . . . . 12 (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
1211eqeq1d 2653 . . . . . . . . . . 11 (𝑗 = 𝐽 → ((𝐼𝑋𝑗) = 0 ↔ (𝐼𝑋𝐽) = 0 ))
1310, 12imbi12d 333 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝐼𝑗 → (𝐼𝑋𝑗) = 0 ) ↔ (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
149, 13rspc2v 3353 . . . . . . . . 9 ((𝐼𝑁𝐽𝑁) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
1514com23 86 . . . . . . . 8 ((𝐼𝑁𝐽𝑁) → (𝐼𝐽 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 )))
16153impia 1280 . . . . . . 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 446 . . . 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 1280 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 ))
2221imp 444 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝐼𝑁𝐽𝑁𝐼𝐽)) → (𝐼𝑋𝐽) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  cfv 5926  (class class class)co 6690  Fincfn 7997  Basecbs 15904  0gc0g 16147  Ringcrg 18593   Mat cmat 20261   DMat cdmat 20342
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-dmat 20344
This theorem is referenced by:  dmatmul  20351  dmatsubcl  20352
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