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Theorem dmatelnd 21098
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 21095 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 ↔ (𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ))))
6 neeq1 3076 . . . . . . . . . . 11 (𝑖 = 𝐼 → (𝑖𝑗𝐼𝑗))
7 oveq1 7152 . . . . . . . . . . . 12 (𝑖 = 𝐼 → (𝑖𝑋𝑗) = (𝐼𝑋𝑗))
87eqeq1d 2826 . . . . . . . . . . 11 (𝑖 = 𝐼 → ((𝑖𝑋𝑗) = 0 ↔ (𝐼𝑋𝑗) = 0 ))
96, 8imbi12d 348 . . . . . . . . . 10 (𝑖 = 𝐼 → ((𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) ↔ (𝐼𝑗 → (𝐼𝑋𝑗) = 0 )))
10 neeq2 3077 . . . . . . . . . . 11 (𝑗 = 𝐽 → (𝐼𝑗𝐼𝐽))
11 oveq2 7153 . . . . . . . . . . . 12 (𝑗 = 𝐽 → (𝐼𝑋𝑗) = (𝐼𝑋𝐽))
1211eqeq1d 2826 . . . . . . . . . . 11 (𝑗 = 𝐽 → ((𝐼𝑋𝑗) = 0 ↔ (𝐼𝑋𝐽) = 0 ))
1310, 12imbi12d 348 . . . . . . . . . 10 (𝑗 = 𝐽 → ((𝐼𝑗 → (𝐼𝑋𝑗) = 0 ) ↔ (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
149, 13rspc2v 3619 . . . . . . . . 9 ((𝐼𝑁𝐽𝑁) → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝐽 → (𝐼𝑋𝐽) = 0 )))
1514com23 86 . . . . . . . 8 ((𝐼𝑁𝐽𝑁) → (𝐼𝐽 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 ) → (𝐼𝑋𝐽) = 0 )))
16153impia 1114 . . . . . . 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 414 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑋𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑋𝑗) = 0 )) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 )))
205, 19sylbid 243 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑋𝐷 → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 )))
21203impia 1114 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) → ((𝐼𝑁𝐽𝑁𝐼𝐽) → (𝐼𝑋𝐽) = 0 ))
2221imp 410 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝐼𝑁𝐽𝑁𝐼𝐽)) → (𝐼𝑋𝐽) = 0 )
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2115  wne 3014  wral 3133  cfv 6343  (class class class)co 7145  Fincfn 8499  Basecbs 16479  0gc0g 16709  Ringcrg 19293   Mat cmat 21009   DMat cdmat 21090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-iota 6302  df-fun 6345  df-fv 6351  df-ov 7148  df-oprab 7149  df-mpo 7150  df-dmat 21092
This theorem is referenced by:  dmatmul  21099  dmatsubcl  21100
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