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Theorem dmatel 21995
Description: A 𝑁 x 𝑁 diagonal matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.)
Hypotheses
Ref Expression
dmatval.a 𝐴 = (𝑁 Mat 𝑅)
dmatval.b 𝐵 = (Base‘𝐴)
dmatval.0 0 = (0g𝑅)
dmatval.d 𝐷 = (𝑁 DMat 𝑅)
Assertion
Ref Expression
dmatel ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))
Distinct variable groups:   𝑖,𝑁,𝑗   𝑅,𝑖,𝑗   𝑖,𝑀,𝑗
Allowed substitution hints:   𝐴(𝑖,𝑗)   𝐵(𝑖,𝑗)   𝐷(𝑖,𝑗)   𝑉(𝑖,𝑗)   0 (𝑖,𝑗)

Proof of Theorem dmatel
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 dmatval.a . . . 4 𝐴 = (𝑁 Mat 𝑅)
2 dmatval.b . . . 4 𝐵 = (Base‘𝐴)
3 dmatval.0 . . . 4 0 = (0g𝑅)
4 dmatval.d . . . 4 𝐷 = (𝑁 DMat 𝑅)
51, 2, 3, 4dmatval 21994 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
65eleq2d 2820 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
7 oveq 7415 . . . . . 6 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
87eqeq1d 2735 . . . . 5 (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = 0 ↔ (𝑖𝑀𝑗) = 0 ))
98imbi2d 341 . . . 4 (𝑚 = 𝑀 → ((𝑖𝑗 → (𝑖𝑚𝑗) = 0 ) ↔ (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )))
1092ralbidv 3219 . . 3 (𝑚 = 𝑀 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 ) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )))
1110elrab 3684 . 2 (𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )))
126, 11bitrdi 287 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2941  wral 3062  {crab 3433  cfv 6544  (class class class)co 7409  Fincfn 8939  Basecbs 17144  0gc0g 17385   Mat cmat 21907   DMat cdmat 21990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-dmat 21992
This theorem is referenced by:  dmatmat  21996  dmatid  21997  dmatelnd  21998  dmatsubcl  22000  dmatscmcl  22005
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