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Theorem dmatel 20118
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 20117 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})
65eleq2d 2673 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )}))
7 oveq 6555 . . . . . 6 (𝑚 = 𝑀 → (𝑖𝑚𝑗) = (𝑖𝑀𝑗))
87eqeq1d 2612 . . . . 5 (𝑚 = 𝑀 → ((𝑖𝑚𝑗) = 0 ↔ (𝑖𝑀𝑗) = 0 ))
98imbi2d 329 . . . 4 (𝑚 = 𝑀 → ((𝑖𝑗 → (𝑖𝑚𝑗) = 0 ) ↔ (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )))
1092ralbidv 2972 . . 3 (𝑚 = 𝑀 → (∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 ) ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )))
1110elrab 3331 . 2 (𝑀 ∈ {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )} ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 )))
126, 11syl6bb 275 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383   = wceq 1475  wcel 1977  wne 2780  wral 2896  {crab 2900  cfv 5804  (class class class)co 6549  Fincfn 7841  Basecbs 15695  0gc0g 15923   Mat cmat 20032   DMat cdmat 20113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-dmat 20115
This theorem is referenced by:  dmatmat  20119  dmatid  20120  dmatelnd  20121  dmatsubcl  20123  dmatscmcl  20128
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