Theorem dmatbas 44465

Description: The set of all 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅 is the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅. (Contributed by AV, 8-Dec-2019.)

Hypotheses

Ref	Expression
dmatbas.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
dmatbas.b	⊢ 𝐵 = (Base‘𝐴)
dmatbas.0	⊢ 0 = (0g‘𝑅)
dmatbas.d	⊢ 𝐷 = (𝑁 DMat 𝑅)

Assertion

Ref	Expression
dmatbas	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))

Proof of Theorem dmatbas

Dummy variables 𝑚 𝑖 𝑗 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmatbas.a	. . 3 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		dmatbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐴)
3		dmatbas.0	. . 3 ⊢ 0 = (0g‘𝑅)
4		dmatbas.d	. . 3 ⊢ 𝐷 = (𝑁 DMat 𝑅)
5	1, 2, 3, 4	dmatval 21104	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
6		elex 3515	. . 3 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
7		eqid 2824	. . . 4 ⊢ (𝑁 DMatALT 𝑅) = (𝑁 DMatALT 𝑅)
8	1, 2, 3, 7	dmatALTbas 44463	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 DMatALT 𝑅)) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
9	6, 8	sylan2 594	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (Base‘(𝑁 DMatALT 𝑅)) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
10	5, 9	eqtr4d 2862	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1536   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  {crab 3145  Vcvv 3497  ‘cfv 6358  (class class class)co 7159  Fincfn 8512  Basecbs 16486  0_gc0g 16716   Mat cmat 21019   DMat cdmat 21100   DMatALT cdmatalt 44458

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-cnex 10596  ax-1cn 10598  ax-addcl 10600

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-ral 3146  df-rex 3147  df-reu 3148  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-iun 4924  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-nn 11642  df-ndx 16489  df-slot 16490  df-base 16492  df-sets 16493  df-ress 16494  df-dmat 21102  df-dmatalt 44460

This theorem is referenced by: (None)