Theorem dmatALTbas 46472

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

Hypotheses

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

Assertion

Ref	Expression
dmatALTbas	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})

Distinct variable groups:   𝐵,𝑚   𝑖,𝑁,𝑗,𝑚   𝑅,𝑖,𝑗,𝑚

Allowed substitution hints:   𝐴(𝑖,𝑗,𝑚)   𝐵(𝑖,𝑗)   𝐷(𝑖,𝑗,𝑚)   0 (𝑖,𝑗,𝑚)

Proof of Theorem dmatALTbas

Step	Hyp	Ref	Expression
1		dmatALTval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
2		dmatALTval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
3		dmatALTval.0	. . . 4 ⊢ 0 = (0g‘𝑅)
4		dmatALTval.d	. . . 4 ⊢ 𝐷 = (𝑁 DMatALT 𝑅)
5	1, 2, 3, 4	dmatALTval 46471	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → 𝐷 = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}))
6	5	fveq2d 6846	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = (Base‘(𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})))
7	2	fvexi 6856	. . . 4 ⊢ 𝐵 ∈ V
8		rabexg 5288	. . . 4 ⊢ (𝐵 ∈ V → {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V)
9	7, 8	mp1i 13	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V)
10		eqid 2736	. . . 4 ⊢ (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}) = (𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
11	10, 2	ressbas 17118	. . 3 ⊢ ({𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∈ V → ({𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = (Base‘(𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})))
12	9, 11	syl 17	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → ({𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = (Base‘(𝐴 ↾s {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})))
13		inrab2 4267	. . 3 ⊢ ({𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = {𝑚 ∈ (𝐵 ∩ 𝐵) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )}
14		inidm 4178	. . . 4 ⊢ (𝐵 ∩ 𝐵) = 𝐵
15		rabeq 3421	. . . 4 ⊢ ((𝐵 ∩ 𝐵) = 𝐵 → {𝑚 ∈ (𝐵 ∩ 𝐵) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
16	14, 15	mp1i 13	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → {𝑚 ∈ (𝐵 ∩ 𝐵) ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
17	13, 16	eqtrid 2788	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → ({𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )} ∩ 𝐵) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})
18	6, 12, 17	3eqtr2d 2782	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘𝐷) = {𝑚 ∈ 𝐵 ∣ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑁 (𝑖 ≠ 𝑗 → (𝑖𝑚𝑗) = 0 )})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  {crab 3407  Vcvv 3445   ∩ cin 3909  ‘cfv 6496  (class class class)co 7357  Fincfn 8883  Basecbs 17083   ↾_s cress 17112  0_gc0g 17321   Mat cmat 21754   DMatALT cdmatalt 46467

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-1cn 11109  ax-addcl 11111

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-nn 12154  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-dmatalt 46469

This theorem is referenced by:  dmatALTbasel  46473  dmatbas  46474