Theorem scmatel 22430

Description: An 𝑁 x 𝑁 scalar matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.)

Hypotheses

Ref	Expression
scmatval.k	⊢ 𝐾 = (Base‘𝑅)
scmatval.a	⊢ 𝐴 = (𝑁 Mat 𝑅)
scmatval.b	⊢ 𝐵 = (Base‘𝐴)
scmatval.1	⊢ 1 = (1r‘𝐴)
scmatval.t	⊢ · = ( ·𝑠 ‘𝐴)
scmatval.s	⊢ 𝑆 = (𝑁 ScMat 𝑅)

Assertion

Ref	Expression
scmatel	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 ))))

Distinct variable groups:   𝐾,𝑐   𝑁,𝑐   𝑅,𝑐   𝑀,𝑐

Allowed substitution hints:   𝐴(𝑐)   𝐵(𝑐)   𝑆(𝑐)   · (𝑐)   1 (𝑐)   𝑉(𝑐)

Proof of Theorem scmatel

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		scmatval.k	. . . 4 ⊢ 𝐾 = (Base‘𝑅)
2		scmatval.a	. . . 4 ⊢ 𝐴 = (𝑁 Mat 𝑅)
3		scmatval.b	. . . 4 ⊢ 𝐵 = (Base‘𝐴)
4		scmatval.1	. . . 4 ⊢ 1 = (1r‘𝐴)
5		scmatval.t	. . . 4 ⊢ · = ( ·𝑠 ‘𝐴)
6		scmatval.s	. . . 4 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
7	1, 2, 3, 4, 5, 6	scmatval 22429	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})
8	7	eleq2d 2819	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )}))
9		eqeq1 2737	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 )))
10	9	rexbidv 3158	. . 3 ⊢ (𝑚 = 𝑀 → (∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))
11	10	elrab 3644	. 2 ⊢ (𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )} ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))
12	8, 11	bitrdi 287	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ (𝑀 ∈ 𝐵 ∧ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 ))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  {crab 3397  ‘cfv 6489  (class class class)co 7355  Fincfn 8878  Basecbs 17130   ·_𝑠 cvsca 17175  1_rcur 20109   Mat cmat 22332   ScMat cscmat 22414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-scmat 22416

This theorem is referenced by:  scmatscmid  22431  scmatmat  22434  scmatid  22439  scmataddcl  22441  scmatsubcl  22442  scmatmulcl  22443  smatvscl  22449  scmatrhmcl  22453  mat0scmat  22463  mat1scmat  22464  chmaidscmat  22773