Theorem scmatel 22527

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 22526	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})
8	7	eleq2d 2825	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑀 ∈ 𝑆 ↔ 𝑀 ∈ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )}))
9		eqeq1 2739	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑚 = (𝑐 · 1 ) ↔ 𝑀 = (𝑐 · 1 )))
10	9	rexbidv 3177	. . 3 ⊢ (𝑚 = 𝑀 → (∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 ) ↔ ∃𝑐 ∈ 𝐾 𝑀 = (𝑐 · 1 )))
11	10	elrab 3695	. 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 1537   ∈ wcel 2106  ∃wrex 3068  {crab 3433  ‘cfv 6563  (class class class)co 7431  Fincfn 8984  Basecbs 17245   ·_𝑠 cvsca 17302  1_rcur 20199   Mat cmat 22427   ScMat cscmat 22511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-scmat 22513

This theorem is referenced by:  scmatscmid  22528  scmatmat  22531  scmatid  22536  scmataddcl  22538  scmatsubcl  22539  scmatmulcl  22540  smatvscl  22546  scmatrhmcl  22550  mat0scmat  22560  mat1scmat  22561  chmaidscmat  22870