Theorem scmatval 22005

Description: The set of 𝑁 x 𝑁 scalar matrices 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
scmatval	⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})

Distinct variable groups:   𝐵,𝑚   𝐾,𝑐   𝑁,𝑐,𝑚   𝑅,𝑐,𝑚

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

Proof of Theorem scmatval

Dummy variables 𝑛 𝑟 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		scmatval.s	. 2 ⊢ 𝑆 = (𝑁 ScMat 𝑅)
2		df-scmat 21992	. . . 4 ⊢ ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))})
3	2	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → ScMat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))}))
4		ovexd 7443	. . . . 5 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → (𝑛 Mat 𝑟) ∈ V)
5		fveq2 6891	. . . . . . 7 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (Base‘𝑎) = (Base‘(𝑛 Mat 𝑟)))
6		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑎 = (𝑛 Mat 𝑟) → ( ·𝑠 ‘𝑎) = ( ·𝑠 ‘(𝑛 Mat 𝑟)))
7		eqidd 2733	. . . . . . . . . 10 ⊢ (𝑎 = (𝑛 Mat 𝑟) → 𝑐 = 𝑐)
8		fveq2 6891	. . . . . . . . . 10 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (1r‘𝑎) = (1r‘(𝑛 Mat 𝑟)))
9	6, 7, 8	oveq123d 7429	. . . . . . . . 9 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎)) = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))))
10	9	eqeq2d 2743	. . . . . . . 8 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎)) ↔ 𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))))
11	10	rexbidv 3178	. . . . . . 7 ⊢ (𝑎 = (𝑛 Mat 𝑟) → (∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎)) ↔ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))))
12	5, 11	rabeqbidv 3449	. . . . . 6 ⊢ (𝑎 = (𝑛 Mat 𝑟) → {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))})
13	12	adantl 482	. . . . 5 ⊢ ((((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) ∧ 𝑎 = (𝑛 Mat 𝑟)) → {𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))})
14	4, 13	csbied 3931	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))} = {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))})
15		oveq12 7417	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
16	15	fveq2d 6895	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = (Base‘(𝑁 Mat 𝑅)))
17		scmatval.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐴)
18		scmatval.a	. . . . . . . . 9 ⊢ 𝐴 = (𝑁 Mat 𝑅)
19	18	fveq2i 6894	. . . . . . . 8 ⊢ (Base‘𝐴) = (Base‘(𝑁 Mat 𝑅))
20	17, 19	eqtri 2760	. . . . . . 7 ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))
21	16, 20	eqtr4di 2790	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘(𝑛 Mat 𝑟)) = 𝐵)
22		fveq2 6891	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
23		scmatval.k	. . . . . . . . 9 ⊢ 𝐾 = (Base‘𝑅)
24	22, 23	eqtr4di 2790	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐾)
25	24	adantl 482	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐾)
26	15	fveq2d 6895	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = ( ·𝑠 ‘(𝑁 Mat 𝑅)))
27		scmatval.t	. . . . . . . . . . 11 ⊢ · = ( ·𝑠 ‘𝐴)
28	18	fveq2i 6894	. . . . . . . . . . 11 ⊢ ( ·𝑠 ‘𝐴) = ( ·𝑠 ‘(𝑁 Mat 𝑅))
29	27, 28	eqtri 2760	. . . . . . . . . 10 ⊢ · = ( ·𝑠 ‘(𝑁 Mat 𝑅))
30	26, 29	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → ( ·𝑠 ‘(𝑛 Mat 𝑟)) = · )
31		eqidd 2733	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → 𝑐 = 𝑐)
32	15	fveq2d 6895	. . . . . . . . . 10 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = (1r‘(𝑁 Mat 𝑅)))
33		scmatval.1	. . . . . . . . . . 11 ⊢ 1 = (1r‘𝐴)
34	18	fveq2i 6894	. . . . . . . . . . 11 ⊢ (1r‘𝐴) = (1r‘(𝑁 Mat 𝑅))
35	33, 34	eqtri 2760	. . . . . . . . . 10 ⊢ 1 = (1r‘(𝑁 Mat 𝑅))
36	32, 35	eqtr4di 2790	. . . . . . . . 9 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (1r‘(𝑛 Mat 𝑟)) = 1 )
37	30, 31, 36	oveq123d 7429	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) = (𝑐 · 1 ))
38	37	eqeq2d 2743	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ 𝑚 = (𝑐 · 1 )))
39	25, 38	rexeqbidv 3343	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → (∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟))) ↔ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )))
40	21, 39	rabeqbidv 3449	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑟 = 𝑅) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))} = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})
41	40	adantl 482	. . . 4 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘(𝑛 Mat 𝑟))(1r‘(𝑛 Mat 𝑟)))} = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})
42	14, 41	eqtrd 2772	. . 3 ⊢ (((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) ∧ (𝑛 = 𝑁 ∧ 𝑟 = 𝑅)) → ⦋(𝑛 Mat 𝑟) / 𝑎⦌{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠 ‘𝑎)(1r‘𝑎))} = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})
43		simpl 483	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑁 ∈ Fin)
44		elex 3492	. . . 4 ⊢ (𝑅 ∈ 𝑉 → 𝑅 ∈ V)
45	44	adantl 482	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑅 ∈ V)
46	17	fvexi 6905	. . . . 5 ⊢ 𝐵 ∈ V
47	46	rabex 5332	. . . 4 ⊢ {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )} ∈ V
48	47	a1i 11	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )} ∈ V)
49	3, 42, 43, 45, 48	ovmpod 7559	. 2 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → (𝑁 ScMat 𝑅) = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})
50	1, 49	eqtrid 2784	1 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉) → 𝑆 = {𝑚 ∈ 𝐵 ∣ ∃𝑐 ∈ 𝐾 𝑚 = (𝑐 · 1 )})

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∃wrex 3070  {crab 3432  Vcvv 3474  ⦋csb 3893  ‘cfv 6543  (class class class)co 7408   ∈ cmpo 7410  Fincfn 8938  Basecbs 17143   ·_𝑠 cvsca 17200  1_rcur 20003   Mat cmat 21906   ScMat cscmat 21990

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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-scmat 21992

This theorem is referenced by:  scmatel  22006  scmatmats  22012  scmatlss  22026