Theorem d1mat2pmat 22624

Description: The transformation of a matrix of dimenson 1. (Contributed by AV, 4-Aug-2019.)

Hypotheses

Ref	Expression
d1mat2pmat.t	⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
d1mat2pmat.b	⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))
d1mat2pmat.p	⊢ 𝑃 = (Poly1‘𝑅)
d1mat2pmat.s	⊢ 𝑆 = (algSc‘𝑃)

Assertion

Ref	Expression
d1mat2pmat	⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})

Proof of Theorem d1mat2pmat

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

Step	Hyp	Ref	Expression
1		snfi 8968	. . . . . 6 ⊢ {𝐴} ∈ Fin
2		eleq1 2816	. . . . . 6 ⊢ (𝑁 = {𝐴} → (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
3	1, 2	mpbiri 258	. . . . 5 ⊢ (𝑁 = {𝐴} → 𝑁 ∈ Fin)
4	3	adantr 480	. . . 4 ⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → 𝑁 ∈ Fin)
5	4	3ad2ant2 1134	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
6		simp1 1136	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ 𝑉)
7		simp3 1138	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
8		d1mat2pmat.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
9		eqid 2729	. . . 4 ⊢ (𝑁 Mat 𝑅) = (𝑁 Mat 𝑅)
10		d1mat2pmat.b	. . . 4 ⊢ 𝐵 = (Base‘(𝑁 Mat 𝑅))
11		d1mat2pmat.p	. . . 4 ⊢ 𝑃 = (Poly1‘𝑅)
12		d1mat2pmat.s	. . . 4 ⊢ 𝑆 = (algSc‘𝑃)
13	8, 9, 10, 11, 12	mat2pmatval 22609	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
14	5, 6, 7, 13	syl3anc 1373	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
15		id 22	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
16		fvexd 6837	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V)
17	15, 15, 16	3jca 1128	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
18	17	adantl 481	. . . . 5 ⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
19	18	3ad2ant2 1134	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
20		eqid 2729	. . . . 5 ⊢ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))
21		fvoveq1 7372	. . . . 5 ⊢ (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗)))
22		oveq2 7357	. . . . . 6 ⊢ (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
23	22	fveq2d 6826	. . . . 5 ⊢ (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴)))
24	20, 21, 23	mposn 8036	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
25	19, 24	syl 17	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
26		mpoeq12 7422	. . . . . . 7 ⊢ ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))))
27	26	eqeq1d 2731	. . . . . 6 ⊢ ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
28	27	anidms 566	. . . . 5 ⊢ (𝑁 = {𝐴} → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
29	28	adantr 480	. . . 4 ⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
30	29	3ad2ant2 1134	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
31	25, 30	mpbird 257	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
32	14, 31	eqtrd 2764	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3436  {csn 4577  ⟨cop 4583  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351  Fincfn 8872  Basecbs 17120  algSccascl 21759  Poly₁cpl1 22059   Mat cmat 22292   matToPolyMat cmat2pmat 22589

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-tr 5200  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-ord 6310  df-on 6311  df-lim 6312  df-suc 6313  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-om 7800  df-1st 7924  df-2nd 7925  df-1o 8388  df-en 8873  df-fin 8876  df-mat2pmat 22592

This theorem is referenced by: (None)