Theorem d1mat2pmat 22761

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 9082	. . . . . 6 ⊢ {𝐴} ∈ Fin
2		eleq1 2827	. . . . . 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 1133	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
6		simp1 1135	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ 𝑉)
7		simp3 1137	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
8		d1mat2pmat.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
9		eqid 2735	. . . 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 22746	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
14	5, 6, 7, 13	syl3anc 1370	. 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 6922	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V)
17	15, 15, 16	3jca 1127	. . . . . 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 1133	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
20		eqid 2735	. . . . 5 ⊢ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))
21		fvoveq1 7454	. . . . 5 ⊢ (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗)))
22		oveq2 7439	. . . . . 6 ⊢ (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
23	22	fveq2d 6911	. . . . 5 ⊢ (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴)))
24	20, 21, 23	mposn 8127	. . . 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 7506	. . . . . . 7 ⊢ ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))))
27	26	eqeq1d 2737	. . . . . 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 1133	. . 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 2775	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 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631  ⟨cop 4637  ‘cfv 6563  (class class class)co 7431   ∈ cmpo 7433  Fincfn 8984  Basecbs 17245  algSccascl 21890  Poly₁cpl1 22194   Mat cmat 22427   matToPolyMat cmat2pmat 22726

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-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-reu 3379  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-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-en 8985  df-fin 8988  df-mat2pmat 22729

This theorem is referenced by: (None)