Theorem d1mat2pmat 23112

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 9109	. . . . . 6 ⊢ {𝐴} ∈ Fin
2		eleq1 2852	. . . . . 6 ⊢ (𝑁 = {𝐴} → (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
3	1, 2	mpbiri 261	. . . . 5 ⊢ (𝑁 = {𝐴} → 𝑁 ∈ Fin)
4	3	adantr 486	. . . 4 ⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → 𝑁 ∈ Fin)
5	4	3ad2ant2 1152	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
6		simp1 1154	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ 𝑉)
7		simp3 1156	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
8		d1mat2pmat.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
9		eqid 2763	. . . 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 23097	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
14	5, 6, 7, 13	syl3anc 1399	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
15		id 23	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝑉)
16		fvexd 6931	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V)
17	15, 15, 16	3jca 1146	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
18	17	adantl 487	. . . . 5 ⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
19	18	3ad2ant2 1152	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
20		eqid 2763	. . . . 5 ⊢ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))
21		fvoveq1 7474	. . . . 5 ⊢ (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗)))
22		oveq2 7459	. . . . . 6 ⊢ (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
23	22	fveq2d 6920	. . . . 5 ⊢ (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴)))
24	20, 21, 23	mposn 8151	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
25	19, 24	syl 18	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
26		mpoeq12 7524	. . . . . . 7 ⊢ ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))))
27	26	eqeq1d 2765	. . . . . 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 577	. . . . 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 486	. . . 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 1152	. . 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 260	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
32	14, 31	eqtrd 2798	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588  ⟨cop 4840  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453  Fincfn 9011  Basecbs 17431  algSccascl 22215  Poly₁cpl1 22550   Mat cmat 22777   matToPolyMat cmat2pmat 23077

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-1o 8514  df-en 9012  df-fin 9015  df-mat2pmat 23080

This theorem is used by: (None)