Theorem d1mat2pmat 20544

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 8038	. . . . . 6 ⊢ {𝐴} ∈ Fin
2		eleq1 2689	. . . . . 6 ⊢ (𝑁 = {𝐴} → (𝑁 ∈ Fin ↔ {𝐴} ∈ Fin))
3	1, 2	mpbiri 248	. . . . 5 ⊢ (𝑁 = {𝐴} → 𝑁 ∈ Fin)
4	3	adantr 481	. . . 4 ⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → 𝑁 ∈ Fin)
5	4	3ad2ant2 1083	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑁 ∈ Fin)
6		simp1 1061	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑅 ∈ 𝑉)
7		simp3 1063	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → 𝑀 ∈ 𝐵)
8		d1mat2pmat.t	. . . 4 ⊢ 𝑇 = (𝑁 matToPolyMat 𝑅)
9		eqid 2622	. . . 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 20529	. . 3 ⊢ ((𝑁 ∈ Fin ∧ 𝑅 ∈ 𝑉 ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))))
14	5, 6, 7, 13	syl3anc 1326	. 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 6203	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → (𝑆‘(𝐴𝑀𝐴)) ∈ V)
17	15, 15, 16	3jca 1242	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
18	17	adantl 482	. . . . 5 ⊢ ((𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
19	18	3ad2ant2 1083	. . . 4 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V))
20		eqid 2622	. . . . 5 ⊢ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗)))
21		oveq1 6657	. . . . . 6 ⊢ (𝑖 = 𝐴 → (𝑖𝑀𝑗) = (𝐴𝑀𝑗))
22	21	fveq2d 6195	. . . . 5 ⊢ (𝑖 = 𝐴 → (𝑆‘(𝑖𝑀𝑗)) = (𝑆‘(𝐴𝑀𝑗)))
23		oveq2 6658	. . . . . 6 ⊢ (𝑗 = 𝐴 → (𝐴𝑀𝑗) = (𝐴𝑀𝐴))
24	23	fveq2d 6195	. . . . 5 ⊢ (𝑗 = 𝐴 → (𝑆‘(𝐴𝑀𝑗)) = (𝑆‘(𝐴𝑀𝐴)))
25	20, 22, 24	mpt2sn 7268	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ (𝑆‘(𝐴𝑀𝐴)) ∈ V) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
26	19, 25	syl 17	. . 3 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
27		mpt2eq12 6715	. . . . . . 7 ⊢ ((𝑁 = {𝐴} ∧ 𝑁 = {𝐴}) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))))
28	27	eqeq1d 2624	. . . . . 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))⟩}))
29	28	anidms 677	. . . . 5 ⊢ (𝑁 = {𝐴} → ((𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩} ↔ (𝑖 ∈ {𝐴}, 𝑗 ∈ {𝐴} ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩}))
30	29	adantr 481	. . . 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))⟩}))
31	30	3ad2ant2 1083	. . 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))⟩}))
32	26, 31	mpbird 247	. 2 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑖 ∈ 𝑁, 𝑗 ∈ 𝑁 ↦ (𝑆‘(𝑖𝑀𝑗))) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})
33	14, 32	eqtrd 2656	1 ⊢ ((𝑅 ∈ 𝑉 ∧ (𝑁 = {𝐴} ∧ 𝐴 ∈ 𝑉) ∧ 𝑀 ∈ 𝐵) → (𝑇‘𝑀) = {⟨⟨𝐴, 𝐴⟩, (𝑆‘(𝐴𝑀𝐴))⟩})

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200  {csn 4177  ⟨cop 4183  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Fincfn 7955  Basecbs 15857  algSccascl 19311  Poly₁cpl1 19547   Mat cmat 20213   matToPolyMat cmat2pmat 20509

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-1o 7560  df-en 7956  df-fin 7959  df-mat2pmat 20512

This theorem is referenced by: (None)