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Theorem pm2mpval 20798
Description: Value of the transformation of a polynomial matrix into a polynomial over matrices. (Contributed by AV, 5-Dec-2019.)
Hypotheses
Ref Expression
pm2mpval.p 𝑃 = (Poly1𝑅)
pm2mpval.c 𝐶 = (𝑁 Mat 𝑃)
pm2mpval.b 𝐵 = (Base‘𝐶)
pm2mpval.m = ( ·𝑠𝑄)
pm2mpval.e = (.g‘(mulGrp‘𝑄))
pm2mpval.x 𝑋 = (var1𝐴)
pm2mpval.a 𝐴 = (𝑁 Mat 𝑅)
pm2mpval.q 𝑄 = (Poly1𝐴)
pm2mpval.t 𝑇 = (𝑁 pMatToMatPoly 𝑅)
Assertion
Ref Expression
pm2mpval ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
Distinct variable groups:   𝐵,𝑚   𝑘,𝑁,𝑚   𝑅,𝑘,𝑚   𝑚,𝑉
Allowed substitution hints:   𝐴(𝑘,𝑚)   𝐵(𝑘)   𝐶(𝑘,𝑚)   𝑃(𝑘,𝑚)   𝑄(𝑘,𝑚)   𝑇(𝑘,𝑚)   (𝑘,𝑚)   (𝑘,𝑚)   𝑉(𝑘)   𝑋(𝑘,𝑚)

Proof of Theorem pm2mpval
Dummy variables 𝑛 𝑟 𝑎 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm2mpval.t . 2 𝑇 = (𝑁 pMatToMatPoly 𝑅)
2 df-pm2mp 20796 . . . 4 pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))))))
32a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → pMatToMatPoly = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))))))
4 simpl 474 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
5 fveq2 6348 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65adantl 473 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
74, 6oveq12d 6827 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat (Poly1𝑅)))
87fveq2d 6352 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = (Base‘(𝑁 Mat (Poly1𝑅))))
9 pm2mpval.b . . . . . . 7 𝐵 = (Base‘𝐶)
10 pm2mpval.c . . . . . . . . 9 𝐶 = (𝑁 Mat 𝑃)
11 pm2mpval.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
1211oveq2i 6820 . . . . . . . . 9 (𝑁 Mat 𝑃) = (𝑁 Mat (Poly1𝑅))
1310, 12eqtri 2778 . . . . . . . 8 𝐶 = (𝑁 Mat (Poly1𝑅))
1413fveq2i 6351 . . . . . . 7 (Base‘𝐶) = (Base‘(𝑁 Mat (Poly1𝑅)))
159, 14eqtri 2778 . . . . . 6 𝐵 = (Base‘(𝑁 Mat (Poly1𝑅)))
168, 15syl6eqr 2808 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
1716adantl 473 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
18 ovex 6837 . . . . . 6 (𝑛 Mat 𝑟) ∈ V
19 fvexd 6360 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) ∈ V)
20 simpr 479 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑞 = (Poly1𝑎))
21 fveq2 6348 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
2221adantr 472 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (Poly1𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
2320, 22eqtrd 2790 . . . . . . . 8 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑞 = (Poly1‘(𝑛 Mat 𝑟)))
2423fveq2d 6352 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → ( ·𝑠𝑞) = ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))))
25 eqidd 2757 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
2623fveq2d 6352 . . . . . . . . . . . 12 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (mulGrp‘𝑞) = (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))
2726fveq2d 6352 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (.g‘(mulGrp‘𝑞)) = (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))))
28 eqidd 2757 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑘 = 𝑘)
29 fveq2 6348 . . . . . . . . . . . 12 (𝑎 = (𝑛 Mat 𝑟) → (var1𝑎) = (var1‘(𝑛 Mat 𝑟)))
3029adantr 472 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (var1𝑎) = (var1‘(𝑛 Mat 𝑟)))
3127, 28, 30oveq123d 6830 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)) = (𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))
3224, 25, 31oveq123d 6830 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))) = ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))
3332mpteq2dv 4893 . . . . . . . 8 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))) = (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))))
3423, 33oveq12d 6827 . . . . . . 7 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))))
3519, 34csbied 3697 . . . . . 6 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))))
3618, 35csbie 3696 . . . . 5 (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))) = ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))))
37 oveq12 6818 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3837fveq2d 6352 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1‘(𝑛 Mat 𝑟)) = (Poly1‘(𝑁 Mat 𝑅)))
39 pm2mpval.q . . . . . . . . 9 𝑄 = (Poly1𝐴)
40 pm2mpval.a . . . . . . . . . 10 𝐴 = (𝑁 Mat 𝑅)
4140fveq2i 6351 . . . . . . . . 9 (Poly1𝐴) = (Poly1‘(𝑁 Mat 𝑅))
4239, 41eqtri 2778 . . . . . . . 8 𝑄 = (Poly1‘(𝑁 Mat 𝑅))
4338, 42syl6eqr 2808 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1‘(𝑛 Mat 𝑟)) = 𝑄)
4438fveq2d 6352 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅))))
45 pm2mpval.m . . . . . . . . . . 11 = ( ·𝑠𝑄)
4642fveq2i 6351 . . . . . . . . . . 11 ( ·𝑠𝑄) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
4745, 46eqtri 2778 . . . . . . . . . 10 = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
4844, 47syl6eqr 2808 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = )
49 eqidd 2757 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
5038fveq2d 6352 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5150fveq2d 6352 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))))
52 pm2mpval.e . . . . . . . . . . . 12 = (.g‘(mulGrp‘𝑄))
5342fveq2i 6351 . . . . . . . . . . . . 13 (mulGrp‘𝑄) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))
5453fveq2i 6351 . . . . . . . . . . . 12 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5552, 54eqtri 2778 . . . . . . . . . . 11 = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5651, 55syl6eqr 2808 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = )
57 eqidd 2757 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑘 = 𝑘)
5837fveq2d 6352 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = (var1‘(𝑁 Mat 𝑅)))
59 pm2mpval.x . . . . . . . . . . . 12 𝑋 = (var1𝐴)
6040fveq2i 6351 . . . . . . . . . . . 12 (var1𝐴) = (var1‘(𝑁 Mat 𝑅))
6159, 60eqtri 2778 . . . . . . . . . . 11 𝑋 = (var1‘(𝑁 Mat 𝑅))
6258, 61syl6eqr 2808 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = 𝑋)
6356, 57, 62oveq123d 6830 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))) = (𝑘 𝑋))
6448, 49, 63oveq123d 6830 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))) = ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))
6564mpteq2dv 4893 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))))) = (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))
6643, 65oveq12d 6827 . . . . . 6 ((𝑛 = 𝑁𝑟 = 𝑅) → ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))))
6766adantl 473 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → ((Poly1‘(𝑛 Mat 𝑟)) Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟)))(𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟)))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))))
6836, 67syl5eq 2802 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎))))) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋)))))
6917, 68mpteq12dv 4881 . . 3 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (𝑚 ∈ (Base‘(𝑛 Mat (Poly1𝑟))) ↦ (𝑛 Mat 𝑟) / 𝑎(Poly1𝑎) / 𝑞(𝑞 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘)( ·𝑠𝑞)(𝑘(.g‘(mulGrp‘𝑞))(var1𝑎)))))) = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
70 simpl 474 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
71 elex 3348 . . . 4 (𝑅𝑉𝑅 ∈ V)
7271adantl 473 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
73 fvex 6358 . . . . . 6 (Base‘𝐶) ∈ V
749, 73eqeltri 2831 . . . . 5 𝐵 ∈ V
7574mptex 6646 . . . 4 (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))) ∈ V
7675a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))) ∈ V)
773, 69, 70, 72, 76ovmpt2d 6949 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 pMatToMatPoly 𝑅) = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
781, 77syl5eq 2802 1 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑇 = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1628  wcel 2135  Vcvv 3336  csb 3670  cmpt 4877  cfv 6045  (class class class)co 6809  cmpt2 6811  Fincfn 8117  0cn0 11480  Basecbs 16055   ·𝑠 cvsca 16143   Σg cgsu 16299  .gcmg 17737  mulGrpcmgp 18685  var1cv1 19744  Poly1cpl1 19745   Mat cmat 20411   decompPMat cdecpmat 20765   pMatToMatPoly cpm2mp 20795
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pr 5051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-pm2mp 20796
This theorem is referenced by:  pm2mpfval  20799  pm2mpf  20801
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