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Theorem pm2mpval 20819
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 20817 . . . 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 468 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑛 = 𝑁)
5 fveq2 6332 . . . . . . . . 9 (𝑟 = 𝑅 → (Poly1𝑟) = (Poly1𝑅))
65adantl 467 . . . . . . . 8 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1𝑟) = (Poly1𝑅))
74, 6oveq12d 6810 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat (Poly1𝑟)) = (𝑁 Mat (Poly1𝑅)))
87fveq2d 6336 . . . . . 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 6803 . . . . . . . . 9 (𝑁 Mat 𝑃) = (𝑁 Mat (Poly1𝑅))
1310, 12eqtri 2792 . . . . . . . 8 𝐶 = (𝑁 Mat (Poly1𝑅))
1413fveq2i 6335 . . . . . . 7 (Base‘𝐶) = (Base‘(𝑁 Mat (Poly1𝑅)))
159, 14eqtri 2792 . . . . . 6 𝐵 = (Base‘(𝑁 Mat (Poly1𝑅)))
168, 15syl6eqr 2822 . . . . 5 ((𝑛 = 𝑁𝑟 = 𝑅) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
1716adantl 467 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅𝑉) ∧ (𝑛 = 𝑁𝑟 = 𝑅)) → (Base‘(𝑛 Mat (Poly1𝑟))) = 𝐵)
18 ovex 6822 . . . . . 6 (𝑛 Mat 𝑟) ∈ V
19 fvexd 6344 . . . . . . 7 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) ∈ V)
20 simpr 471 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑞 = (Poly1𝑎))
21 fveq2 6332 . . . . . . . . . 10 (𝑎 = (𝑛 Mat 𝑟) → (Poly1𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
2221adantr 466 . . . . . . . . 9 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (Poly1𝑎) = (Poly1‘(𝑛 Mat 𝑟)))
2320, 22eqtrd 2804 . . . . . . . 8 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑞 = (Poly1‘(𝑛 Mat 𝑟)))
2423fveq2d 6336 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → ( ·𝑠𝑞) = ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))))
25 eqidd 2771 . . . . . . . . . 10 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
2623fveq2d 6336 . . . . . . . . . . . 12 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (mulGrp‘𝑞) = (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))
2726fveq2d 6336 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (.g‘(mulGrp‘𝑞)) = (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))))
28 eqidd 2771 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → 𝑘 = 𝑘)
29 fveq2 6332 . . . . . . . . . . . 12 (𝑎 = (𝑛 Mat 𝑟) → (var1𝑎) = (var1‘(𝑛 Mat 𝑟)))
3029adantr 466 . . . . . . . . . . 11 ((𝑎 = (𝑛 Mat 𝑟) ∧ 𝑞 = (Poly1𝑎)) → (var1𝑎) = (var1‘(𝑛 Mat 𝑟)))
3127, 28, 30oveq123d 6813 . . . . . . . . . 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 6813 . . . . . . . . 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 4877 . . . . . . . 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 6810 . . . . . . 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 3707 . . . . . 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 3706 . . . . 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 6801 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑛 Mat 𝑟) = (𝑁 Mat 𝑅))
3837fveq2d 6336 . . . . . . . 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 6335 . . . . . . . . 9 (Poly1𝐴) = (Poly1‘(𝑁 Mat 𝑅))
4239, 41eqtri 2792 . . . . . . . 8 𝑄 = (Poly1‘(𝑁 Mat 𝑅))
4338, 42syl6eqr 2822 . . . . . . 7 ((𝑛 = 𝑁𝑟 = 𝑅) → (Poly1‘(𝑛 Mat 𝑟)) = 𝑄)
4438fveq2d 6336 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅))))
45 pm2mpval.m . . . . . . . . . . 11 = ( ·𝑠𝑄)
4642fveq2i 6335 . . . . . . . . . . 11 ( ·𝑠𝑄) = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
4745, 46eqtri 2792 . . . . . . . . . 10 = ( ·𝑠 ‘(Poly1‘(𝑁 Mat 𝑅)))
4844, 47syl6eqr 2822 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → ( ·𝑠 ‘(Poly1‘(𝑛 Mat 𝑟))) = )
49 eqidd 2771 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑚 decompPMat 𝑘) = (𝑚 decompPMat 𝑘))
5038fveq2d 6336 . . . . . . . . . . . 12 ((𝑛 = 𝑁𝑟 = 𝑅) → (mulGrp‘(Poly1‘(𝑛 Mat 𝑟))) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5150fveq2d 6336 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))))
52 pm2mpval.e . . . . . . . . . . . 12 = (.g‘(mulGrp‘𝑄))
5342fveq2i 6335 . . . . . . . . . . . . 13 (mulGrp‘𝑄) = (mulGrp‘(Poly1‘(𝑁 Mat 𝑅)))
5453fveq2i 6335 . . . . . . . . . . . 12 (.g‘(mulGrp‘𝑄)) = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5552, 54eqtri 2792 . . . . . . . . . . 11 = (.g‘(mulGrp‘(Poly1‘(𝑁 Mat 𝑅))))
5651, 55syl6eqr 2822 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟)))) = )
57 eqidd 2771 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → 𝑘 = 𝑘)
5837fveq2d 6336 . . . . . . . . . . 11 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = (var1‘(𝑁 Mat 𝑅)))
59 pm2mpval.x . . . . . . . . . . . 12 𝑋 = (var1𝐴)
6040fveq2i 6335 . . . . . . . . . . . 12 (var1𝐴) = (var1‘(𝑁 Mat 𝑅))
6159, 60eqtri 2792 . . . . . . . . . . 11 𝑋 = (var1‘(𝑁 Mat 𝑅))
6258, 61syl6eqr 2822 . . . . . . . . . 10 ((𝑛 = 𝑁𝑟 = 𝑅) → (var1‘(𝑛 Mat 𝑟)) = 𝑋)
6356, 57, 62oveq123d 6813 . . . . . . . . 9 ((𝑛 = 𝑁𝑟 = 𝑅) → (𝑘(.g‘(mulGrp‘(Poly1‘(𝑛 Mat 𝑟))))(var1‘(𝑛 Mat 𝑟))) = (𝑘 𝑋))
6448, 49, 63oveq123d 6813 . . . . . . . 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 4877 . . . . . . 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 6810 . . . . . 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 467 . . . . 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 2816 . . . 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 4865 . . 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 468 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑁 ∈ Fin)
71 elex 3361 . . . 4 (𝑅𝑉𝑅 ∈ V)
7271adantl 467 . . 3 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 ∈ V)
73 fvex 6342 . . . . . 6 (Base‘𝐶) ∈ V
749, 73eqeltri 2845 . . . . 5 𝐵 ∈ V
7574mptex 6629 . . . 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 6934 . 2 ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑁 pMatToMatPoly 𝑅) = (𝑚𝐵 ↦ (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑚 decompPMat 𝑘) (𝑘 𝑋))))))
781, 77syl5eq 2816 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 382   = wceq 1630  wcel 2144  Vcvv 3349  csb 3680  cmpt 4861  cfv 6031  (class class class)co 6792  cmpt2 6794  Fincfn 8108  0cn0 11493  Basecbs 16063   ·𝑠 cvsca 16152   Σg cgsu 16308  .gcmg 17747  mulGrpcmgp 18696  var1cv1 19760  Poly1cpl1 19761   Mat cmat 20429   decompPMat cdecpmat 20786   pMatToMatPoly cpm2mp 20816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-pm2mp 20817
This theorem is referenced by:  pm2mpfval  20820  pm2mpf  20822
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