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Theorem pm2mpf1 20652
 Description: The transformation of polynomial matrices into polynomials over matrices is a 1-1 function mapping polynomial matrices to polynomials over matrices. (Contributed by AV, 14-Oct-2019.) (Revised by AV, 6-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)
pm2mpcl.l 𝐿 = (Base‘𝑄)
Assertion
Ref Expression
pm2mpf1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝐿)

Proof of Theorem pm2mpf1
Dummy variables 𝑛 𝑘 𝑎 𝑏 𝑖 𝑗 𝑢 𝑤 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pm2mpval.p . . 3 𝑃 = (Poly1𝑅)
2 pm2mpval.c . . 3 𝐶 = (𝑁 Mat 𝑃)
3 pm2mpval.b . . 3 𝐵 = (Base‘𝐶)
4 pm2mpval.m . . 3 = ( ·𝑠𝑄)
5 pm2mpval.e . . 3 = (.g‘(mulGrp‘𝑄))
6 pm2mpval.x . . 3 𝑋 = (var1𝐴)
7 pm2mpval.a . . 3 𝐴 = (𝑁 Mat 𝑅)
8 pm2mpval.q . . 3 𝑄 = (Poly1𝐴)
9 pm2mpval.t . . 3 𝑇 = (𝑁 pMatToMatPoly 𝑅)
10 pm2mpcl.l . . 3 𝐿 = (Base‘𝑄)
111, 2, 3, 4, 5, 6, 7, 8, 9, 10pm2mpf 20651 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵𝐿)
127matring 20297 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
1312adantr 480 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝐴 ∈ Ring)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10pm2mpcl 20650 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑇𝑢) ∈ 𝐿)
15143expa 1284 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑢𝐵) → (𝑇𝑢) ∈ 𝐿)
1615adantrr 753 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑢) ∈ 𝐿)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10pm2mpcl 20650 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑤𝐵) → (𝑇𝑤) ∈ 𝐿)
18173expia 1286 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑤𝐵 → (𝑇𝑤) ∈ 𝐿))
1918adantld 482 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑢𝐵𝑤𝐵) → (𝑇𝑤) ∈ 𝐿))
2019imp 444 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑤) ∈ 𝐿)
21 eqid 2651 . . . . . . 7 (coe1‘(𝑇𝑢)) = (coe1‘(𝑇𝑢))
22 eqid 2651 . . . . . . 7 (coe1‘(𝑇𝑤)) = (coe1‘(𝑇𝑤))
238, 10, 21, 22ply1coe1eq 19716 . . . . . 6 ((𝐴 ∈ Ring ∧ (𝑇𝑢) ∈ 𝐿 ∧ (𝑇𝑤) ∈ 𝐿) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) ↔ (𝑇𝑢) = (𝑇𝑤)))
2423bicomd 213 . . . . 5 ((𝐴 ∈ Ring ∧ (𝑇𝑢) ∈ 𝐿 ∧ (𝑇𝑤) ∈ 𝐿) → ((𝑇𝑢) = (𝑇𝑤) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)))
2513, 16, 20, 24syl3anc 1366 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → ((𝑇𝑢) = (𝑇𝑤) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)))
26 simpll 805 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑁 ∈ Fin)
27 simplr 807 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑅 ∈ Ring)
28 simprl 809 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑢𝐵)
291, 2, 3, 4, 5, 6, 7, 8, 9pm2mpfval 20649 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3026, 27, 28, 29syl3anc 1366 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3130ad2antrr 762 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3231fveq2d 6233 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝑇𝑢)) = (coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋))))))
3332fveq1d 6231 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛))
34 simplll 813 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
3528adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑢𝐵)
3635anim1i 591 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑢𝐵𝑛 ∈ ℕ0))
371, 2, 3, 4, 5, 6, 7, 8pm2mpf1lem 20647 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑛 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛) = (𝑢 decompPMat 𝑛))
3834, 36, 37syl2anc 694 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛) = (𝑢 decompPMat 𝑛))
3933, 38eqtrd 2685 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑢))‘𝑛) = (𝑢 decompPMat 𝑛))
40 simprr 811 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑤𝐵)
411, 2, 3, 4, 5, 6, 7, 8, 9pm2mpfval 20649 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑤𝐵) → (𝑇𝑤) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))
4226, 27, 40, 41syl3anc 1366 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑤) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))
4342fveq2d 6233 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (coe1‘(𝑇𝑤)) = (coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋))))))
4443fveq1d 6231 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → ((coe1‘(𝑇𝑤))‘𝑛) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛))
4544ad2antrr 762 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑤))‘𝑛) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛))
4640adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑤𝐵)
4746anim1i 591 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑤𝐵𝑛 ∈ ℕ0))
481, 2, 3, 4, 5, 6, 7, 8pm2mpf1lem 20647 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑤𝐵𝑛 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛) = (𝑤 decompPMat 𝑛))
4934, 47, 48syl2anc 694 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛) = (𝑤 decompPMat 𝑛))
5045, 49eqtrd 2685 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑤))‘𝑛) = (𝑤 decompPMat 𝑛))
5139, 50eqeq12d 2666 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) ↔ (𝑢 decompPMat 𝑛) = (𝑤 decompPMat 𝑛)))
522, 3decpmatval 20618 . . . . . . . . . . . . . . . . 17 ((𝑢𝐵𝑛 ∈ ℕ0) → (𝑢 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
5328, 52sylan 487 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑢 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
542, 3decpmatval 20618 . . . . . . . . . . . . . . . . 17 ((𝑤𝐵𝑛 ∈ ℕ0) → (𝑤 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
5540, 54sylan 487 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑤 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
5653, 55eqeq12d 2666 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑢 decompPMat 𝑛) = (𝑤 decompPMat 𝑛) ↔ (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))))
57 eqid 2651 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
58 eqid 2651 . . . . . . . . . . . . . . . . 17 (Base‘𝐴) = (Base‘𝐴)
59 simplll 813 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
60 simpllr 815 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
61 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
62 simp2 1082 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
63 simp3 1083 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
643eleq2i 2722 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢𝐵𝑢 ∈ (Base‘𝐶))
6564biimpi 206 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢𝐵𝑢 ∈ (Base‘𝐶))
6665adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢𝐵𝑤𝐵) → 𝑢 ∈ (Base‘𝐶))
6766ad2antlr 763 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑢 ∈ (Base‘𝐶))
68673ad2ant1 1102 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑢 ∈ (Base‘𝐶))
6968, 3syl6eleqr 2741 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑢𝐵)
702, 61, 3, 62, 63, 69matecld 20280 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑢𝑗) ∈ (Base‘𝑃))
71 simp1r 1106 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑛 ∈ ℕ0)
72 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (coe1‘(𝑖𝑢𝑗)) = (coe1‘(𝑖𝑢𝑗))
7372, 61, 1, 57coe1fvalcl 19630 . . . . . . . . . . . . . . . . . 18 (((𝑖𝑢𝑗) ∈ (Base‘𝑃) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) ∈ (Base‘𝑅))
7470, 71, 73syl2anc 694 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) ∈ (Base‘𝑅))
757, 57, 58, 59, 60, 74matbas2d 20277 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) ∈ (Base‘𝐴))
763eleq2i 2722 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤𝐵𝑤 ∈ (Base‘𝐶))
7776biimpi 206 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤𝐵𝑤 ∈ (Base‘𝐶))
7877ad2antll 765 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑤 ∈ (Base‘𝐶))
7978adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑤 ∈ (Base‘𝐶))
80793ad2ant1 1102 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑤 ∈ (Base‘𝐶))
8180, 3syl6eleqr 2741 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑤𝐵)
822, 61, 3, 62, 63, 81matecld 20280 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑤𝑗) ∈ (Base‘𝑃))
83 eqid 2651 . . . . . . . . . . . . . . . . . . 19 (coe1‘(𝑖𝑤𝑗)) = (coe1‘(𝑖𝑤𝑗))
8483, 61, 1, 57coe1fvalcl 19630 . . . . . . . . . . . . . . . . . 18 (((𝑖𝑤𝑗) ∈ (Base‘𝑃) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) ∈ (Base‘𝑅))
8582, 71, 84syl2anc 694 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) ∈ (Base‘𝑅))
867, 57, 58, 59, 60, 85matbas2d 20277 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) ∈ (Base‘𝐴))
877, 58eqmat 20278 . . . . . . . . . . . . . . . 16 (((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) ∈ (Base‘𝐴) ∧ (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) ∈ (Base‘𝐴)) → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦)))
8875, 86, 87syl2anc 694 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦)))
8956, 88bitrd 268 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑢 decompPMat 𝑛) = (𝑤 decompPMat 𝑛) ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦)))
9089adantlr 751 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((𝑢 decompPMat 𝑛) = (𝑤 decompPMat 𝑛) ↔ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦)))
91 oveq1 6697 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦))
92 oveq1 6697 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦))
9391, 92eqeq12d 2666 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑎 → ((𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) ↔ (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦)))
94 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑏 → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏))
95 oveq2 6698 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑏 → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏))
9694, 95eqeq12d 2666 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑏 → ((𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) ↔ (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏)))
9793, 96rspc2va 3354 . . . . . . . . . . . . . . . . . . 19 (((𝑎𝑁𝑏𝑁) ∧ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦)) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏))
98 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
99 oveq12 6699 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑢𝑗) = (𝑎𝑢𝑏))
10099fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑢𝑗)) = (coe1‘(𝑎𝑢𝑏)))
101100fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 = 𝑎𝑗 = 𝑏) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) = ((coe1‘(𝑎𝑢𝑏))‘𝑛))
102101adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) ∧ (𝑖 = 𝑎𝑗 = 𝑏)) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) = ((coe1‘(𝑎𝑢𝑏))‘𝑛))
103 simplll 813 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → 𝑎𝑁)
104 simpllr 815 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → 𝑏𝑁)
105 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) ∈ V)
10698, 102, 103, 104, 105ovmpt2d 6830 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = ((coe1‘(𝑎𝑢𝑏))‘𝑛))
107 eqidd 2652 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
108 oveq12 6699 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑤𝑗) = (𝑎𝑤𝑏))
109108fveq2d 6233 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑤𝑗)) = (coe1‘(𝑎𝑤𝑏)))
110109fveq1d 6231 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 = 𝑎𝑗 = 𝑏) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
111110adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) ∧ (𝑖 = 𝑎𝑗 = 𝑏)) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
112 fvexd 6241 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑎𝑤𝑏))‘𝑛) ∈ V)
113107, 111, 103, 104, 112ovmpt2d 6830 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
114106, 113eqeq12d 2666 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → ((𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏) ↔ ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
115114biimpd 219 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → ((𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
116115exp31 629 . . . . . . . . . . . . . . . . . . . 20 ((𝑎𝑁𝑏𝑁) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))))
117116com14 96 . . . . . . . . . . . . . . . . . . 19 ((𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎𝑁𝑏𝑁) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))))
11897, 117syl 17 . . . . . . . . . . . . . . . . . 18 (((𝑎𝑁𝑏𝑁) ∧ ∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦)) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎𝑁𝑏𝑁) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))))
119118ex 449 . . . . . . . . . . . . . . . . 17 ((𝑎𝑁𝑏𝑁) → (∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑛 ∈ ℕ0 → ((𝑎𝑁𝑏𝑁) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))))))
120119com25 99 . . . . . . . . . . . . . . . 16 ((𝑎𝑁𝑏𝑁) → ((𝑎𝑁𝑏𝑁) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑛 ∈ ℕ0 → (∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))))))
121120pm2.43i 52 . . . . . . . . . . . . . . 15 ((𝑎𝑁𝑏𝑁) → (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑛 ∈ ℕ0 → (∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))))
122121impcom 445 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑛 ∈ ℕ0 → (∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))))
123122imp 444 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (∀𝑥𝑁𝑦𝑁 (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
12490, 123sylbid 230 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((𝑢 decompPMat 𝑛) = (𝑤 decompPMat 𝑛) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
12551, 124sylbid 230 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
126125ralimdva 2991 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
127126impancom 455 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → ((𝑎𝑁𝑏𝑁) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
128127imp 444 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
12927ad2antrr 762 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑅 ∈ Ring)
130 simprl 809 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑎𝑁)
131 simprr 811 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑏𝑁)
13266ad2antlr 763 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → 𝑢 ∈ (Base‘𝐶))
133132adantr 480 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑢 ∈ (Base‘𝐶))
134133, 3syl6eleqr 2741 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑢𝐵)
1352, 61, 3, 130, 131, 134matecld 20280 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑢𝑏) ∈ (Base‘𝑃))
13678ad2antrr 762 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑤 ∈ (Base‘𝐶))
137136, 3syl6eleqr 2741 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑤𝐵)
1382, 61, 3, 130, 131, 137matecld 20280 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑤𝑏) ∈ (Base‘𝑃))
139 eqid 2651 . . . . . . . . . . 11 (coe1‘(𝑎𝑢𝑏)) = (coe1‘(𝑎𝑢𝑏))
140 eqid 2651 . . . . . . . . . . 11 (coe1‘(𝑎𝑤𝑏)) = (coe1‘(𝑎𝑤𝑏))
1411, 61, 139, 140ply1coe1eq 19716 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (𝑎𝑢𝑏) ∈ (Base‘𝑃) ∧ (𝑎𝑤𝑏) ∈ (Base‘𝑃)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛) ↔ (𝑎𝑢𝑏) = (𝑎𝑤𝑏)))
142141bicomd 213 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑎𝑢𝑏) ∈ (Base‘𝑃) ∧ (𝑎𝑤𝑏) ∈ (Base‘𝑃)) → ((𝑎𝑢𝑏) = (𝑎𝑤𝑏) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
143129, 135, 138, 142syl3anc 1366 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → ((𝑎𝑢𝑏) = (𝑎𝑤𝑏) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
144128, 143mpbird 247 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑢𝑏) = (𝑎𝑤𝑏))
145144ralrimivva 3000 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → ∀𝑎𝑁𝑏𝑁 (𝑎𝑢𝑏) = (𝑎𝑤𝑏))
1462, 3eqmat 20278 . . . . . . 7 ((𝑢𝐵𝑤𝐵) → (𝑢 = 𝑤 ↔ ∀𝑎𝑁𝑏𝑁 (𝑎𝑢𝑏) = (𝑎𝑤𝑏)))
147146ad2antlr 763 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → (𝑢 = 𝑤 ↔ ∀𝑎𝑁𝑏𝑁 (𝑎𝑢𝑏) = (𝑎𝑤𝑏)))
148145, 147mpbird 247 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → 𝑢 = 𝑤)
149148ex 449 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) → 𝑢 = 𝑤))
15025, 149sylbid 230 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → ((𝑇𝑢) = (𝑇𝑤) → 𝑢 = 𝑤))
151150ralrimivva 3000 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑢𝐵𝑤𝐵 ((𝑇𝑢) = (𝑇𝑤) → 𝑢 = 𝑤))
152 dff13 6552 . 2 (𝑇:𝐵1-1𝐿 ↔ (𝑇:𝐵𝐿 ∧ ∀𝑢𝐵𝑤𝐵 ((𝑇𝑢) = (𝑇𝑤) → 𝑢 = 𝑤)))
15311, 151, 152sylanbrc 699 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝐿)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∀wral 2941  Vcvv 3231   ↦ cmpt 4762  ⟶wf 5922  –1-1→wf1 5923  ‘cfv 5926  (class class class)co 6690   ↦ cmpt2 6692  Fincfn 7997  ℕ0cn0 11330  Basecbs 15904   ·𝑠 cvsca 15992   Σg cgsu 16148  .gcmg 17587  mulGrpcmgp 18535  Ringcrg 18593  var1cv1 19594  Poly1cpl1 19595  coe1cco1 19596   Mat cmat 20261   decompPMat cdecpmat 20615   pMatToMatPoly cpm2mp 20645 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-srg 18552  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601  df-dsmm 20124  df-frlm 20139  df-mamu 20238  df-mat 20262  df-decpmat 20616  df-pm2mp 20646 This theorem is referenced by:  pm2mpf1o  20668
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