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Theorem pm2mpf1 20544
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 20543 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵𝐿)
127matring 20189 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
1312adantr 481 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝐴 ∈ Ring)
141, 2, 3, 4, 5, 6, 7, 8, 9, 10pm2mpcl 20542 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑇𝑢) ∈ 𝐿)
15143expa 1262 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑢𝐵) → (𝑇𝑢) ∈ 𝐿)
1615adantrr 752 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑢) ∈ 𝐿)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10pm2mpcl 20542 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑤𝐵) → (𝑇𝑤) ∈ 𝐿)
18173expia 1264 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑤𝐵 → (𝑇𝑤) ∈ 𝐿))
1918adantld 483 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑢𝐵𝑤𝐵) → (𝑇𝑤) ∈ 𝐿))
2019imp 445 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑤) ∈ 𝐿)
21 eqid 2621 . . . . . . 7 (coe1‘(𝑇𝑢)) = (coe1‘(𝑇𝑢))
22 eqid 2621 . . . . . . 7 (coe1‘(𝑇𝑤)) = (coe1‘(𝑇𝑤))
238, 10, 21, 22ply1coe1eq 19608 . . . . . 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 1323 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → ((𝑇𝑢) = (𝑇𝑤) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)))
26 simpll 789 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑁 ∈ Fin)
27 simplr 791 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑅 ∈ Ring)
28 simprl 793 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑢𝐵)
291, 2, 3, 4, 5, 6, 7, 8, 9pm2mpfval 20541 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3026, 27, 28, 29syl3anc 1323 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3130ad2antrr 761 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3231fveq2d 6162 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝑇𝑢)) = (coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋))))))
3332fveq1d 6160 . . . . . . . . . . . . . 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 797 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
3528adantr 481 . . . . . . . . . . . . . . . 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 20539 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑛 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛) = (𝑢 decompPMat 𝑛))
3834, 36, 37syl2anc 692 . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑢))‘𝑛) = (𝑢 decompPMat 𝑛))
40 simprr 795 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑤𝐵)
411, 2, 3, 4, 5, 6, 7, 8, 9pm2mpfval 20541 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑤𝐵) → (𝑇𝑤) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))
4226, 27, 40, 41syl3anc 1323 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑤) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))
4342fveq2d 6162 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (coe1‘(𝑇𝑤)) = (coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋))))))
4443fveq1d 6160 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → ((coe1‘(𝑇𝑤))‘𝑛) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛))
4544ad2antrr 761 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑤))‘𝑛) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛))
4640adantr 481 . . . . . . . . . . . . . . . 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 20539 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑤𝐵𝑛 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛) = (𝑤 decompPMat 𝑛))
4934, 47, 48syl2anc 692 . . . . . . . . . . . . . 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 2655 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑤))‘𝑛) = (𝑤 decompPMat 𝑛))
5139, 50eqeq12d 2636 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) ↔ (𝑢 decompPMat 𝑛) = (𝑤 decompPMat 𝑛)))
522, 3decpmatval 20510 . . . . . . . . . . . . . . . . 17 ((𝑢𝐵𝑛 ∈ ℕ0) → (𝑢 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
5328, 52sylan 488 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑢 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
542, 3decpmatval 20510 . . . . . . . . . . . . . . . . 17 ((𝑤𝐵𝑛 ∈ ℕ0) → (𝑤 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
5540, 54sylan 488 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑤 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
5653, 55eqeq12d 2636 . . . . . . . . . . . . . . 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 2621 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
58 eqid 2621 . . . . . . . . . . . . . . . . 17 (Base‘𝐴) = (Base‘𝐴)
59 simplll 797 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
60 simpllr 798 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
61 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
62 simp2 1060 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
63 simp3 1061 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
643eleq2i 2690 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢𝐵𝑢 ∈ (Base‘𝐶))
6564biimpi 206 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢𝐵𝑢 ∈ (Base‘𝐶))
6665adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢𝐵𝑤𝐵) → 𝑢 ∈ (Base‘𝐶))
6766ad2antlr 762 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑢 ∈ (Base‘𝐶))
68673ad2ant1 1080 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑢 ∈ (Base‘𝐶))
6968, 3syl6eleqr 2709 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑢𝐵)
702, 61, 3, 62, 63, 69matecld 20172 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑢𝑗) ∈ (Base‘𝑃))
71 simp1r 1084 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑛 ∈ ℕ0)
72 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (coe1‘(𝑖𝑢𝑗)) = (coe1‘(𝑖𝑢𝑗))
7372, 61, 1, 57coe1fvalcl 19522 . . . . . . . . . . . . . . . . . 18 (((𝑖𝑢𝑗) ∈ (Base‘𝑃) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) ∈ (Base‘𝑅))
7470, 71, 73syl2anc 692 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) ∈ (Base‘𝑅))
757, 57, 58, 59, 60, 74matbas2d 20169 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) ∈ (Base‘𝐴))
763eleq2i 2690 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤𝐵𝑤 ∈ (Base‘𝐶))
7776biimpi 206 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤𝐵𝑤 ∈ (Base‘𝐶))
7877ad2antll 764 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑤 ∈ (Base‘𝐶))
7978adantr 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑤 ∈ (Base‘𝐶))
80793ad2ant1 1080 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑤 ∈ (Base‘𝐶))
8180, 3syl6eleqr 2709 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑤𝐵)
822, 61, 3, 62, 63, 81matecld 20172 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑤𝑗) ∈ (Base‘𝑃))
83 eqid 2621 . . . . . . . . . . . . . . . . . . 19 (coe1‘(𝑖𝑤𝑗)) = (coe1‘(𝑖𝑤𝑗))
8483, 61, 1, 57coe1fvalcl 19522 . . . . . . . . . . . . . . . . . 18 (((𝑖𝑤𝑗) ∈ (Base‘𝑃) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) ∈ (Base‘𝑅))
8582, 71, 84syl2anc 692 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) ∈ (Base‘𝑅))
867, 57, 58, 59, 60, 85matbas2d 20169 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) ∈ (Base‘𝐴))
877, 58eqmat 20170 . . . . . . . . . . . . . . . 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 692 . . . . . . . . . . . . . . 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 750 . . . . . . . . . . . . 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 6622 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦))
92 oveq1 6622 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦))
9391, 92eqeq12d 2636 . . . . . . . . . . . . . . . . . . . 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 6623 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑏 → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏))
95 oveq2 6623 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑏 → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏))
9694, 95eqeq12d 2636 . . . . . . . . . . . . . . . . . . . 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 3312 . . . . . . . . . . . . . . . . . . 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 2622 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
99 oveq12 6624 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑢𝑗) = (𝑎𝑢𝑏))
10099fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑢𝑗)) = (coe1‘(𝑎𝑢𝑏)))
101100fveq1d 6160 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 = 𝑎𝑗 = 𝑏) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) = ((coe1‘(𝑎𝑢𝑏))‘𝑛))
102101adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) ∧ (𝑖 = 𝑎𝑗 = 𝑏)) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) = ((coe1‘(𝑎𝑢𝑏))‘𝑛))
103 simplll 797 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → 𝑎𝑁)
104 simpllr 798 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → 𝑏𝑁)
105 fvexd 6170 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) ∈ V)
10698, 102, 103, 104, 105ovmpt2d 6753 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = ((coe1‘(𝑎𝑢𝑏))‘𝑛))
107 eqidd 2622 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
108 oveq12 6624 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑤𝑗) = (𝑎𝑤𝑏))
109108fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑤𝑗)) = (coe1‘(𝑎𝑤𝑏)))
110109fveq1d 6160 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑖 = 𝑎𝑗 = 𝑏) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
111110adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) ∧ (𝑖 = 𝑎𝑗 = 𝑏)) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
112 fvexd 6170 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑎𝑤𝑏))‘𝑛) ∈ V)
113107, 111, 103, 104, 112ovmpt2d 6753 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
114106, 113eqeq12d 2636 . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . 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 446 . . . . . . . . . . . . . 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 445 . . . . . . . . . . . . 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 2958 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
127126impancom 456 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → ((𝑎𝑁𝑏𝑁) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
128127imp 445 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
12927ad2antrr 761 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑅 ∈ Ring)
130 simprl 793 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑎𝑁)
131 simprr 795 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑏𝑁)
13266ad2antlr 762 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → 𝑢 ∈ (Base‘𝐶))
133132adantr 481 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑢 ∈ (Base‘𝐶))
134133, 3syl6eleqr 2709 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑢𝐵)
1352, 61, 3, 130, 131, 134matecld 20172 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑢𝑏) ∈ (Base‘𝑃))
13678ad2antrr 761 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑤 ∈ (Base‘𝐶))
137136, 3syl6eleqr 2709 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑤𝐵)
1382, 61, 3, 130, 131, 137matecld 20172 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑤𝑏) ∈ (Base‘𝑃))
139 eqid 2621 . . . . . . . . . . 11 (coe1‘(𝑎𝑢𝑏)) = (coe1‘(𝑎𝑢𝑏))
140 eqid 2621 . . . . . . . . . . 11 (coe1‘(𝑎𝑤𝑏)) = (coe1‘(𝑎𝑤𝑏))
1411, 61, 139, 140ply1coe1eq 19608 . . . . . . . . . 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 1323 . . . . . . . 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 2967 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → ∀𝑎𝑁𝑏𝑁 (𝑎𝑢𝑏) = (𝑎𝑤𝑏))
1462, 3eqmat 20170 . . . . . . 7 ((𝑢𝐵𝑤𝐵) → (𝑢 = 𝑤 ↔ ∀𝑎𝑁𝑏𝑁 (𝑎𝑢𝑏) = (𝑎𝑤𝑏)))
147146ad2antlr 762 . . . . . 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 450 . . . 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 2967 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑢𝐵𝑤𝐵 ((𝑇𝑢) = (𝑇𝑤) → 𝑢 = 𝑤))
152 dff13 6477 . 2 (𝑇:𝐵1-1𝐿 ↔ (𝑇:𝐵𝐿 ∧ ∀𝑢𝐵𝑤𝐵 ((𝑇𝑢) = (𝑇𝑤) → 𝑢 = 𝑤)))
15311, 151, 152sylanbrc 697 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  cmpt 4683  wf 5853  1-1wf1 5854  cfv 5857  (class class class)co 6615  cmpt2 6617  Fincfn 7915  0cn0 11252  Basecbs 15800   ·𝑠 cvsca 15885   Σg cgsu 16041  .gcmg 17480  mulGrpcmgp 18429  Ringcrg 18487  var1cv1 19486  Poly1cpl1 19487  coe1cco1 19488   Mat cmat 20153   decompPMat cdecpmat 20507   pMatToMatPoly cpm2mp 20537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-ot 4164  df-uni 4410  df-int 4448  df-iun 4494  df-iin 4495  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-of 6862  df-ofr 6863  df-om 7028  df-1st 7128  df-2nd 7129  df-supp 7256  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-pm 7820  df-ixp 7869  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-fsupp 8236  df-sup 8308  df-oi 8375  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-dec 11454  df-uz 11648  df-fz 12285  df-fzo 12423  df-seq 12758  df-hash 13074  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-mulr 15895  df-sca 15897  df-vsca 15898  df-ip 15899  df-tset 15900  df-ple 15901  df-ds 15904  df-hom 15906  df-cco 15907  df-0g 16042  df-gsum 16043  df-prds 16048  df-pws 16050  df-mre 16186  df-mrc 16187  df-acs 16189  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-mhm 17275  df-submnd 17276  df-grp 17365  df-minusg 17366  df-sbg 17367  df-mulg 17481  df-subg 17531  df-ghm 17598  df-cntz 17690  df-cmn 18135  df-abl 18136  df-mgp 18430  df-ur 18442  df-srg 18446  df-ring 18489  df-subrg 18718  df-lmod 18805  df-lss 18873  df-sra 19112  df-rgmod 19113  df-psr 19296  df-mvr 19297  df-mpl 19298  df-opsr 19300  df-psr1 19490  df-vr1 19491  df-ply1 19492  df-coe1 19493  df-dsmm 20016  df-frlm 20031  df-mamu 20130  df-mat 20154  df-decpmat 20508  df-pm2mp 20538
This theorem is referenced by:  pm2mpf1o  20560
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