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Theorem pm2mpf1 22821
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 22820 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵𝐿)
127matring 22465 . . . . . 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 22819 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑇𝑢) ∈ 𝐿)
15143expa 1117 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝑢𝐵) → (𝑇𝑢) ∈ 𝐿)
1615adantrr 717 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑢) ∈ 𝐿)
171, 2, 3, 4, 5, 6, 7, 8, 9, 10pm2mpcl 22819 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑤𝐵) → (𝑇𝑤) ∈ 𝐿)
18173expia 1120 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑤𝐵 → (𝑇𝑤) ∈ 𝐿))
1918adantld 490 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ((𝑢𝐵𝑤𝐵) → (𝑇𝑤) ∈ 𝐿))
2019imp 406 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑤) ∈ 𝐿)
21 eqid 2735 . . . . . . 7 (coe1‘(𝑇𝑢)) = (coe1‘(𝑇𝑢))
22 eqid 2735 . . . . . . 7 (coe1‘(𝑇𝑤)) = (coe1‘(𝑇𝑤))
238, 10, 21, 22ply1coe1eq 22320 . . . . . 6 ((𝐴 ∈ Ring ∧ (𝑇𝑢) ∈ 𝐿 ∧ (𝑇𝑤) ∈ 𝐿) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) ↔ (𝑇𝑢) = (𝑇𝑤)))
2423bicomd 223 . . . . 5 ((𝐴 ∈ Ring ∧ (𝑇𝑢) ∈ 𝐿 ∧ (𝑇𝑤) ∈ 𝐿) → ((𝑇𝑢) = (𝑇𝑤) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)))
2513, 16, 20, 24syl3anc 1370 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → ((𝑇𝑢) = (𝑇𝑤) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)))
26 simpll 767 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑁 ∈ Fin)
27 simplr 769 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑅 ∈ Ring)
28 simprl 771 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑢𝐵)
291, 2, 3, 4, 5, 6, 7, 8, 9pm2mpfval 22818 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑢𝐵) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3026, 27, 28, 29syl3anc 1370 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3130ad2antrr 726 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑇𝑢) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))
3231fveq2d 6911 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (coe1‘(𝑇𝑢)) = (coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋))))))
3332fveq1d 6909 . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑅 ∈ Ring))
3528adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑢𝐵)
3635anim1i 615 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑢𝐵𝑛 ∈ ℕ0))
371, 2, 3, 4, 5, 6, 7, 8pm2mpf1lem 22816 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑛 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑢 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛) = (𝑢 decompPMat 𝑛))
3834, 36, 37syl2anc 584 . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑢))‘𝑛) = (𝑢 decompPMat 𝑛))
40 simprr 773 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑤𝐵)
411, 2, 3, 4, 5, 6, 7, 8, 9pm2mpfval 22818 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑤𝐵) → (𝑇𝑤) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))
4226, 27, 40, 41syl3anc 1370 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (𝑇𝑤) = (𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))
4342fveq2d 6911 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (coe1‘(𝑇𝑤)) = (coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋))))))
4443fveq1d 6909 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → ((coe1‘(𝑇𝑤))‘𝑛) = ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛))
4544ad2antrr 726 . . . . . . . . . . . . . 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 615 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (𝑤𝐵𝑛 ∈ ℕ0))
481, 2, 3, 4, 5, 6, 7, 8pm2mpf1lem 22816 . . . . . . . . . . . . . . 15 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑤𝐵𝑛 ∈ ℕ0)) → ((coe1‘(𝑄 Σg (𝑘 ∈ ℕ0 ↦ ((𝑤 decompPMat 𝑘) (𝑘 𝑋)))))‘𝑛) = (𝑤 decompPMat 𝑛))
4934, 47, 48syl2anc 584 . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑇𝑤))‘𝑛) = (𝑤 decompPMat 𝑛))
5139, 50eqeq12d 2751 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) ↔ (𝑢 decompPMat 𝑛) = (𝑤 decompPMat 𝑛)))
522, 3decpmatval 22787 . . . . . . . . . . . . . . . . 17 ((𝑢𝐵𝑛 ∈ ℕ0) → (𝑢 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
5328, 52sylan 580 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑢 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
542, 3decpmatval 22787 . . . . . . . . . . . . . . . . 17 ((𝑤𝐵𝑛 ∈ ℕ0) → (𝑤 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
5540, 54sylan 580 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑤 decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
5653, 55eqeq12d 2751 . . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
58 eqid 2735 . . . . . . . . . . . . . . . . 17 (Base‘𝐴) = (Base‘𝐴)
59 simplll 775 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑁 ∈ Fin)
60 simpllr 776 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
61 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (Base‘𝑃) = (Base‘𝑃)
62 simp2 1136 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑖𝑁)
63 simp3 1137 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑗𝑁)
643eleq2i 2831 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢𝐵𝑢 ∈ (Base‘𝐶))
6564biimpi 216 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢𝐵𝑢 ∈ (Base‘𝐶))
6665adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑢𝐵𝑤𝐵) → 𝑢 ∈ (Base‘𝐶))
6766ad2antlr 727 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑢 ∈ (Base‘𝐶))
68673ad2ant1 1132 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑢 ∈ (Base‘𝐶))
6968, 3eleqtrrdi 2850 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑢𝐵)
702, 61, 3, 62, 63, 69matecld 22448 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑢𝑗) ∈ (Base‘𝑃))
71 simp1r 1197 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑛 ∈ ℕ0)
72 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (coe1‘(𝑖𝑢𝑗)) = (coe1‘(𝑖𝑢𝑗))
7372, 61, 1, 57coe1fvalcl 22230 . . . . . . . . . . . . . . . . . 18 (((𝑖𝑢𝑗) ∈ (Base‘𝑃) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) ∈ (Base‘𝑅))
7470, 71, 73syl2anc 584 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑢𝑗))‘𝑛) ∈ (Base‘𝑅))
757, 57, 58, 59, 60, 74matbas2d 22445 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) ∈ (Base‘𝐴))
763eleq2i 2831 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑤𝐵𝑤 ∈ (Base‘𝐶))
7776biimpi 216 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑤𝐵𝑤 ∈ (Base‘𝐶))
7877ad2antll 729 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → 𝑤 ∈ (Base‘𝐶))
7978adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑤 ∈ (Base‘𝐶))
80793ad2ant1 1132 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑤 ∈ (Base‘𝐶))
8180, 3eleqtrrdi 2850 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → 𝑤𝐵)
822, 61, 3, 62, 63, 81matecld 22448 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → (𝑖𝑤𝑗) ∈ (Base‘𝑃))
83 eqid 2735 . . . . . . . . . . . . . . . . . . 19 (coe1‘(𝑖𝑤𝑗)) = (coe1‘(𝑖𝑤𝑗))
8483, 61, 1, 57coe1fvalcl 22230 . . . . . . . . . . . . . . . . . 18 (((𝑖𝑤𝑗) ∈ (Base‘𝑃) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) ∈ (Base‘𝑅))
8582, 71, 84syl2anc 584 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖𝑤𝑗))‘𝑛) ∈ (Base‘𝑅))
867, 57, 58, 59, 60, 85matbas2d 22445 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) ∈ (Base‘𝐴))
877, 58eqmat 22446 . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . 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 279 . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . 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 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦))
92 oveq1 7438 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑎 → (𝑥(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦))
9391, 92eqeq12d 2751 . . . . . . . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑏 → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏))
95 oveq2 7439 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑏 → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑦) = (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏))
9694, 95eqeq12d 2751 . . . . . . . . . . . . . . . . . . . 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 3634 . . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛)))
99 oveq12 7440 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑢𝑗) = (𝑎𝑢𝑏))
10099fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑢𝑗)) = (coe1‘(𝑎𝑢𝑏)))
101100fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → 𝑎𝑁)
104 simpllr 776 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → 𝑏𝑁)
105 fvexd 6922 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) ∈ V)
10698, 102, 103, 104, 105ovmpod 7585 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑢𝑗))‘𝑛))𝑏) = ((coe1‘(𝑎𝑢𝑏))‘𝑛))
107 eqidd 2736 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛)))
108 oveq12 7440 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑖 = 𝑎𝑗 = 𝑏) → (𝑖𝑤𝑗) = (𝑎𝑤𝑏))
109108fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑖 = 𝑎𝑗 = 𝑏) → (coe1‘(𝑖𝑤𝑗)) = (coe1‘(𝑎𝑤𝑏)))
110109fveq1d 6909 . . . . . . . . . . . . . . . . . . . . . . . . 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 6922 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → ((coe1‘(𝑎𝑤𝑏))‘𝑛) ∈ V)
113107, 111, 103, 104, 112ovmpod 7585 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑎𝑁𝑏𝑁) ∧ ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵))) ∧ 𝑛 ∈ ℕ0) → (𝑎(𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖𝑤𝑗))‘𝑛))𝑏) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
114106, 113eqeq12d 2751 . . . . . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . . . . . 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 419 . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . 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 240 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → ((𝑢 decompPMat 𝑛) = (𝑤 decompPMat 𝑛) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
12551, 124sylbid 240 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) ∧ 𝑛 ∈ ℕ0) → (((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) → ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
126125ralimdva 3165 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ (𝑎𝑁𝑏𝑁)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
127126impancom 451 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → ((𝑎𝑁𝑏𝑁) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
128127imp 406 . . . . . . . 8 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛))
12927ad2antrr 726 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑅 ∈ Ring)
130 simprl 771 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑎𝑁)
131 simprr 773 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑏𝑁)
13266ad2antlr 727 . . . . . . . . . . . 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, 3eleqtrrdi 2850 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑢𝐵)
1352, 61, 3, 130, 131, 134matecld 22448 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑢𝑏) ∈ (Base‘𝑃))
13678ad2antrr 726 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑤 ∈ (Base‘𝐶))
137136, 3eleqtrrdi 2850 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → 𝑤𝐵)
1382, 61, 3, 130, 131, 137matecld 22448 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑤𝑏) ∈ (Base‘𝑃))
139 eqid 2735 . . . . . . . . . . 11 (coe1‘(𝑎𝑢𝑏)) = (coe1‘(𝑎𝑢𝑏))
140 eqid 2735 . . . . . . . . . . 11 (coe1‘(𝑎𝑤𝑏)) = (coe1‘(𝑎𝑤𝑏))
1411, 61, 139, 140ply1coe1eq 22320 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (𝑎𝑢𝑏) ∈ (Base‘𝑃) ∧ (𝑎𝑤𝑏) ∈ (Base‘𝑃)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛) ↔ (𝑎𝑢𝑏) = (𝑎𝑤𝑏)))
142141bicomd 223 . . . . . . . . 9 ((𝑅 ∈ Ring ∧ (𝑎𝑢𝑏) ∈ (Base‘𝑃) ∧ (𝑎𝑤𝑏) ∈ (Base‘𝑃)) → ((𝑎𝑢𝑏) = (𝑎𝑤𝑏) ↔ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑎𝑢𝑏))‘𝑛) = ((coe1‘(𝑎𝑤𝑏))‘𝑛)))
143129, 135, 138, 142syl3anc 1370 . . . . . . . 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 257 . . . . . . 7 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) ∧ (𝑎𝑁𝑏𝑁)) → (𝑎𝑢𝑏) = (𝑎𝑤𝑏))
145144ralrimivva 3200 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → ∀𝑎𝑁𝑏𝑁 (𝑎𝑢𝑏) = (𝑎𝑤𝑏))
1462, 3eqmat 22446 . . . . . . 7 ((𝑢𝐵𝑤𝐵) → (𝑢 = 𝑤 ↔ ∀𝑎𝑁𝑏𝑁 (𝑎𝑢𝑏) = (𝑎𝑤𝑏)))
147146ad2antlr 727 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → (𝑢 = 𝑤 ↔ ∀𝑎𝑁𝑏𝑁 (𝑎𝑢𝑏) = (𝑎𝑤𝑏)))
148145, 147mpbird 257 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) ∧ ∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛)) → 𝑢 = 𝑤)
149148ex 412 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → (∀𝑛 ∈ ℕ0 ((coe1‘(𝑇𝑢))‘𝑛) = ((coe1‘(𝑇𝑤))‘𝑛) → 𝑢 = 𝑤))
15025, 149sylbid 240 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑢𝐵𝑤𝐵)) → ((𝑇𝑢) = (𝑇𝑤) → 𝑢 = 𝑤))
151150ralrimivva 3200 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → ∀𝑢𝐵𝑤𝐵 ((𝑇𝑢) = (𝑇𝑤) → 𝑢 = 𝑤))
152 dff13 7275 . 2 (𝑇:𝐵1-1𝐿 ↔ (𝑇:𝐵𝐿 ∧ ∀𝑢𝐵𝑤𝐵 ((𝑇𝑢) = (𝑇𝑤) → 𝑢 = 𝑤)))
15311, 151, 152sylanbrc 583 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑇:𝐵1-1𝐿)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cmpt 5231  wf 6559  1-1wf1 6560  cfv 6563  (class class class)co 7431  cmpo 7433  Fincfn 8984  0cn0 12524  Basecbs 17245   ·𝑠 cvsca 17302   Σg cgsu 17487  .gcmg 19098  mulGrpcmgp 20152  Ringcrg 20251  var1cv1 22193  Poly1cpl1 22194  coe1cco1 22195   Mat cmat 22427   decompPMat cdecpmat 22784   pMatToMatPoly cpm2mp 22814
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-fz 13545  df-fzo 13692  df-seq 14040  df-hash 14367  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-subrng 20563  df-subrg 20587  df-lmod 20877  df-lss 20948  df-sra 21190  df-rgmod 21191  df-dsmm 21770  df-frlm 21785  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-mamu 22411  df-mat 22428  df-decpmat 22785  df-pm2mp 22815
This theorem is referenced by:  pm2mpf1o  22837
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