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Theorem pm2mpmhmlem1 22839
Description: Lemma 1 for pm2mpmhm 22841. (Contributed by AV, 21-Oct-2019.) (Revised by AV, 6-Dec-2019.)
Hypotheses
Ref Expression
pm2mpfo.p 𝑃 = (Poly1𝑅)
pm2mpfo.c 𝐶 = (𝑁 Mat 𝑃)
pm2mpfo.b 𝐵 = (Base‘𝐶)
pm2mpfo.m = ( ·𝑠𝑄)
pm2mpfo.e = (.g‘(mulGrp‘𝑄))
pm2mpfo.x 𝑋 = (var1𝐴)
pm2mpfo.a 𝐴 = (𝑁 Mat 𝑅)
pm2mpfo.q 𝑄 = (Poly1𝐴)
pm2mpfo.l 𝐿 = (Base‘𝑄)
pm2mpfo.t 𝑇 = (𝑁 pMatToMatPoly 𝑅)
Assertion
Ref Expression
pm2mpmhmlem1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))))) (𝑙 𝑋))) finSupp (0g𝑄))
Distinct variable groups:   𝐴,𝑘   𝐵,𝑘,𝑙   𝐶,𝑘,𝑙   𝑘,𝐿   𝑘,𝑁,𝑙   𝑄,𝑘   𝑅,𝑘   ,𝑘   𝐴,𝑙   𝑃,𝑘   𝑅,𝑙   𝑋,𝑙   ,𝑙   ,𝑙   𝑥,𝑦,𝑘,𝑙
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝑃(𝑥,𝑦,𝑙)   𝑄(𝑥,𝑦,𝑙)   𝑅(𝑥,𝑦)   𝑇(𝑥,𝑦,𝑘,𝑙)   (𝑥,𝑦,𝑘)   (𝑥,𝑦)   𝐿(𝑥,𝑦,𝑙)   𝑁(𝑥,𝑦)   𝑋(𝑥,𝑦,𝑘)

Proof of Theorem pm2mpmhmlem1
Dummy variables 𝑎 𝑏 𝑖 𝑗 𝑛 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fvexd 6921 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (0g𝑄) ∈ V)
2 ovexd 7465 . 2 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑙 ∈ ℕ0) → ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))))) (𝑙 𝑋)) ∈ V)
3 oveq2 7438 . . . . 5 (𝑙 = 𝑛 → (0...𝑙) = (0...𝑛))
4 oveq1 7437 . . . . . . 7 (𝑙 = 𝑛 → (𝑙𝑘) = (𝑛𝑘))
54oveq2d 7446 . . . . . 6 (𝑙 = 𝑛 → (𝑦 decompPMat (𝑙𝑘)) = (𝑦 decompPMat (𝑛𝑘)))
65oveq2d 7446 . . . . 5 (𝑙 = 𝑛 → ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))) = ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))
73, 6mpteq12dv 5238 . . . 4 (𝑙 = 𝑛 → (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘)))))
87oveq2d 7446 . . 3 (𝑙 = 𝑛 → (𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))))) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))))
9 oveq1 7437 . . 3 (𝑙 = 𝑛 → (𝑙 𝑋) = (𝑛 𝑋))
108, 9oveq12d 7448 . 2 (𝑙 = 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))))) (𝑙 𝑋)) = ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))) (𝑛 𝑋)))
11 simpll 767 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑁 ∈ Fin)
12 simplr 769 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
13 pm2mpfo.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
14 pm2mpfo.c . . . . . . . . . 10 𝐶 = (𝑁 Mat 𝑃)
1513, 14pmatring 22713 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
1615anim1i 615 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝐶 ∈ Ring ∧ (𝑥𝐵𝑦𝐵)))
17 3anass 1094 . . . . . . . 8 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) ↔ (𝐶 ∈ Ring ∧ (𝑥𝐵𝑦𝐵)))
1816, 17sylibr 234 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵))
19 pm2mpfo.b . . . . . . . 8 𝐵 = (Base‘𝐶)
20 eqid 2734 . . . . . . . 8 (.r𝐶) = (.r𝐶)
2119, 20ringcl 20267 . . . . . . 7 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
2218, 21syl 17 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
23 eqid 2734 . . . . . . 7 (0g𝑅) = (0g𝑅)
2413, 14, 19, 23pmatcoe1fsupp 22722 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ (𝑥(.r𝐶)𝑦) ∈ 𝐵) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)))
2511, 12, 22, 24syl3anc 1370 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)))
26 fvoveq1 7453 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑖 → (coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏)))
2726fveq1d 6908 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛))
2827eqeq1d 2736 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑖 → (((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) ↔ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)))
29 oveq2 7438 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑗 → (𝑖(𝑥(.r𝐶)𝑦)𝑏) = (𝑖(𝑥(.r𝐶)𝑦)𝑗))
3029fveq2d 6910 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑗 → (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗)))
3130fveq1d 6908 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑗 → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛))
3231eqeq1d 2736 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑗 → (((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) ↔ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅)))
3328, 32rspc2va 3633 . . . . . . . . . . . . . . . 16 (((𝑖𝑁𝑗𝑁) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅))
3433expcom 413 . . . . . . . . . . . . . . 15 (∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) → ((𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅)))
3534adantl 481 . . . . . . . . . . . . . 14 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ((𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅)))
36353impib 1115 . . . . . . . . . . . . 13 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) ∧ 𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅))
3736mpoeq3dva 7509 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
38 pm2mpfo.a . . . . . . . . . . . . . 14 𝐴 = (𝑁 Mat 𝑅)
3938, 23mat0op 22440 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
4039ad3antrrr 730 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
4138matring 22464 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
42 pm2mpfo.q . . . . . . . . . . . . . . . 16 𝑄 = (Poly1𝐴)
4342ply1sca 22269 . . . . . . . . . . . . . . 15 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
4441, 43syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
4544ad3antrrr 730 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → 𝐴 = (Scalar‘𝑄))
4645fveq2d 6910 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (0g𝐴) = (0g‘(Scalar‘𝑄)))
4737, 40, 463eqtr2d 2780 . . . . . . . . . . 11 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) = (0g‘(Scalar‘𝑄)))
4847oveq1d 7445 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)))
4942ply1lmod 22268 . . . . . . . . . . . . . 14 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
5041, 49syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
5150adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ LMod)
5241adantr 480 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐴 ∈ Ring)
53 pm2mpfo.x . . . . . . . . . . . . . 14 𝑋 = (var1𝐴)
54 eqid 2734 . . . . . . . . . . . . . 14 (mulGrp‘𝑄) = (mulGrp‘𝑄)
55 pm2mpfo.e . . . . . . . . . . . . . 14 = (.g‘(mulGrp‘𝑄))
56 pm2mpfo.l . . . . . . . . . . . . . 14 𝐿 = (Base‘𝑄)
5742, 53, 54, 55, 56ply1moncl 22289 . . . . . . . . . . . . 13 ((𝐴 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ 𝐿)
5852, 57sylan 580 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ 𝐿)
59 eqid 2734 . . . . . . . . . . . . 13 (Scalar‘𝑄) = (Scalar‘𝑄)
60 pm2mpfo.m . . . . . . . . . . . . 13 = ( ·𝑠𝑄)
61 eqid 2734 . . . . . . . . . . . . 13 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
62 eqid 2734 . . . . . . . . . . . . 13 (0g𝑄) = (0g𝑄)
6356, 59, 60, 61, 62lmod0vs 20909 . . . . . . . . . . . 12 ((𝑄 ∈ LMod ∧ (𝑛 𝑋) ∈ 𝐿) → ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)) = (0g𝑄))
6451, 58, 63syl2an2r 685 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)) = (0g𝑄))
6564adantr 480 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)) = (0g𝑄))
6648, 65eqtrd 2774 . . . . . . . . 9 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄))
6766ex 412 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄)))
6867imim2d 57 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (𝑠 < 𝑛 → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄))))
6968ralimdva 3164 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄))))
7069reximdv 3167 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄))))
7125, 70mpd 15 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄)))
7214, 19decpmatval 22786 . . . . . . . . . 10 (((𝑥(.r𝐶)𝑦) ∈ 𝐵𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)))
7322, 72sylan 580 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)))
7473oveq1d 7445 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)))
7574eqeq1d 2736 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄) ↔ ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄)))
7675imbi2d 340 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄)) ↔ (𝑠 < 𝑛 → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄))))
7776ralbidva 3173 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄)) ↔ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄))))
7877rexbidv 3176 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄)) ↔ ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄))))
7971, 78mpbird 257 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄)))
8013, 14, 19, 38decpmatmul 22793 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))))
8180ad4ant234 1174 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))))
8281eqcomd 2740 . . . . . . . 8 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))) = ((𝑥(.r𝐶)𝑦) decompPMat 𝑛))
8382oveq1d 7445 . . . . . . 7 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))) (𝑛 𝑋)) = (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)))
8483eqeq1d 2736 . . . . . 6 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))) (𝑛 𝑋)) = (0g𝑄) ↔ (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄)))
8584imbi2d 340 . . . . 5 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))) (𝑛 𝑋)) = (0g𝑄)) ↔ (𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄))))
8685ralbidva 3173 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))) (𝑛 𝑋)) = (0g𝑄)) ↔ ∀𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄))))
8786rexbidv 3176 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))) (𝑛 𝑋)) = (0g𝑄)) ↔ ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄))))
8879, 87mpbird 257 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))) (𝑛 𝑋)) = (0g𝑄)))
891, 2, 10, 88mptnn0fsuppd 14035 1 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑙 ∈ ℕ0 ↦ ((𝐴 Σg (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))))) (𝑙 𝑋))) finSupp (0g𝑄))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  Vcvv 3477   class class class wbr 5147  cmpt 5230  cfv 6562  (class class class)co 7430  cmpo 7432  Fincfn 8983   finSupp cfsupp 9398  0cc0 11152   < clt 11292  cmin 11489  0cn0 12523  ...cfz 13543  Basecbs 17244  .rcmulr 17298  Scalarcsca 17300   ·𝑠 cvsca 17301  0gc0g 17485   Σg cgsu 17486  .gcmg 19097  mulGrpcmgp 20151  Ringcrg 20250  LModclmod 20874  var1cv1 22192  Poly1cpl1 22193  coe1cco1 22194   Mat cmat 22426   decompPMat cdecpmat 22783   pMatToMatPoly cpm2mp 22813
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-ot 4639  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-se 5641  df-we 5642  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-isom 6571  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-of 7696  df-ofr 7697  df-om 7887  df-1st 8012  df-2nd 8013  df-supp 8184  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-er 8743  df-map 8866  df-pm 8867  df-ixp 8936  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-fsupp 9399  df-sup 9479  df-oi 9547  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-3 12327  df-4 12328  df-5 12329  df-6 12330  df-7 12331  df-8 12332  df-9 12333  df-n0 12524  df-z 12611  df-dec 12731  df-uz 12876  df-fz 13544  df-fzo 13691  df-seq 14039  df-hash 14366  df-struct 17180  df-sets 17197  df-slot 17215  df-ndx 17227  df-base 17245  df-ress 17274  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18665  df-sgrp 18744  df-mnd 18760  df-mhm 18808  df-submnd 18809  df-grp 18966  df-minusg 18967  df-sbg 18968  df-mulg 19098  df-subg 19153  df-ghm 19243  df-cntz 19347  df-cmn 19814  df-abl 19815  df-mgp 20152  df-rng 20170  df-ur 20199  df-ring 20252  df-subrng 20562  df-subrg 20586  df-lmod 20876  df-lss 20947  df-sra 21189  df-rgmod 21190  df-dsmm 21769  df-frlm 21784  df-psr 21946  df-mvr 21947  df-mpl 21948  df-opsr 21950  df-psr1 22196  df-vr1 22197  df-ply1 22198  df-coe1 22199  df-mamu 22410  df-mat 22427  df-decpmat 22784
This theorem is referenced by:  pm2mpmhmlem2  22840
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