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Theorem pm2mpmhmlem1 20842
Description: Lemma 1 for pm2mpmhm 20844. (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 6344 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (0g𝑄) ∈ V)
2 ovexd 6824 . 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 6800 . . . . 5 (𝑙 = 𝑛 → (0...𝑙) = (0...𝑛))
4 oveq1 6799 . . . . . . 7 (𝑙 = 𝑛 → (𝑙𝑘) = (𝑛𝑘))
54oveq2d 6808 . . . . . 6 (𝑙 = 𝑛 → (𝑦 decompPMat (𝑙𝑘)) = (𝑦 decompPMat (𝑛𝑘)))
65oveq2d 6808 . . . . 5 (𝑙 = 𝑛 → ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))) = ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))
73, 6mpteq12dv 4865 . . . 4 (𝑙 = 𝑛 → (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘)))))
87oveq2d 6808 . . 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 6799 . . 3 (𝑙 = 𝑛 → (𝑙 𝑋) = (𝑛 𝑋))
108, 9oveq12d 6810 . 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 742 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑁 ∈ Fin)
12 simplr 744 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
13 pm2mpfo.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
14 pm2mpfo.c . . . . . . . . . 10 𝐶 = (𝑁 Mat 𝑃)
1513, 14pmatring 20717 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
1615anim1i 594 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝐶 ∈ Ring ∧ (𝑥𝐵𝑦𝐵)))
17 3anass 1079 . . . . . . . 8 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) ↔ (𝐶 ∈ Ring ∧ (𝑥𝐵𝑦𝐵)))
1816, 17sylibr 224 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵))
19 pm2mpfo.b . . . . . . . 8 𝐵 = (Base‘𝐶)
20 eqid 2770 . . . . . . . 8 (.r𝐶) = (.r𝐶)
2119, 20ringcl 18768 . . . . . . 7 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
2218, 21syl 17 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
23 eqid 2770 . . . . . . 7 (0g𝑅) = (0g𝑅)
2413, 14, 19, 23pmatcoe1fsupp 20725 . . . . . 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 1475 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)))
26 fvoveq1 6815 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑖 → (coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏)))
2726fveq1d 6334 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛))
2827eqeq1d 2772 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑖 → (((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) ↔ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)))
29 oveq2 6800 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑗 → (𝑖(𝑥(.r𝐶)𝑦)𝑏) = (𝑖(𝑥(.r𝐶)𝑦)𝑗))
3029fveq2d 6336 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑗 → (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗)))
3130fveq1d 6334 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑗 → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛))
3231eqeq1d 2772 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑗 → (((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) ↔ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅)))
3328, 32rspc2va 3471 . . . . . . . . . . . . . . . 16 (((𝑖𝑁𝑗𝑁) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅))
3433expcom 398 . . . . . . . . . . . . . . 15 (∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) → ((𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅)))
3534adantl 467 . . . . . . . . . . . . . 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 1107 . . . . . . . . . . . . 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))
3736mpt2eq3dva 6865 . . . . . . . . . . . 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 20441 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
4039ad3antrrr 701 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
4138matring 20465 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
42 pm2mpfo.q . . . . . . . . . . . . . . . 16 𝑄 = (Poly1𝐴)
4342ply1sca 19837 . . . . . . . . . . . . . . 15 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
4441, 43syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
4544ad3antrrr 701 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → 𝐴 = (Scalar‘𝑄))
4645fveq2d 6336 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (0g𝐴) = (0g‘(Scalar‘𝑄)))
4737, 40, 463eqtr2d 2810 . . . . . . . . . . 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 6807 . . . . . . . . . 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 19836 . . . . . . . . . . . . . . 15 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
5041, 49syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
5150adantr 466 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ LMod)
5251adantr 466 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑄 ∈ LMod)
5341adantr 466 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐴 ∈ Ring)
54 pm2mpfo.x . . . . . . . . . . . . . 14 𝑋 = (var1𝐴)
55 eqid 2770 . . . . . . . . . . . . . 14 (mulGrp‘𝑄) = (mulGrp‘𝑄)
56 pm2mpfo.e . . . . . . . . . . . . . 14 = (.g‘(mulGrp‘𝑄))
57 pm2mpfo.l . . . . . . . . . . . . . 14 𝐿 = (Base‘𝑄)
5842, 54, 55, 56, 57ply1moncl 19855 . . . . . . . . . . . . 13 ((𝐴 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ 𝐿)
5953, 58sylan 561 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ 𝐿)
60 eqid 2770 . . . . . . . . . . . . 13 (Scalar‘𝑄) = (Scalar‘𝑄)
61 pm2mpfo.m . . . . . . . . . . . . 13 = ( ·𝑠𝑄)
62 eqid 2770 . . . . . . . . . . . . 13 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
63 eqid 2770 . . . . . . . . . . . . 13 (0g𝑄) = (0g𝑄)
6457, 60, 61, 62, 63lmod0vs 19105 . . . . . . . . . . . 12 ((𝑄 ∈ LMod ∧ (𝑛 𝑋) ∈ 𝐿) → ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)) = (0g𝑄))
6552, 59, 64syl2anc 565 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)) = (0g𝑄))
6665adantr 466 . . . . . . . . . 10 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)) = (0g𝑄))
6748, 66eqtrd 2804 . . . . . . . . 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))
6867ex 397 . . . . . . . 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)))
6968imim2d 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))))
7069ralimdva 3110 . . . . . 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))))
7170reximdv 3163 . . . . 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))))
7225, 71mpd 15 . . . 4 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ((𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)) (𝑛 𝑋)) = (0g𝑄)))
7314, 19decpmatval 20789 . . . . . . . . . 10 (((𝑥(.r𝐶)𝑦) ∈ 𝐵𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)))
7422, 73sylan 561 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)))
7574oveq1d 6807 . . . . . . . 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)))
7675eqeq1d 2772 . . . . . . 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)))
7776imbi2d 329 . . . . . 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))))
7877ralbidva 3133 . . . . 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))))
7978rexbidv 3199 . . . 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))))
8072, 79mpbird 247 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄)))
81 simpllr 752 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑅 ∈ Ring)
82 simplr 744 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑥𝐵𝑦𝐵))
83 simpr 471 . . . . . . . . . 10 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈ ℕ0)
8413, 14, 19, 38decpmatmul 20796 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))))
8581, 82, 83, 84syl3anc 1475 . . . . . . . . 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))))))
8685eqcomd 2776 . . . . . . . 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))
8786oveq1d 6807 . . . . . . 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)))
8887eqeq1d 2772 . . . . . 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)))
8988imbi2d 329 . . . . 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))))
9089ralbidva 3133 . . . 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))))
9190rexbidv 3199 . . 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))))
9280, 91mpbird 247 . 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)))
931, 2, 10, 92mptnn0fsuppd 13004 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 382  w3a 1070   = wceq 1630  wcel 2144  wral 3060  wrex 3061  Vcvv 3349   class class class wbr 4784  cmpt 4861  cfv 6031  (class class class)co 6792  cmpt2 6794  Fincfn 8108   finSupp cfsupp 8430  0cc0 10137   < clt 10275  cmin 10467  0cn0 11493  ...cfz 12532  Basecbs 16063  .rcmulr 16149  Scalarcsca 16151   ·𝑠 cvsca 16152  0gc0g 16307   Σg cgsu 16308  .gcmg 17747  mulGrpcmgp 18696  Ringcrg 18754  LModclmod 19072  var1cv1 19760  Poly1cpl1 19761  coe1cco1 19762   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-8 2146  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-pow 4971  ax-pr 5034  ax-un 7095  ax-inf2 8701  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  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-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  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-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-ot 4323  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  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-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  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-isom 6040  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-ofr 7044  df-om 7212  df-1st 7314  df-2nd 7315  df-supp 7446  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-pm 8011  df-ixp 8062  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-fsupp 8431  df-sup 8503  df-oi 8570  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-dec 11695  df-uz 11888  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-ip 16166  df-tset 16167  df-ple 16168  df-ds 16171  df-hom 16173  df-cco 16174  df-0g 16309  df-gsum 16310  df-prds 16315  df-pws 16317  df-mre 16453  df-mrc 16454  df-acs 16456  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-mhm 17542  df-submnd 17543  df-grp 17632  df-minusg 17633  df-sbg 17634  df-mulg 17748  df-subg 17798  df-ghm 17865  df-cntz 17956  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-subrg 18987  df-lmod 19074  df-lss 19142  df-sra 19386  df-rgmod 19387  df-psr 19570  df-mvr 19571  df-mpl 19572  df-opsr 19574  df-psr1 19764  df-vr1 19765  df-ply1 19766  df-coe1 19767  df-dsmm 20292  df-frlm 20307  df-mamu 20406  df-mat 20430  df-decpmat 20787
This theorem is referenced by:  pm2mpmhmlem2  20843
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