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Theorem pm2mpmhmlem1 21531
 Description: Lemma 1 for pm2mpmhm 21533. (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 6678 . 2 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (0g𝑄) ∈ V)
2 ovexd 7191 . 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 7164 . . . . 5 (𝑙 = 𝑛 → (0...𝑙) = (0...𝑛))
4 oveq1 7163 . . . . . . 7 (𝑙 = 𝑛 → (𝑙𝑘) = (𝑛𝑘))
54oveq2d 7172 . . . . . 6 (𝑙 = 𝑛 → (𝑦 decompPMat (𝑙𝑘)) = (𝑦 decompPMat (𝑛𝑘)))
65oveq2d 7172 . . . . 5 (𝑙 = 𝑛 → ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘))) = ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))
73, 6mpteq12dv 5121 . . . 4 (𝑙 = 𝑛 → (𝑘 ∈ (0...𝑙) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑙𝑘)))) = (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘)))))
87oveq2d 7172 . . 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 7163 . . 3 (𝑙 = 𝑛 → (𝑙 𝑋) = (𝑛 𝑋))
108, 9oveq12d 7174 . 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 766 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑁 ∈ Fin)
12 simplr 768 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑅 ∈ Ring)
13 pm2mpfo.p . . . . . . . . . 10 𝑃 = (Poly1𝑅)
14 pm2mpfo.c . . . . . . . . . 10 𝐶 = (𝑁 Mat 𝑃)
1513, 14pmatring 21405 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐶 ∈ Ring)
1615anim1i 617 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝐶 ∈ Ring ∧ (𝑥𝐵𝑦𝐵)))
17 3anass 1092 . . . . . . . 8 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) ↔ (𝐶 ∈ Ring ∧ (𝑥𝐵𝑦𝐵)))
1816, 17sylibr 237 . . . . . . 7 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵))
19 pm2mpfo.b . . . . . . . 8 𝐵 = (Base‘𝐶)
20 eqid 2758 . . . . . . . 8 (.r𝐶) = (.r𝐶)
2119, 20ringcl 19395 . . . . . . 7 ((𝐶 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
2218, 21syl 17 . . . . . 6 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(.r𝐶)𝑦) ∈ 𝐵)
23 eqid 2758 . . . . . . 7 (0g𝑅) = (0g𝑅)
2413, 14, 19, 23pmatcoe1fsupp 21414 . . . . . 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 1368 . . . . 5 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)))
26 fvoveq1 7179 . . . . . . . . . . . . . . . . . . 19 (𝑎 = 𝑖 → (coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏)))
2726fveq1d 6665 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑖 → ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛))
2827eqeq1d 2760 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑖 → (((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) ↔ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)))
29 oveq2 7164 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = 𝑗 → (𝑖(𝑥(.r𝐶)𝑦)𝑏) = (𝑖(𝑥(.r𝐶)𝑦)𝑗))
3029fveq2d 6667 . . . . . . . . . . . . . . . . . . 19 (𝑏 = 𝑗 → (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏)) = (coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗)))
3130fveq1d 6665 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑗 → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛))
3231eqeq1d 2760 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑗 → (((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) ↔ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅)))
3328, 32rspc2va 3554 . . . . . . . . . . . . . . . 16 (((𝑖𝑁𝑗𝑁) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅))
3433expcom 417 . . . . . . . . . . . . . . 15 (∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅) → ((𝑖𝑁𝑗𝑁) → ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛) = (0g𝑅)))
3534adantl 485 . . . . . . . . . . . . . 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 1113 . . . . . . . . . . . . 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 7231 . . . . . . . . . . . 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 21132 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
4039ad3antrrr 729 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑅)))
4138matring 21156 . . . . . . . . . . . . . . 15 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)
42 pm2mpfo.q . . . . . . . . . . . . . . . 16 𝑄 = (Poly1𝐴)
4342ply1sca 20990 . . . . . . . . . . . . . . 15 (𝐴 ∈ Ring → 𝐴 = (Scalar‘𝑄))
4441, 43syl 17 . . . . . . . . . . . . . 14 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 = (Scalar‘𝑄))
4544ad3antrrr 729 . . . . . . . . . . . . 13 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → 𝐴 = (Scalar‘𝑄))
4645fveq2d 6667 . . . . . . . . . . . 12 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) ∧ ∀𝑎𝑁𝑏𝑁 ((coe1‘(𝑎(𝑥(.r𝐶)𝑦)𝑏))‘𝑛) = (0g𝑅)) → (0g𝐴) = (0g‘(Scalar‘𝑄)))
4737, 40, 463eqtr2d 2799 . . . . . . . . . . 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 7171 . . . . . . . . . 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 20989 . . . . . . . . . . . . . 14 (𝐴 ∈ Ring → 𝑄 ∈ LMod)
5041, 49syl 17 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑄 ∈ LMod)
5150adantr 484 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝑄 ∈ LMod)
5241adantr 484 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → 𝐴 ∈ Ring)
53 pm2mpfo.x . . . . . . . . . . . . . 14 𝑋 = (var1𝐴)
54 eqid 2758 . . . . . . . . . . . . . 14 (mulGrp‘𝑄) = (mulGrp‘𝑄)
55 pm2mpfo.e . . . . . . . . . . . . . 14 = (.g‘(mulGrp‘𝑄))
56 pm2mpfo.l . . . . . . . . . . . . . 14 𝐿 = (Base‘𝑄)
5742, 53, 54, 55, 56ply1moncl 21008 . . . . . . . . . . . . 13 ((𝐴 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ 𝐿)
5852, 57sylan 583 . . . . . . . . . . . 12 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → (𝑛 𝑋) ∈ 𝐿)
59 eqid 2758 . . . . . . . . . . . . 13 (Scalar‘𝑄) = (Scalar‘𝑄)
60 pm2mpfo.m . . . . . . . . . . . . 13 = ( ·𝑠𝑄)
61 eqid 2758 . . . . . . . . . . . . 13 (0g‘(Scalar‘𝑄)) = (0g‘(Scalar‘𝑄))
62 eqid 2758 . . . . . . . . . . . . 13 (0g𝑄) = (0g𝑄)
6356, 59, 60, 61, 62lmod0vs 19748 . . . . . . . . . . . 12 ((𝑄 ∈ LMod ∧ (𝑛 𝑋) ∈ 𝐿) → ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)) = (0g𝑄))
6451, 58, 63syl2an2r 684 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((0g‘(Scalar‘𝑄)) (𝑛 𝑋)) = (0g𝑄))
6564adantr 484 . . . . . . . . . 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 2793 . . . . . . . . 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 416 . . . . . . . 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 3108 . . . . . 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 3197 . . . . 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 21478 . . . . . . . . . 10 (((𝑥(.r𝐶)𝑦) ∈ 𝐵𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)))
7322, 72sylan 583 . . . . . . . . 9 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝑖𝑁, 𝑗𝑁 ↦ ((coe1‘(𝑖(𝑥(.r𝐶)𝑦)𝑗))‘𝑛)))
7473oveq1d 7171 . . . . . . . 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 2760 . . . . . . 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 344 . . . . . 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 3125 . . . . 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 3221 . . . 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 260 . . 3 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑠 ∈ ℕ0𝑛 ∈ ℕ0 (𝑠 < 𝑛 → (((𝑥(.r𝐶)𝑦) decompPMat 𝑛) (𝑛 𝑋)) = (0g𝑄)))
8013, 14, 19, 38decpmatmul 21485 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵) ∧ 𝑛 ∈ ℕ0) → ((𝑥(.r𝐶)𝑦) decompPMat 𝑛) = (𝐴 Σg (𝑘 ∈ (0...𝑛) ↦ ((𝑥 decompPMat 𝑘)(.r𝐴)(𝑦 decompPMat (𝑛𝑘))))))
8180ad4ant234 1172 . . . . . . . . 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 2764 . . . . . . . 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 7171 . . . . . . 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 2760 . . . . . 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 344 . . . . 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 3125 . . . 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 3221 . . 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 260 . 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 13428 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 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ∃wrex 3071  Vcvv 3409   class class class wbr 5036   ↦ cmpt 5116  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  Fincfn 8540   finSupp cfsupp 8879  0cc0 10588   < clt 10726   − cmin 10921  ℕ0cn0 11947  ...cfz 12952  Basecbs 16554  .rcmulr 16637  Scalarcsca 16639   ·𝑠 cvsca 16640  0gc0g 16784   Σg cgsu 16785  .gcmg 18304  mulGrpcmgp 19320  Ringcrg 19378  LModclmod 19715  var1cv1 20913  Poly1cpl1 20914  coe1cco1 20915   Mat cmat 21120   decompPMat cdecpmat 21475   pMatToMatPoly cpm2mp 21505 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465  ax-cnex 10644  ax-resscn 10645  ax-1cn 10646  ax-icn 10647  ax-addcl 10648  ax-addrcl 10649  ax-mulcl 10650  ax-mulrcl 10651  ax-mulcom 10652  ax-addass 10653  ax-mulass 10654  ax-distr 10655  ax-i2m1 10656  ax-1ne0 10657  ax-1rid 10658  ax-rnegex 10659  ax-rrecex 10660  ax-cnre 10661  ax-pre-lttri 10662  ax-pre-lttrn 10663  ax-pre-ltadd 10664  ax-pre-mulgt0 10665 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rmo 3078  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-ot 4534  df-uni 4802  df-int 4842  df-iun 4888  df-iin 4889  df-br 5037  df-opab 5099  df-mpt 5117  df-tr 5143  df-id 5434  df-eprel 5439  df-po 5447  df-so 5448  df-fr 5487  df-se 5488  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-isom 6349  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-of 7411  df-ofr 7412  df-om 7586  df-1st 7699  df-2nd 7700  df-supp 7842  df-wrecs 7963  df-recs 8024  df-rdg 8062  df-1o 8118  df-er 8305  df-map 8424  df-pm 8425  df-ixp 8493  df-en 8541  df-dom 8542  df-sdom 8543  df-fin 8544  df-fsupp 8880  df-sup 8952  df-oi 9020  df-card 9414  df-pnf 10728  df-mnf 10729  df-xr 10730  df-ltxr 10731  df-le 10732  df-sub 10923  df-neg 10924  df-nn 11688  df-2 11750  df-3 11751  df-4 11752  df-5 11753  df-6 11754  df-7 11755  df-8 11756  df-9 11757  df-n0 11948  df-z 12034  df-dec 12151  df-uz 12296  df-fz 12953  df-fzo 13096  df-seq 13432  df-hash 13754  df-struct 16556  df-ndx 16557  df-slot 16558  df-base 16560  df-sets 16561  df-ress 16562  df-plusg 16649  df-mulr 16650  df-sca 16652  df-vsca 16653  df-ip 16654  df-tset 16655  df-ple 16656  df-ds 16658  df-hom 16660  df-cco 16661  df-0g 16786  df-gsum 16787  df-prds 16792  df-pws 16794  df-mre 16928  df-mrc 16929  df-acs 16931  df-mgm 17931  df-sgrp 17980  df-mnd 17991  df-mhm 18035  df-submnd 18036  df-grp 18185  df-minusg 18186  df-sbg 18187  df-mulg 18305  df-subg 18356  df-ghm 18436  df-cntz 18527  df-cmn 18988  df-abl 18989  df-mgp 19321  df-ur 19333  df-ring 19380  df-subrg 19614  df-lmod 19717  df-lss 19785  df-sra 20025  df-rgmod 20026  df-dsmm 20510  df-frlm 20525  df-psr 20684  df-mvr 20685  df-mpl 20686  df-opsr 20688  df-psr1 20917  df-vr1 20918  df-ply1 20919  df-coe1 20920  df-mamu 21099  df-mat 21121  df-decpmat 21476 This theorem is referenced by:  pm2mpmhmlem2  21532
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