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Theorem pmatcollpw2lem 20861
Description: Lemma for pmatcollpw2 20862. (Contributed by AV, 3-Oct-2019.) (Revised by AV, 3-Dec-2019.)
Hypotheses
Ref Expression
pmatcollpw1.p 𝑃 = (Poly1𝑅)
pmatcollpw1.c 𝐶 = (𝑁 Mat 𝑃)
pmatcollpw1.b 𝐵 = (Base‘𝐶)
pmatcollpw1.m × = ( ·𝑠𝑃)
pmatcollpw1.e = (.g‘(mulGrp‘𝑃))
pmatcollpw1.x 𝑋 = (var1𝑅)
Assertion
Ref Expression
pmatcollpw2lem ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) finSupp (0g𝐶))
Distinct variable groups:   𝐵,𝑛   𝑛,𝑀   𝑛,𝑁   𝑅,𝑛   𝑛,𝑋   × ,𝑛   ,𝑛   𝑃,𝑛   𝐵,𝑖,𝑗   𝑖,𝑀,𝑗   𝑖,𝑁,𝑗   𝑃,𝑖,𝑗,𝑛   𝑅,𝑖,𝑗   𝑖,𝑋,𝑗   × ,𝑖,𝑗   ,𝑖,𝑗
Allowed substitution hints:   𝐶(𝑖,𝑗,𝑛)

Proof of Theorem pmatcollpw2lem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 1166 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
2 mpt2exga 7447 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
31, 1, 2syl2anc 579 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
43ralrimivw 3114 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∀𝑛 ∈ ℕ0 (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
5 eqid 2765 . . . . . 6 (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) = (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))
65fnmpt 6198 . . . . 5 (∀𝑛 ∈ ℕ0 (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) Fn ℕ0)
74, 6syl 17 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) Fn ℕ0)
8 nn0ex 11545 . . . . 5 0 ∈ V
98a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ℕ0 ∈ V)
10 fvexd 6390 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝐶) ∈ V)
11 suppvalfn 7504 . . . 4 (((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) Fn ℕ0 ∧ ℕ0 ∈ V ∧ (0g𝐶) ∈ V) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) = {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)})
127, 9, 10, 11syl3anc 1490 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) = {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)})
13 pmatcollpw1.p . . . . . . . . . . 11 𝑃 = (Poly1𝑅)
14 pmatcollpw1.c . . . . . . . . . . 11 𝐶 = (𝑁 Mat 𝑃)
15 pmatcollpw1.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
16 eqid 2765 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1713, 14, 15, 16pmatcoe1fsupp 20785 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)))
18 oveq1 6849 . . . . . . . . . . . . . . . . 17 (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = ((0g𝑅) × (𝑥 𝑋)))
19 pmatcollpw1.m . . . . . . . . . . . . . . . . . . . . 21 × = ( ·𝑠𝑃)
2019a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → × = ( ·𝑠𝑃))
2113ply1sca 19896 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
22213ad2ant2 1164 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝑃))
2322fveq2d 6379 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
24 eqidd 2766 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑥 𝑋) = (𝑥 𝑋))
2520, 23, 24oveq123d 6863 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2625ad3antrrr 721 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2722eqcomd 2771 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (Scalar‘𝑃) = 𝑅)
2827ad3antrrr 721 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (Scalar‘𝑃) = 𝑅)
2928fveq2d 6379 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (0g‘(Scalar‘𝑃)) = (0g𝑅))
3029oveq1d 6857 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)) = ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)))
31 simpl2 1244 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑅 ∈ Ring)
32 pmatcollpw1.x . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = (var1𝑅)
33 eqid 2765 . . . . . . . . . . . . . . . . . . . . . . . 24 (mulGrp‘𝑃) = (mulGrp‘𝑃)
34 pmatcollpw1.e . . . . . . . . . . . . . . . . . . . . . . . 24 = (.g‘(mulGrp‘𝑃))
35 eqid 2765 . . . . . . . . . . . . . . . . . . . . . . . 24 (Base‘𝑃) = (Base‘𝑃)
3613, 32, 33, 34, 35ply1moncl 19914 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
37363ad2antl2 1237 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
3831, 37jca 507 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
3938adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
4039adantr 472 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
41 eqid 2765 . . . . . . . . . . . . . . . . . . . 20 ( ·𝑠𝑃) = ( ·𝑠𝑃)
4213, 35, 41, 16ply10s0 19899 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)) → ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)) = (0g𝑃))
4340, 42syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)) = (0g𝑃))
4426, 30, 433eqtrd 2803 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = (0g𝑃))
4518, 44sylan9eqr 2821 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) ∧ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))
4645ex 401 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4746anasss 458 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4847ralimdvva 3111 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4948imim2d 57 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → ((𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)) → (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))))
5049ralimdva 3109 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)) → ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))))
5150reximdv 3162 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))))
5217, 51mpd 15 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
53 simpl3 1246 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑀𝐵)
54 simpr 477 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
5531, 53, 543jca 1158 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0))
5613, 14, 15decpmate 20850 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5755, 56sylan 575 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5857oveq1d 6857 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)))
5958eqeq1d 2767 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
60592ralbidva 3135 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
6160imbi2d 331 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → ((𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))))
6261ralbidva 3132 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))))
6362rexbidv 3199 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))))
6452, 63mpbird 248 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))
65 eqid 2765 . . . . . . . . . . . . 13 𝑁 = 𝑁
6665biantrur 526 . . . . . . . . . . . 12 (∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ (𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))))
6765biantrur 526 . . . . . . . . . . . . . 14 (∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))
6867bicomi 215 . . . . . . . . . . . . 13 ((𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
6968ralbii 3127 . . . . . . . . . . . 12 (∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
7066, 69bitr3i 268 . . . . . . . . . . 11 ((𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))) ↔ ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
7170a1i 11 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))) ↔ ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))
7271imbi2d 331 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))) ↔ (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))))
7372rexralbidv 3205 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))) ↔ ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))))
7464, 73mpbird 248 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))))
75 mpt2eq123 6912 . . . . . . . . . 10 ((𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃)))
7675imim2i 16 . . . . . . . . 9 ((𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))) → (𝑦 < 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃))))
7776ralimi 3099 . . . . . . . 8 (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))) → ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃))))
7877reximi 3157 . . . . . . 7 (∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑁 = 𝑁 ∧ ∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃))))
7974, 78syl 17 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃))))
80 eqidd 2766 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) = (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))))
81 oveq2 6850 . . . . . . . . . . . . . . 15 (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
8281oveqd 6859 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
83 oveq1 6849 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑛 𝑋) = (𝑥 𝑋))
8482, 83oveq12d 6860 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)))
8584mpt2eq3dv 6919 . . . . . . . . . . . 12 (𝑛 = 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))))
8685adantl 473 . . . . . . . . . . 11 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑛 = 𝑥) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))))
87 id 22 . . . . . . . . . . . . . . 15 (𝑁 ∈ Fin → 𝑁 ∈ Fin)
8887ancri 545 . . . . . . . . . . . . . 14 (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
89883ad2ant1 1163 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9089adantr 472 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91 mpt2exga 7447 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) ∈ V)
9290, 91syl 17 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) ∈ V)
9380, 86, 54, 92fvmptd 6477 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))))
9413ply1ring 19891 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
9594anim2i 610 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
96953adant3 1162 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
9796adantr 472 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
98 eqid 2765 . . . . . . . . . . . 12 (0g𝑃) = (0g𝑃)
9914, 98mat0op 20501 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → (0g𝐶) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃)))
10097, 99syl 17 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (0g𝐶) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃)))
10193, 100eqeq12d 2780 . . . . . . . . 9 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶) ↔ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃))))
102101imbi2d 331 . . . . . . . 8 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → ((𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶)) ↔ (𝑦 < 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃)))))
103102ralbidva 3132 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶)) ↔ ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃)))))
104103rexbidv 3199 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶)) ↔ ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃)))))
10579, 104mpbird 248 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶)))
106 nne 2941 . . . . . . . 8 (¬ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶) ↔ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶))
107106imbi2i 327 . . . . . . 7 ((𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)) ↔ (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶)))
108107ralbii 3127 . . . . . 6 (∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)) ↔ ∀𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶)))
109108rexbii 3188 . . . . 5 (∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)) ↔ ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) = (0g𝐶)))
110105, 109sylibr 225 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)))
111 rabssnn0fi 12993 . . . 4 ({𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)} ∈ Fin ↔ ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ¬ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)))
112110, 111sylibr 225 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)} ∈ Fin)
11312, 112eqeltrd 2844 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) ∈ Fin)
114 funmpt 6106 . . . 4 Fun (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))
115114a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → Fun (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))))
1168mptex 6679 . . . 4 (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) ∈ V
117116a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) ∈ V)
118 funisfsupp 8487 . . 3 ((Fun (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) ∧ (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) ∈ V ∧ (0g𝐶) ∈ V) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) finSupp (0g𝐶) ↔ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) ∈ Fin))
119115, 117, 10, 118syl3anc 1490 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) finSupp (0g𝐶) ↔ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) ∈ Fin))
120113, 119mpbird 248 1 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) finSupp (0g𝐶))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350   class class class wbr 4809  cmpt 4888  Fun wfun 6062   Fn wfn 6063  cfv 6068  (class class class)co 6842  cmpt2 6844   supp csupp 7497  Fincfn 8160   finSupp cfsupp 8482   < clt 10328  0cn0 11538  Basecbs 16130  Scalarcsca 16217   ·𝑠 cvsca 16218  0gc0g 16366  .gcmg 17807  mulGrpcmgp 18756  Ringcrg 18814  var1cv1 19819  Poly1cpl1 19820  coe1cco1 19821   Mat cmat 20489   decompPMat cdecpmat 20846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-ot 4343  df-uni 4595  df-int 4634  df-iun 4678  df-iin 4679  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-of 7095  df-ofr 7096  df-om 7264  df-1st 7366  df-2nd 7367  df-supp 7498  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-2o 7765  df-oadd 7768  df-er 7947  df-map 8062  df-pm 8063  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-fsupp 8483  df-sup 8555  df-oi 8622  df-card 9016  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-fzo 12674  df-seq 13009  df-hash 13322  df-struct 16132  df-ndx 16133  df-slot 16134  df-base 16136  df-sets 16137  df-ress 16138  df-plusg 16227  df-mulr 16228  df-sca 16230  df-vsca 16231  df-ip 16232  df-tset 16233  df-ple 16234  df-ds 16236  df-hom 16238  df-cco 16239  df-0g 16368  df-gsum 16369  df-prds 16374  df-pws 16376  df-mre 16512  df-mrc 16513  df-acs 16515  df-mgm 17508  df-sgrp 17550  df-mnd 17561  df-mhm 17601  df-submnd 17602  df-grp 17692  df-minusg 17693  df-sbg 17694  df-mulg 17808  df-subg 17855  df-ghm 17922  df-cntz 18013  df-cmn 18461  df-abl 18462  df-mgp 18757  df-ur 18769  df-ring 18816  df-subrg 19047  df-lmod 19134  df-lss 19202  df-sra 19446  df-rgmod 19447  df-psr 19630  df-mvr 19631  df-mpl 19632  df-opsr 19634  df-psr1 19823  df-vr1 19824  df-ply1 19825  df-coe1 19826  df-dsmm 20352  df-frlm 20367  df-mat 20490  df-decpmat 20847
This theorem is referenced by:  pmatcollpw2  20862
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