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Theorem pmatcollpw2lem 20630
Description: Lemma for pmatcollpw2 20631. (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 1081 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
2 mpt2exga 7291 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
31, 1, 2syl2anc 694 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
43ralrimivw 2996 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∀𝑛 ∈ ℕ0 (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
5 eqid 2651 . . . . . 6 (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) = (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))
65fnmpt 6058 . . . . 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 11336 . . . . 5 0 ∈ V
98a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ℕ0 ∈ V)
10 fvexd 6241 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝐶) ∈ V)
11 suppvalfn 7347 . . . 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 1366 . . 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 2651 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1713, 14, 15, 16pmatcoe1fsupp 20554 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)))
18 oveq1 6697 . . . . . . . . . . . . . . . . 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 19671 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
22213ad2ant2 1103 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝑃))
2322fveq2d 6233 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
24 eqidd 2652 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑥 𝑋) = (𝑥 𝑋))
2520, 23, 24oveq123d 6711 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2625ad3antrrr 766 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2722eqcomd 2657 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (Scalar‘𝑃) = 𝑅)
2827ad3antrrr 766 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (Scalar‘𝑃) = 𝑅)
2928fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (0g‘(Scalar‘𝑃)) = (0g𝑅))
3029oveq1d 6705 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)) = ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)))
31 simpl2 1085 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑅 ∈ Ring)
32 pmatcollpw1.x . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = (var1𝑅)
33 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . 24 (mulGrp‘𝑃) = (mulGrp‘𝑃)
34 pmatcollpw1.e . . . . . . . . . . . . . . . . . . . . . . . 24 = (.g‘(mulGrp‘𝑃))
35 eqid 2651 . . . . . . . . . . . . . . . . . . . . . . . 24 (Base‘𝑃) = (Base‘𝑃)
3613, 32, 33, 34, 35ply1moncl 19689 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
37363ad2antl2 1244 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
3831, 37jca 553 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
3938adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
4039adantr 480 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
41 eqid 2651 . . . . . . . . . . . . . . . . . . . 20 ( ·𝑠𝑃) = ( ·𝑠𝑃)
4213, 35, 41, 16ply10s0 19674 . . . . . . . . . . . . . . . . . . 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 2689 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = (0g𝑃))
4518, 44sylan9eqr 2707 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) ∧ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))
4645ex 449 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4746anasss 680 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4847ralimdvva 2993 . . . . . . . . . . . . 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 2991 . . . . . . . . . . 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 3045 . . . . . . . . . 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 1086 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑀𝐵)
54 simpr 476 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
5531, 53, 543jca 1261 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0))
5613, 14, 15decpmate 20619 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5755, 56sylan 487 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5857oveq1d 6705 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)))
5958eqeq1d 2653 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
60592ralbidva 3017 . . . . . . . . . . . 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 329 . . . . . . . . . . 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 3014 . . . . . . . . . 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 3081 . . . . . . . . 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 247 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))
65 eqid 2651 . . . . . . . . . . . . 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 214 . . . . . . . . . . . . 13 ((𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
6968ralbii 3009 . . . . . . . . . . . 12 (∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
7066, 69bitr3i 266 . . . . . . . . . . 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 329 . . . . . . . . 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 3087 . . . . . . . 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 247 . . . . . . 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 6756 . . . . . . . . . 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 2981 . . . . . . . 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 3040 . . . . . . 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 2652 . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . 15 (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
8281oveqd 6707 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
83 oveq1 6697 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑛 𝑋) = (𝑥 𝑋))
8482, 83oveq12d 6708 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)))
8584mpt2eq3dv 6763 . . . . . . . . . . . 12 (𝑛 = 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))))
8685adantl 481 . . . . . . . . . . 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 574 . . . . . . . . . . . . . 14 (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
89883ad2ant1 1102 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9089adantr 480 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91 mpt2exga 7291 . . . . . . . . . . . 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 6327 . . . . . . . . . 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 19666 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
9594anim2i 592 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
96953adant3 1101 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
9796adantr 480 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
98 eqid 2651 . . . . . . . . . . . 12 (0g𝑃) = (0g𝑃)
9914, 98mat0op 20273 . . . . . . . . . . 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 2666 . . . . . . . . 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 329 . . . . . . . 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 3014 . . . . . . 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 3081 . . . . . 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 247 . . . . 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 2827 . . . . . . . 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 325 . . . . . . 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 3009 . . . . . 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 3070 . . . . 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 224 . . . 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 12825 . . . 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 224 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)} ∈ Fin)
11312, 112eqeltrd 2730 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) ∈ Fin)
114 funmpt 5964 . . . 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 6527 . . . 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 8321 . . 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 1366 . 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 247 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231   class class class wbr 4685  cmpt 4762  Fun wfun 5920   Fn wfn 5921  cfv 5926  (class class class)co 6690  cmpt2 6692   supp csupp 7340  Fincfn 7997   finSupp cfsupp 8316   < clt 10112  0cn0 11330  Basecbs 15904  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  .gcmg 17587  mulGrpcmgp 18535  Ringcrg 18593  var1cv1 19594  Poly1cpl1 19595  coe1cco1 19596   Mat cmat 20261   decompPMat cdecpmat 20615
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-ot 4219  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-ofr 6940  df-om 7108  df-1st 7210  df-2nd 7211  df-supp 7341  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-er 7787  df-map 7901  df-pm 7902  df-ixp 7951  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-fsupp 8317  df-sup 8389  df-oi 8456  df-card 8803  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-fzo 12505  df-seq 12842  df-hash 13158  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-ip 16006  df-tset 16007  df-ple 16008  df-ds 16011  df-hom 16013  df-cco 16014  df-0g 16149  df-gsum 16150  df-prds 16155  df-pws 16157  df-mre 16293  df-mrc 16294  df-acs 16296  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-mhm 17382  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-subg 17638  df-ghm 17705  df-cntz 17796  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-subrg 18826  df-lmod 18913  df-lss 18981  df-sra 19220  df-rgmod 19221  df-psr 19404  df-mvr 19405  df-mpl 19406  df-opsr 19408  df-psr1 19598  df-vr1 19599  df-ply1 19600  df-coe1 19601  df-dsmm 20124  df-frlm 20139  df-mat 20262  df-decpmat 20616
This theorem is referenced by:  pmatcollpw2  20631
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