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Theorem pmatcollpw2lem 22208
Description: Lemma for pmatcollpw2 22209. (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 1136 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
2 mpoexga 8046 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
31, 1, 2syl2anc 584 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
43ralrimivw 3149 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∀𝑛 ∈ ℕ0 (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
5 eqid 2731 . . . . . 6 (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) = (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))
65fnmpt 6677 . . . . 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 12460 . . . . 5 0 ∈ V
98a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ℕ0 ∈ V)
10 fvexd 6893 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝐶) ∈ V)
11 suppvalfn 8136 . . . 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 1371 . . 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 2731 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1713, 14, 15, 16pmatcoe1fsupp 22132 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)))
18 oveq1 7400 . . . . . . . . . . . . . . . . 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 21706 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
22213ad2ant2 1134 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝑃))
2322fveq2d 6882 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
24 eqidd 2732 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑥 𝑋) = (𝑥 𝑋))
2520, 23, 24oveq123d 7414 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2625ad3antrrr 728 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2722eqcomd 2737 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (Scalar‘𝑃) = 𝑅)
2827ad3antrrr 728 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (Scalar‘𝑃) = 𝑅)
2928fveq2d 6882 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (0g‘(Scalar‘𝑃)) = (0g𝑅))
3029oveq1d 7408 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)) = ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)))
31 simpl2 1192 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑅 ∈ Ring)
32 pmatcollpw1.x . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = (var1𝑅)
33 eqid 2731 . . . . . . . . . . . . . . . . . . . . . . . 24 (mulGrp‘𝑃) = (mulGrp‘𝑃)
34 pmatcollpw1.e . . . . . . . . . . . . . . . . . . . . . . . 24 = (.g‘(mulGrp‘𝑃))
35 eqid 2731 . . . . . . . . . . . . . . . . . . . . . . . 24 (Base‘𝑃) = (Base‘𝑃)
3613, 32, 33, 34, 35ply1moncl 21724 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
37363ad2antl2 1186 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
3831, 37jca 512 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
3938adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
4039adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
41 eqid 2731 . . . . . . . . . . . . . . . . . . . 20 ( ·𝑠𝑃) = ( ·𝑠𝑃)
4213, 35, 41, 16ply10s0 21709 . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = (0g𝑃))
4518, 44sylan9eqr 2793 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) ∧ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))
4645ex 413 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4746anasss 467 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4847ralimdvva 3203 . . . . . . . . . . . . 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 3166 . . . . . . . . . . 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 3169 . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑀𝐵)
54 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
5531, 53, 543jca 1128 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0))
5613, 14, 15decpmate 22197 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5755, 56sylan 580 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5857oveq1d 7408 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)))
5958eqeq1d 2733 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
60592ralbidva 3215 . . . . . . . . . . . 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 340 . . . . . . . . . . 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 3174 . . . . . . . . . 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 3177 . . . . . . . . 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 256 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))
65 eqid 2731 . . . . . . . . . . . . 13 𝑁 = 𝑁
6665biantrur 531 . . . . . . . . . . . 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 531 . . . . . . . . . . . . . 14 (∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))
6867bicomi 223 . . . . . . . . . . . . 13 ((𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
6968ralbii 3092 . . . . . . . . . . . 12 (∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
7066, 69bitr3i 276 . . . . . . . . . . 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 340 . . . . . . . . 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 3219 . . . . . . . 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 256 . . . . . . 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 mpoeq123 7465 . . . . . . . . . 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 3082 . . . . . . . 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 3083 . . . . . . 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 2732 . . . . . . . . . . 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 7401 . . . . . . . . . . . . . . 15 (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
8281oveqd 7410 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
83 oveq1 7400 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑛 𝑋) = (𝑥 𝑋))
8482, 83oveq12d 7411 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)))
8584mpoeq3dv 7472 . . . . . . . . . . . 12 (𝑛 = 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))))
8685adantl 482 . . . . . . . . . . 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 550 . . . . . . . . . . . . . 14 (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
89883ad2ant1 1133 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9089adantr 481 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
91 mpoexga 8046 . . . . . . . . . . . 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 6991 . . . . . . . . . 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 21701 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
9594anim2i 617 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
96953adant3 1132 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
9796adantr 481 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
98 eqid 2731 . . . . . . . . . . . 12 (0g𝑃) = (0g𝑃)
9914, 98mat0op 21850 . . . . . . . . . . 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 2747 . . . . . . . . 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 340 . . . . . . . 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 3174 . . . . . . 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 3177 . . . . . 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 256 . . . . 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 2943 . . . . . . . 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 335 . . . . . . 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 3092 . . . . . 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 3093 . . . . 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 233 . . . 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 13933 . . . 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 233 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)} ∈ Fin)
11312, 112eqeltrd 2832 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) ∈ Fin)
114 funmpt 6575 . . 3 Fun (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))
1158mptex 7209 . . 3 (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) ∈ V
116 funisfsupp 9350 . . 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))
117114, 115, 10, 116mp3an12i 1465 . 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))
118113, 117mpbird 256 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 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939  wral 3060  wrex 3069  {crab 3431  Vcvv 3473   class class class wbr 5141  cmpt 5224  Fun wfun 6526   Fn wfn 6527  cfv 6532  (class class class)co 7393  cmpo 7395   supp csupp 8128  Fincfn 8922   finSupp cfsupp 9344   < clt 11230  0cn0 12454  Basecbs 17126  Scalarcsca 17182   ·𝑠 cvsca 17183  0gc0g 17367  .gcmg 18922  mulGrpcmgp 19946  Ringcrg 20014  var1cv1 21629  Poly1cpl1 21630  coe1cco1 21631   Mat cmat 21836   decompPMat cdecpmat 22193
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-cnex 11148  ax-resscn 11149  ax-1cn 11150  ax-icn 11151  ax-addcl 11152  ax-addrcl 11153  ax-mulcl 11154  ax-mulrcl 11155  ax-mulcom 11156  ax-addass 11157  ax-mulass 11158  ax-distr 11159  ax-i2m1 11160  ax-1ne0 11161  ax-1rid 11162  ax-rnegex 11163  ax-rrecex 11164  ax-cnre 11165  ax-pre-lttri 11166  ax-pre-lttrn 11167  ax-pre-ltadd 11168  ax-pre-mulgt0 11169
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-tp 4627  df-op 4629  df-ot 4631  df-uni 4902  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-se 5625  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-isom 6541  df-riota 7349  df-ov 7396  df-oprab 7397  df-mpo 7398  df-of 7653  df-ofr 7654  df-om 7839  df-1st 7957  df-2nd 7958  df-supp 8129  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-1o 8448  df-er 8686  df-map 8805  df-pm 8806  df-ixp 8875  df-en 8923  df-dom 8924  df-sdom 8925  df-fin 8926  df-fsupp 9345  df-sup 9419  df-oi 9487  df-card 9916  df-pnf 11232  df-mnf 11233  df-xr 11234  df-ltxr 11235  df-le 11236  df-sub 11428  df-neg 11429  df-nn 12195  df-2 12257  df-3 12258  df-4 12259  df-5 12260  df-6 12261  df-7 12262  df-8 12263  df-9 12264  df-n0 12455  df-z 12541  df-dec 12660  df-uz 12805  df-fz 13467  df-fzo 13610  df-seq 13949  df-hash 14273  df-struct 17062  df-sets 17079  df-slot 17097  df-ndx 17109  df-base 17127  df-ress 17156  df-plusg 17192  df-mulr 17193  df-sca 17195  df-vsca 17196  df-ip 17197  df-tset 17198  df-ple 17199  df-ds 17201  df-hom 17203  df-cco 17204  df-0g 17369  df-gsum 17370  df-prds 17375  df-pws 17377  df-mre 17512  df-mrc 17513  df-acs 17515  df-mgm 18543  df-sgrp 18592  df-mnd 18603  df-mhm 18647  df-submnd 18648  df-grp 18797  df-minusg 18798  df-sbg 18799  df-mulg 18923  df-subg 18975  df-ghm 19056  df-cntz 19147  df-cmn 19614  df-abl 19615  df-mgp 19947  df-ur 19964  df-ring 20016  df-subrg 20310  df-lmod 20422  df-lss 20492  df-sra 20734  df-rgmod 20735  df-dsmm 21220  df-frlm 21235  df-psr 21393  df-mvr 21394  df-mpl 21395  df-opsr 21397  df-psr1 21633  df-vr1 21634  df-ply1 21635  df-coe1 21636  df-mat 21837  df-decpmat 22194
This theorem is referenced by:  pmatcollpw2  22209
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