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Theorem pmatcollpw2lem 20784
Description: Lemma for pmatcollpw2 20785. (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 1131 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
2 mpt2exga 7414 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
31, 1, 2syl2anc 696 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
43ralrimivw 3105 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∀𝑛 ∈ ℕ0 (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
5 eqid 2760 . . . . . 6 (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) = (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))
65fnmpt 6181 . . . . 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 11490 . . . . 5 0 ∈ V
98a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ℕ0 ∈ V)
10 fvexd 6364 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝐶) ∈ V)
11 suppvalfn 7470 . . . 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 1477 . . 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 2760 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1713, 14, 15, 16pmatcoe1fsupp 20708 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)))
18 oveq1 6820 . . . . . . . . . . . . . . . . 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 19825 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
22213ad2ant2 1129 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝑃))
2322fveq2d 6356 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
24 eqidd 2761 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑥 𝑋) = (𝑥 𝑋))
2520, 23, 24oveq123d 6834 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2625ad3antrrr 768 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2722eqcomd 2766 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (Scalar‘𝑃) = 𝑅)
2827ad3antrrr 768 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (Scalar‘𝑃) = 𝑅)
2928fveq2d 6356 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (0g‘(Scalar‘𝑃)) = (0g𝑅))
3029oveq1d 6828 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)) = ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)))
31 simpl2 1230 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑅 ∈ Ring)
32 pmatcollpw1.x . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = (var1𝑅)
33 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . . 24 (mulGrp‘𝑃) = (mulGrp‘𝑃)
34 pmatcollpw1.e . . . . . . . . . . . . . . . . . . . . . . . 24 = (.g‘(mulGrp‘𝑃))
35 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . . 24 (Base‘𝑃) = (Base‘𝑃)
3613, 32, 33, 34, 35ply1moncl 19843 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
37363ad2antl2 1202 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
3831, 37jca 555 . . . . . . . . . . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . . . . . . . 20 ( ·𝑠𝑃) = ( ·𝑠𝑃)
4213, 35, 41, 16ply10s0 19828 . . . . . . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = (0g𝑃))
4518, 44sylan9eqr 2816 . . . . . . . . . . . . . . . 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 682 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4847ralimdvva 3102 . . . . . . . . . . . . 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 3100 . . . . . . . . . . 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 3154 . . . . . . . . . 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 1232 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑀𝐵)
54 simpr 479 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
5531, 53, 543jca 1123 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0))
5613, 14, 15decpmate 20773 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5755, 56sylan 489 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5857oveq1d 6828 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)))
5958eqeq1d 2762 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
60592ralbidva 3126 . . . . . . . . . . . 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 3123 . . . . . . . . . 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 3190 . . . . . . . . 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 2760 . . . . . . . . . . . . 13 𝑁 = 𝑁
6665biantrur 528 . . . . . . . . . . . 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 528 . . . . . . . . . . . . . 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 3118 . . . . . . . . . . . 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 3196 . . . . . . . 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 6879 . . . . . . . . . 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 3090 . . . . . . . 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 3149 . . . . . . 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 2761 . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . 15 (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
8281oveqd 6830 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
83 oveq1 6820 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑛 𝑋) = (𝑥 𝑋))
8482, 83oveq12d 6831 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)))
8584mpt2eq3dv 6886 . . . . . . . . . . . 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 576 . . . . . . . . . . . . . 14 (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
89883ad2ant1 1128 . . . . . . . . . . . . 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 7414 . . . . . . . . . . . 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 6450 . . . . . . . . . 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 19820 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
9594anim2i 594 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
96953adant3 1127 . . . . . . . . . . . 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 2760 . . . . . . . . . . . 12 (0g𝑃) = (0g𝑃)
9914, 98mat0op 20427 . . . . . . . . . . 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 2775 . . . . . . . . 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 3123 . . . . . . 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 3190 . . . . . 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 2936 . . . . . . . 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 3118 . . . . . 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 3179 . . . . 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 12979 . . . 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 2839 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) ∈ Fin)
114 funmpt 6087 . . . 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 6650 . . . 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 8445 . . 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 1477 . 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 1072   = wceq 1632  wcel 2139  wne 2932  wral 3050  wrex 3051  {crab 3054  Vcvv 3340   class class class wbr 4804  cmpt 4881  Fun wfun 6043   Fn wfn 6044  cfv 6049  (class class class)co 6813  cmpt2 6815   supp csupp 7463  Fincfn 8121   finSupp cfsupp 8440   < clt 10266  0cn0 11484  Basecbs 16059  Scalarcsca 16146   ·𝑠 cvsca 16147  0gc0g 16302  .gcmg 17741  mulGrpcmgp 18689  Ringcrg 18747  var1cv1 19748  Poly1cpl1 19749  coe1cco1 19750   Mat cmat 20415   decompPMat cdecpmat 20769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-inf2 8711  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-ot 4330  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-of 7062  df-ofr 7063  df-om 7231  df-1st 7333  df-2nd 7334  df-supp 7464  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-2o 7730  df-oadd 7733  df-er 7911  df-map 8025  df-pm 8026  df-ixp 8075  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-fsupp 8441  df-sup 8513  df-oi 8580  df-card 8955  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-nn 11213  df-2 11271  df-3 11272  df-4 11273  df-5 11274  df-6 11275  df-7 11276  df-8 11277  df-9 11278  df-n0 11485  df-z 11570  df-dec 11686  df-uz 11880  df-fz 12520  df-fzo 12660  df-seq 12996  df-hash 13312  df-struct 16061  df-ndx 16062  df-slot 16063  df-base 16065  df-sets 16066  df-ress 16067  df-plusg 16156  df-mulr 16157  df-sca 16159  df-vsca 16160  df-ip 16161  df-tset 16162  df-ple 16163  df-ds 16166  df-hom 16168  df-cco 16169  df-0g 16304  df-gsum 16305  df-prds 16310  df-pws 16312  df-mre 16448  df-mrc 16449  df-acs 16451  df-mgm 17443  df-sgrp 17485  df-mnd 17496  df-mhm 17536  df-submnd 17537  df-grp 17626  df-minusg 17627  df-sbg 17628  df-mulg 17742  df-subg 17792  df-ghm 17859  df-cntz 17950  df-cmn 18395  df-abl 18396  df-mgp 18690  df-ur 18702  df-ring 18749  df-subrg 18980  df-lmod 19067  df-lss 19135  df-sra 19374  df-rgmod 19375  df-psr 19558  df-mvr 19559  df-mpl 19560  df-opsr 19562  df-psr1 19752  df-vr1 19753  df-ply1 19754  df-coe1 19755  df-dsmm 20278  df-frlm 20293  df-mat 20416  df-decpmat 20770
This theorem is referenced by:  pmatcollpw2  20785
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