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Theorem pmatcollpw2lem 20501
Description: Lemma for pmatcollpw2 20502. (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 1059 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑁 ∈ Fin)
2 mpt2exga 7191 . . . . . . 7 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
31, 1, 2syl2anc 692 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
43ralrimivw 2961 . . . . 5 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∀𝑛 ∈ ℕ0 (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) ∈ V)
5 eqid 2621 . . . . . 6 (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) = (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))
65fnmpt 5977 . . . . 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 11242 . . . . 5 0 ∈ V
98a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ℕ0 ∈ V)
10 fvex 6158 . . . . 5 (0g𝐶) ∈ V
1110a1i 11 . . . 4 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝐶) ∈ V)
12 suppvalfn 7247 . . . 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)})
137, 9, 11, 12syl3anc 1323 . . 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)})
14 pmatcollpw1.p . . . . . . . . . . 11 𝑃 = (Poly1𝑅)
15 pmatcollpw1.c . . . . . . . . . . 11 𝐶 = (𝑁 Mat 𝑃)
16 pmatcollpw1.b . . . . . . . . . . 11 𝐵 = (Base‘𝐶)
17 eqid 2621 . . . . . . . . . . 11 (0g𝑅) = (0g𝑅)
1814, 15, 16, 17pmatcoe1fsupp 20425 . . . . . . . . . 10 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)))
19 oveq1 6611 . . . . . . . . . . . . . . . . 17 (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = ((0g𝑅) × (𝑥 𝑋)))
20 pmatcollpw1.m . . . . . . . . . . . . . . . . . . . . 21 × = ( ·𝑠𝑃)
2120a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → × = ( ·𝑠𝑃))
2214ply1sca 19542 . . . . . . . . . . . . . . . . . . . . . 22 (𝑅 ∈ Ring → 𝑅 = (Scalar‘𝑃))
23223ad2ant2 1081 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → 𝑅 = (Scalar‘𝑃))
2423fveq2d 6152 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
25 eqidd 2622 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑥 𝑋) = (𝑥 𝑋))
2621, 24, 25oveq123d 6625 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2726ad3antrrr 765 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)))
2823eqcomd 2627 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (Scalar‘𝑃) = 𝑅)
2928ad3antrrr 765 . . . . . . . . . . . . . . . . . . . 20 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (Scalar‘𝑃) = 𝑅)
3029fveq2d 6152 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (0g‘(Scalar‘𝑃)) = (0g𝑅))
3130oveq1d 6619 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑥 𝑋)) = ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)))
32 simpl2 1063 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑅 ∈ Ring)
33 pmatcollpw1.x . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = (var1𝑅)
34 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 (mulGrp‘𝑃) = (mulGrp‘𝑃)
35 pmatcollpw1.e . . . . . . . . . . . . . . . . . . . . . . . 24 = (.g‘(mulGrp‘𝑃))
36 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 (Base‘𝑃) = (Base‘𝑃)
3714, 33, 34, 35, 36ply1moncl 19560 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑅 ∈ Ring ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
38373ad2antl2 1222 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑥 𝑋) ∈ (Base‘𝑃))
3932, 38jca 554 . . . . . . . . . . . . . . . . . . . . 21 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
4039adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
4140adantr 481 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)))
42 eqid 2621 . . . . . . . . . . . . . . . . . . . 20 ( ·𝑠𝑃) = ( ·𝑠𝑃)
4314, 36, 42, 17ply10s0 19545 . . . . . . . . . . . . . . . . . . 19 ((𝑅 ∈ Ring ∧ (𝑥 𝑋) ∈ (Base‘𝑃)) → ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)) = (0g𝑃))
4441, 43syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅)( ·𝑠𝑃)(𝑥 𝑋)) = (0g𝑃))
4527, 31, 443eqtrd 2659 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → ((0g𝑅) × (𝑥 𝑋)) = (0g𝑃))
4619, 45sylan9eqr 2677 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) ∧ ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃))
4746ex 450 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ 𝑖𝑁) ∧ 𝑗𝑁) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4847anasss 678 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
4948ralimdvva 2958 . . . . . . . . . . . . 13 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (∀𝑖𝑁𝑗𝑁 ((coe1‘(𝑖𝑀𝑗))‘𝑥) = (0g𝑅) → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
5049imim2d 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))))
5150ralimdva 2956 . . . . . . . . . . 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))))
5251reximdv 3010 . . . . . . . . . 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))))
5318, 52mpd 15 . . . . . . . . 9 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
54 simpl3 1064 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑀𝐵)
55 simpr 477 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → 𝑥 ∈ ℕ0)
5632, 54, 553jca 1240 . . . . . . . . . . . . . . . 16 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0))
5714, 15, 16decpmate 20490 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ Ring ∧ 𝑀𝐵𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5856, 57sylan 488 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (𝑖(𝑀 decompPMat 𝑥)𝑗) = ((coe1‘(𝑖𝑀𝑗))‘𝑥))
5958oveq1d 6619 . . . . . . . . . . . . . 14 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)))
6059eqeq1d 2623 . . . . . . . . . . . . 13 ((((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) ∧ (𝑖𝑁𝑗𝑁)) → (((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
61602ralbidva 2982 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ ∀𝑖𝑁𝑗𝑁 (((coe1‘(𝑖𝑀𝑗))‘𝑥) × (𝑥 𝑋)) = (0g𝑃)))
6261imbi2d 330 . . . . . . . . . . 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))))
6362ralbidva 2979 . . . . . . . . . 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))))
6463rexbidv 3045 . . . . . . . . 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))))
6553, 64mpbird 247 . . . . . . . 8 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))
66 eqid 2621 . . . . . . . . . . . . 13 𝑁 = 𝑁
6766biantrur 527 . . . . . . . . . . . 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))))
6866biantrur 527 . . . . . . . . . . . . . 14 (∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃) ↔ (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)))
6968bicomi 214 . . . . . . . . . . . . 13 ((𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
7069ralbii 2974 . . . . . . . . . . . 12 (∀𝑖𝑁 (𝑁 = 𝑁 ∧ ∀𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃)) ↔ ∀𝑖𝑁𝑗𝑁 ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)) = (0g𝑃))
7167, 70bitr3i 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))
7271a1i 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)))
7372imbi2d 330 . . . . . . . . 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))))
7473rexralbidv 3051 . . . . . . . 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))))
7565, 74mpbird 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)))))
76 mpt2eq123 6667 . . . . . . . . . 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)))
7776imim2i 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))))
7877ralimi 2947 . . . . . . . 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))))
7978reximi 3005 . . . . . . 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))))
8075, 79syl 17 . . . . . 6 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ∃𝑦 ∈ ℕ0𝑥 ∈ ℕ0 (𝑦 < 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃))))
81 eqidd 2622 . . . . . . . . . . 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)))))
82 oveq2 6612 . . . . . . . . . . . . . . 15 (𝑛 = 𝑥 → (𝑀 decompPMat 𝑛) = (𝑀 decompPMat 𝑥))
8382oveqd 6621 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑖(𝑀 decompPMat 𝑛)𝑗) = (𝑖(𝑀 decompPMat 𝑥)𝑗))
84 oveq1 6611 . . . . . . . . . . . . . 14 (𝑛 = 𝑥 → (𝑛 𝑋) = (𝑥 𝑋))
8583, 84oveq12d 6622 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)) = ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋)))
8685mpt2eq3dv 6674 . . . . . . . . . . . 12 (𝑛 = 𝑥 → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))) = (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))))
8786adantl 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))))
88 id 22 . . . . . . . . . . . . . . 15 (𝑁 ∈ Fin → 𝑁 ∈ Fin)
8988ancri 574 . . . . . . . . . . . . . 14 (𝑁 ∈ Fin → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
90893ad2ant1 1080 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
9190adantr 481 . . . . . . . . . . . 12 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑁 ∈ Fin))
92 mpt2exga 7191 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑁 ∈ Fin) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) ∈ V)
9391, 92syl 17 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑥)𝑗) × (𝑥 𝑋))) ∈ V)
9481, 87, 55, 93fvmptd 6245 . . . . . . . . . 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))))
9514ply1ring 19537 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
9695anim2i 592 . . . . . . . . . . . . 13 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
97963adant3 1079 . . . . . . . . . . . 12 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
9897adantr 481 . . . . . . . . . . 11 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (𝑁 ∈ Fin ∧ 𝑃 ∈ Ring))
99 eqid 2621 . . . . . . . . . . . 12 (0g𝑃) = (0g𝑃)
10015, 99mat0op 20144 . . . . . . . . . . 11 ((𝑁 ∈ Fin ∧ 𝑃 ∈ Ring) → (0g𝐶) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃)))
10198, 100syl 17 . . . . . . . . . 10 (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) ∧ 𝑥 ∈ ℕ0) → (0g𝐶) = (𝑖𝑁, 𝑗𝑁 ↦ (0g𝑃)))
10294, 101eqeq12d 2636 . . . . . . . . 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))))
103102imbi2d 330 . . . . . . . 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)))))
104103ralbidva 2979 . . . . . . 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)))))
105104rexbidv 3045 . . . . . 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)))))
10680, 105mpbird 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)))
107 nne 2794 . . . . . . . 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))
108107imbi2i 326 . . . . . . 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)))
109108ralbii 2974 . . . . . 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)))
110109rexbii 3034 . . . . 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)))
111106, 110sylibr 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)))
112 rabssnn0fi 12725 . . . 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)))
113111, 112sylibr 224 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → {𝑥 ∈ ℕ0 ∣ ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))‘𝑥) ≠ (0g𝐶)} ∈ Fin)
11413, 113eqeltrd 2698 . 2 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → ((𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) supp (0g𝐶)) ∈ Fin)
115 funmpt 5884 . . . 4 Fun (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋))))
116115a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → Fun (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))))
1178mptex 6440 . . . 4 (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) ∈ V
118117a1i 11 . . 3 ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝐵) → (𝑛 ∈ ℕ0 ↦ (𝑖𝑁, 𝑗𝑁 ↦ ((𝑖(𝑀 decompPMat 𝑛)𝑗) × (𝑛 𝑋)))) ∈ V)
119 funisfsupp 8224 . . 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))
120116, 118, 11, 119syl3anc 1323 . 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))
121114, 120mpbird 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 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  Vcvv 3186   class class class wbr 4613  cmpt 4673  Fun wfun 5841   Fn wfn 5842  cfv 5847  (class class class)co 6604  cmpt2 6606   supp csupp 7240  Fincfn 7899   finSupp cfsupp 8219   < clt 10018  0cn0 11236  Basecbs 15781  Scalarcsca 15865   ·𝑠 cvsca 15866  0gc0g 16021  .gcmg 17461  mulGrpcmgp 18410  Ringcrg 18468  var1cv1 19465  Poly1cpl1 19466  coe1cco1 19467   Mat cmat 20132   decompPMat cdecpmat 20486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-inf2 8482  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-ot 4157  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-se 5034  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-isom 5856  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-of 6850  df-ofr 6851  df-om 7013  df-1st 7113  df-2nd 7114  df-supp 7241  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-2o 7506  df-oadd 7509  df-er 7687  df-map 7804  df-pm 7805  df-ixp 7853  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-fsupp 8220  df-sup 8292  df-oi 8359  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-4 11025  df-5 11026  df-6 11027  df-7 11028  df-8 11029  df-9 11030  df-n0 11237  df-z 11322  df-dec 11438  df-uz 11632  df-fz 12269  df-fzo 12407  df-seq 12742  df-hash 13058  df-struct 15783  df-ndx 15784  df-slot 15785  df-base 15786  df-sets 15787  df-ress 15788  df-plusg 15875  df-mulr 15876  df-sca 15878  df-vsca 15879  df-ip 15880  df-tset 15881  df-ple 15882  df-ds 15885  df-hom 15887  df-cco 15888  df-0g 16023  df-gsum 16024  df-prds 16029  df-pws 16031  df-mre 16167  df-mrc 16168  df-acs 16170  df-mgm 17163  df-sgrp 17205  df-mnd 17216  df-mhm 17256  df-submnd 17257  df-grp 17346  df-minusg 17347  df-sbg 17348  df-mulg 17462  df-subg 17512  df-ghm 17579  df-cntz 17671  df-cmn 18116  df-abl 18117  df-mgp 18411  df-ur 18423  df-ring 18470  df-subrg 18699  df-lmod 18786  df-lss 18852  df-sra 19091  df-rgmod 19092  df-psr 19275  df-mvr 19276  df-mpl 19277  df-opsr 19279  df-psr1 19469  df-vr1 19470  df-ply1 19471  df-coe1 19472  df-dsmm 19995  df-frlm 20010  df-mat 20133  df-decpmat 20487
This theorem is referenced by:  pmatcollpw2  20502
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