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Theorem ply1degltdimlem 32996
Description: Lemma for ply1degltdim 32997. (Contributed by Thierry Arnoux, 20-Feb-2025.)
Hypotheses
Ref Expression
ply1degltdim.p 𝑃 = (Poly1𝑅)
ply1degltdim.d 𝐷 = ( deg1𝑅)
ply1degltdim.s 𝑆 = (𝐷 “ (-∞[,)𝑁))
ply1degltdim.n (𝜑𝑁 ∈ ℕ0)
ply1degltdim.r (𝜑𝑅 ∈ DivRing)
ply1degltdim.e 𝐸 = (𝑃s 𝑆)
ply1degltdimlem.f 𝐹 = (𝑛 ∈ (0..^𝑁) ↦ (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)))
Assertion
Ref Expression
ply1degltdimlem (𝜑 → ran 𝐹 ∈ (LBasis‘𝐸))
Distinct variable groups:   𝑛,𝐸   𝑛,𝐹   𝑛,𝑁   𝑃,𝑛   𝑅,𝑛   𝑆,𝑛   𝜑,𝑛
Allowed substitution hint:   𝐷(𝑛)

Proof of Theorem ply1degltdimlem
Dummy variables 𝑎 𝑖 𝑗 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ply1degltdim.p . . . . . 6 𝑃 = (Poly1𝑅)
2 eqid 2731 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
3 ply1degltdim.n . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
43ad3antrrr 727 . . . . . 6 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑁 ∈ ℕ0)
5 ply1degltdim.r . . . . . . . 8 (𝜑𝑅 ∈ DivRing)
65drngringd 20509 . . . . . . 7 (𝜑𝑅 ∈ Ring)
76ad3antrrr 727 . . . . . 6 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑅 ∈ Ring)
8 ply1degltdimlem.f . . . . . 6 𝐹 = (𝑛 ∈ (0..^𝑁) ↦ (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)))
9 eqid 2731 . . . . . 6 (0g𝑅) = (0g𝑅)
10 eqid 2731 . . . . . 6 (0g𝑃) = (0g𝑃)
11 elmapi 8847 . . . . . . . . 9 (𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁)) → 𝑎:(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
1211adantl 481 . . . . . . . 8 ((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) → 𝑎:(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
131ply1sca 21996 . . . . . . . . . . . 12 (𝑅 ∈ DivRing → 𝑅 = (Scalar‘𝑃))
145, 13syl 17 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘𝑃))
1514fveq2d 6895 . . . . . . . . . 10 (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
1615adantr 480 . . . . . . . . 9 ((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
1716feq3d 6704 . . . . . . . 8 ((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) → (𝑎:(0..^𝑁)⟶(Base‘𝑅) ↔ 𝑎:(0..^𝑁)⟶(Base‘(Scalar‘𝑃))))
1812, 17mpbird 257 . . . . . . 7 ((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) → 𝑎:(0..^𝑁)⟶(Base‘𝑅))
1918ad2antrr 723 . . . . . 6 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑎:(0..^𝑁)⟶(Base‘𝑅))
20 simpr 484 . . . . . . 7 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸))
21 ovexd 7447 . . . . . . . 8 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → (0..^𝑁) ∈ V)
221, 5ply1lvec 32913 . . . . . . . . . . . 12 (𝜑𝑃 ∈ LVec)
2322lveclmodd 20863 . . . . . . . . . . 11 (𝜑𝑃 ∈ LMod)
24 ply1degltdim.d . . . . . . . . . . . 12 𝐷 = ( deg1𝑅)
25 ply1degltdim.s . . . . . . . . . . . 12 𝑆 = (𝐷 “ (-∞[,)𝑁))
261, 24, 25, 3, 6ply1degltlss 32943 . . . . . . . . . . 11 (𝜑𝑆 ∈ (LSubSp‘𝑃))
27 eqid 2731 . . . . . . . . . . . 12 (LSubSp‘𝑃) = (LSubSp‘𝑃)
2827lsssubg 20713 . . . . . . . . . . 11 ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃)) → 𝑆 ∈ (SubGrp‘𝑃))
2923, 26, 28syl2anc 583 . . . . . . . . . 10 (𝜑𝑆 ∈ (SubGrp‘𝑃))
30 subgsubm 19065 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝑃) → 𝑆 ∈ (SubMnd‘𝑃))
3129, 30syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ (SubMnd‘𝑃))
3231ad3antrrr 727 . . . . . . . 8 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑆 ∈ (SubMnd‘𝑃))
33 eqid 2731 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
3424, 1, 33deg1xrf 25835 . . . . . . . . . . . . . 14 𝐷:(Base‘𝑃)⟶ℝ*
35 ffn 6717 . . . . . . . . . . . . . 14 (𝐷:(Base‘𝑃)⟶ℝ*𝐷 Fn (Base‘𝑃))
3634, 35mp1i 13 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷 Fn (Base‘𝑃))
3723ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑃 ∈ LMod)
38 simplr 766 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑘 ∈ (Base‘(Scalar‘𝑃)))
3933, 27lssss 20692 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (LSubSp‘𝑃) → 𝑆 ⊆ (Base‘𝑃))
4026, 39syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ⊆ (Base‘𝑃))
41 ply1degltdim.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (𝑃s 𝑆)
4241, 33ressbas2 17187 . . . . . . . . . . . . . . . . . 18 (𝑆 ⊆ (Base‘𝑃) → 𝑆 = (Base‘𝐸))
4340, 42syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑆 = (Base‘𝐸))
4443, 40eqsstrrd 4021 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐸) ⊆ (Base‘𝑃))
4544sselda 3982 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝑃))
4645adantlr 712 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝑃))
47 eqid 2731 . . . . . . . . . . . . . . 15 (Scalar‘𝑃) = (Scalar‘𝑃)
48 eqid 2731 . . . . . . . . . . . . . . 15 ( ·𝑠𝑃) = ( ·𝑠𝑃)
49 eqid 2731 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
5033, 47, 48, 49lmodvscl 20633 . . . . . . . . . . . . . 14 ((𝑃 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑘( ·𝑠𝑃)𝑥) ∈ (Base‘𝑃))
5137, 38, 46, 50syl3anc 1370 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·𝑠𝑃)𝑥) ∈ (Base‘𝑃))
52 mnfxr 11276 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
5352a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → -∞ ∈ ℝ*)
543nn0red 12538 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℝ)
5554rexrd 11269 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℝ*)
5655ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑁 ∈ ℝ*)
5734a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷:(Base‘𝑃)⟶ℝ*)
5857, 51ffvelcdmd 7087 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·𝑠𝑃)𝑥)) ∈ ℝ*)
5958mnfled 13120 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → -∞ ≤ (𝐷‘(𝑘( ·𝑠𝑃)𝑥)))
6057, 46ffvelcdmd 7087 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷𝑥) ∈ ℝ*)
616ad2antrr 723 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑅 ∈ Ring)
6215ad2antrr 723 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
6338, 62eleqtrrd 2835 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑘 ∈ (Base‘𝑅))
641, 24, 61, 33, 2, 48, 63, 46deg1vscale 25858 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·𝑠𝑃)𝑥)) ≤ (𝐷𝑥))
65 simpll 764 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝜑)
66 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝐸))
6743ad2antrr 723 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑆 = (Base‘𝐸))
6866, 67eleqtrrd 2835 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥𝑆)
6952a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑆) → -∞ ∈ ℝ*)
7055adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑆) → 𝑁 ∈ ℝ*)
7134, 35mp1i 13 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑆) → 𝐷 Fn (Base‘𝑃))
72 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝑆) → 𝑥𝑆)
7372, 25eleqtrdi 2842 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑆) → 𝑥 ∈ (𝐷 “ (-∞[,)𝑁)))
74 elpreima 7059 . . . . . . . . . . . . . . . . . . 19 (𝐷 Fn (Base‘𝑃) → (𝑥 ∈ (𝐷 “ (-∞[,)𝑁)) ↔ (𝑥 ∈ (Base‘𝑃) ∧ (𝐷𝑥) ∈ (-∞[,)𝑁))))
7574simplbda 499 . . . . . . . . . . . . . . . . . 18 ((𝐷 Fn (Base‘𝑃) ∧ 𝑥 ∈ (𝐷 “ (-∞[,)𝑁))) → (𝐷𝑥) ∈ (-∞[,)𝑁))
7671, 73, 75syl2anc 583 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑆) → (𝐷𝑥) ∈ (-∞[,)𝑁))
77 elico1 13372 . . . . . . . . . . . . . . . . . . 19 ((-∞ ∈ ℝ*𝑁 ∈ ℝ*) → ((𝐷𝑥) ∈ (-∞[,)𝑁) ↔ ((𝐷𝑥) ∈ ℝ* ∧ -∞ ≤ (𝐷𝑥) ∧ (𝐷𝑥) < 𝑁)))
7877biimpa 476 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ*𝑁 ∈ ℝ*) ∧ (𝐷𝑥) ∈ (-∞[,)𝑁)) → ((𝐷𝑥) ∈ ℝ* ∧ -∞ ≤ (𝐷𝑥) ∧ (𝐷𝑥) < 𝑁))
7978simp3d 1143 . . . . . . . . . . . . . . . . 17 (((-∞ ∈ ℝ*𝑁 ∈ ℝ*) ∧ (𝐷𝑥) ∈ (-∞[,)𝑁)) → (𝐷𝑥) < 𝑁)
8069, 70, 76, 79syl21anc 835 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑆) → (𝐷𝑥) < 𝑁)
8165, 68, 80syl2anc 583 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷𝑥) < 𝑁)
8258, 60, 56, 64, 81xrlelttrd 13144 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·𝑠𝑃)𝑥)) < 𝑁)
8353, 56, 58, 59, 82elicod 13379 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·𝑠𝑃)𝑥)) ∈ (-∞[,)𝑁))
8436, 51, 83elpreimad 7060 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·𝑠𝑃)𝑥) ∈ (𝐷 “ (-∞[,)𝑁)))
8584, 25eleqtrrdi 2843 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·𝑠𝑃)𝑥) ∈ 𝑆)
8685anasss 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑥 ∈ (Base‘𝐸))) → (𝑘( ·𝑠𝑃)𝑥) ∈ 𝑆)
8786ad5ant15 756 . . . . . . . . 9 (((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑥 ∈ (Base‘𝐸))) → (𝑘( ·𝑠𝑃)𝑥) ∈ 𝑆)
8812ad2antrr 723 . . . . . . . . 9 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑎:(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
8934, 35mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝐷 Fn (Base‘𝑃))
90 eqid 2731 . . . . . . . . . . . . . . . 16 (mulGrp‘𝑃) = (mulGrp‘𝑃)
9190, 33mgpbas 20035 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
92 eqid 2731 . . . . . . . . . . . . . . 15 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
931ply1ring 21991 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
9490ringmgp 20134 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
956, 93, 943syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
9695adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → (mulGrp‘𝑃) ∈ Mnd)
97 elfzonn0 13682 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ0)
9897adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
99 eqid 2731 . . . . . . . . . . . . . . . . . 18 (var1𝑅) = (var1𝑅)
10099, 1, 33vr1cl 21961 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → (var1𝑅) ∈ (Base‘𝑃))
1016, 100syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (var1𝑅) ∈ (Base‘𝑃))
102101adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → (var1𝑅) ∈ (Base‘𝑃))
10391, 92, 96, 98, 102mulgnn0cld 19012 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
10452a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → -∞ ∈ ℝ*)
10555adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ*)
10624, 1, 33deg1xrcl 25836 . . . . . . . . . . . . . . . 16 ((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ ℝ*)
107103, 106syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ ℝ*)
108107mnfled 13120 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → -∞ ≤ (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))))
10997nn0red 12538 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℝ)
110109rexrd 11269 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℝ*)
111110adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ*)
11224, 1, 99, 90, 92deg1pwle 25873 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ≤ 𝑛)
1136, 97, 112syl2an 595 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ≤ 𝑛)
114 elfzolt2 13646 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
115114adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
116107, 111, 105, 113, 115xrlelttrd 13144 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) < 𝑁)
117104, 105, 107, 108, 116elicod 13379 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ (-∞[,)𝑁))
11889, 103, 117elpreimad 7060 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (𝐷 “ (-∞[,)𝑁)))
119118, 25eleqtrrdi 2843 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ 𝑆)
12043adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑆 = (Base‘𝐸))
121119, 120eleqtrd 2834 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝐸))
122121, 8fmptd 7115 . . . . . . . . . 10 (𝜑𝐹:(0..^𝑁)⟶(Base‘𝐸))
123122ad3antrrr 727 . . . . . . . . 9 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝐹:(0..^𝑁)⟶(Base‘𝐸))
124 inidm 4218 . . . . . . . . 9 ((0..^𝑁) ∩ (0..^𝑁)) = (0..^𝑁)
12587, 88, 123, 21, 21, 124off 7692 . . . . . . . 8 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → (𝑎f ( ·𝑠𝑃)𝐹):(0..^𝑁)⟶𝑆)
12621, 32, 125, 41gsumsubm 18753 . . . . . . 7 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)))
127 ringmnd 20138 . . . . . . . . . 10 (𝑃 ∈ Ring → 𝑃 ∈ Mnd)
1286, 93, 1273syl 18 . . . . . . . . 9 (𝜑𝑃 ∈ Mnd)
12934, 35mp1i 13 . . . . . . . . . . 11 (𝜑𝐷 Fn (Base‘𝑃))
13033, 10mndidcl 18675 . . . . . . . . . . . 12 (𝑃 ∈ Mnd → (0g𝑃) ∈ (Base‘𝑃))
131128, 130syl 17 . . . . . . . . . . 11 (𝜑 → (0g𝑃) ∈ (Base‘𝑃))
13252a1i 11 . . . . . . . . . . . 12 (𝜑 → -∞ ∈ ℝ*)
13324, 1, 33deg1xrcl 25836 . . . . . . . . . . . . 13 ((0g𝑃) ∈ (Base‘𝑃) → (𝐷‘(0g𝑃)) ∈ ℝ*)
134131, 133syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐷‘(0g𝑃)) ∈ ℝ*)
135134mnfled 13120 . . . . . . . . . . . 12 (𝜑 → -∞ ≤ (𝐷‘(0g𝑃)))
13624, 1, 10deg1z 25841 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → (𝐷‘(0g𝑃)) = -∞)
1376, 136syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐷‘(0g𝑃)) = -∞)
13854mnfltd 13109 . . . . . . . . . . . . 13 (𝜑 → -∞ < 𝑁)
139137, 138eqbrtrd 5170 . . . . . . . . . . . 12 (𝜑 → (𝐷‘(0g𝑃)) < 𝑁)
140132, 55, 134, 135, 139elicod 13379 . . . . . . . . . . 11 (𝜑 → (𝐷‘(0g𝑃)) ∈ (-∞[,)𝑁))
141129, 131, 140elpreimad 7060 . . . . . . . . . 10 (𝜑 → (0g𝑃) ∈ (𝐷 “ (-∞[,)𝑁)))
142141, 25eleqtrrdi 2843 . . . . . . . . 9 (𝜑 → (0g𝑃) ∈ 𝑆)
14341, 33, 10ress0g 18688 . . . . . . . . 9 ((𝑃 ∈ Mnd ∧ (0g𝑃) ∈ 𝑆𝑆 ⊆ (Base‘𝑃)) → (0g𝑃) = (0g𝐸))
144128, 142, 40, 143syl3anc 1370 . . . . . . . 8 (𝜑 → (0g𝑃) = (0g𝐸))
145144ad3antrrr 727 . . . . . . 7 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → (0g𝑃) = (0g𝐸))
14620, 126, 1453eqtr4d 2781 . . . . . 6 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝑃))
1471, 2, 4, 7, 8, 9, 10, 19, 146ply1gsumz 32945 . . . . 5 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑎 = ((0..^𝑁) × {(0g𝑅)}))
14814fveq2d 6895 . . . . . . . 8 (𝜑 → (0g𝑅) = (0g‘(Scalar‘𝑃)))
149148sneqd 4640 . . . . . . 7 (𝜑 → {(0g𝑅)} = {(0g‘(Scalar‘𝑃))})
150149xpeq2d 5706 . . . . . 6 (𝜑 → ((0..^𝑁) × {(0g𝑅)}) = ((0..^𝑁) × {(0g‘(Scalar‘𝑃))}))
151150ad3antrrr 727 . . . . 5 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → ((0..^𝑁) × {(0g𝑅)}) = ((0..^𝑁) × {(0g‘(Scalar‘𝑃))}))
152147, 151eqtrd 2771 . . . 4 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑎 = ((0..^𝑁) × {(0g‘(Scalar‘𝑃))}))
153152expl 457 . . 3 ((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) → ((𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑎 = ((0..^𝑁) × {(0g‘(Scalar‘𝑃))})))
154153ralrimiva 3145 . 2 (𝜑 → ∀𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))((𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑎 = ((0..^𝑁) × {(0g‘(Scalar‘𝑃))})))
155119, 8fmptd 7115 . . . . . 6 (𝜑𝐹:(0..^𝑁)⟶𝑆)
156155frnd 6725 . . . . 5 (𝜑 → ran 𝐹𝑆)
157 eqid 2731 . . . . . 6 (LSpan‘𝑃) = (LSpan‘𝑃)
15827, 157lspssp 20744 . . . . 5 ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃) ∧ ran 𝐹𝑆) → ((LSpan‘𝑃)‘ran 𝐹) ⊆ 𝑆)
15923, 26, 156, 158syl3anc 1370 . . . 4 (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) ⊆ 𝑆)
160 fvexd 6906 . . . . . . . 8 ((𝜑𝑥𝑆) → (Base‘(Scalar‘𝑃)) ∈ V)
161 ovexd 7447 . . . . . . . 8 ((𝜑𝑥𝑆) → (0..^𝑁) ∈ V)
16240sselda 3982 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑥 ∈ (Base‘𝑃))
163 eqid 2731 . . . . . . . . . . . 12 (coe1𝑥) = (coe1𝑥)
164163, 33, 1, 2coe1f 21955 . . . . . . . . . . 11 (𝑥 ∈ (Base‘𝑃) → (coe1𝑥):ℕ0⟶(Base‘𝑅))
165162, 164syl 17 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (coe1𝑥):ℕ0⟶(Base‘𝑅))
16615adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
167166feq3d 6704 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ((coe1𝑥):ℕ0⟶(Base‘𝑅) ↔ (coe1𝑥):ℕ0⟶(Base‘(Scalar‘𝑃))))
168165, 167mpbid 231 . . . . . . . . 9 ((𝜑𝑥𝑆) → (coe1𝑥):ℕ0⟶(Base‘(Scalar‘𝑃)))
169 fzo0ssnn0 13718 . . . . . . . . . 10 (0..^𝑁) ⊆ ℕ0
170169a1i 11 . . . . . . . . 9 ((𝜑𝑥𝑆) → (0..^𝑁) ⊆ ℕ0)
171168, 170fssresd 6758 . . . . . . . 8 ((𝜑𝑥𝑆) → ((coe1𝑥) ↾ (0..^𝑁)):(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
172 elmapg 8837 . . . . . . . . 9 (((Base‘(Scalar‘𝑃)) ∈ V ∧ (0..^𝑁) ∈ V) → (((coe1𝑥) ↾ (0..^𝑁)) ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁)) ↔ ((coe1𝑥) ↾ (0..^𝑁)):(0..^𝑁)⟶(Base‘(Scalar‘𝑃))))
173172biimpar 477 . . . . . . . 8 ((((Base‘(Scalar‘𝑃)) ∈ V ∧ (0..^𝑁) ∈ V) ∧ ((coe1𝑥) ↾ (0..^𝑁)):(0..^𝑁)⟶(Base‘(Scalar‘𝑃))) → ((coe1𝑥) ↾ (0..^𝑁)) ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁)))
174160, 161, 171, 173syl21anc 835 . . . . . . 7 ((𝜑𝑥𝑆) → ((coe1𝑥) ↾ (0..^𝑁)) ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁)))
175 breq1 5151 . . . . . . . . 9 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → (𝑎 finSupp (0g‘(Scalar‘𝑃)) ↔ ((coe1𝑥) ↾ (0..^𝑁)) finSupp (0g‘(Scalar‘𝑃))))
176 oveq1 7419 . . . . . . . . . . 11 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → (𝑎f ( ·𝑠𝑃)𝐹) = (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹))
177176oveq2d 7428 . . . . . . . . . 10 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹)))
178177eqeq2d 2742 . . . . . . . . 9 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → (𝑥 = (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹)) ↔ 𝑥 = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹))))
179175, 178anbi12d 630 . . . . . . . 8 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → ((𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹))) ↔ (((coe1𝑥) ↾ (0..^𝑁)) finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹)))))
180179adantl 481 . . . . . . 7 (((𝜑𝑥𝑆) ∧ 𝑎 = ((coe1𝑥) ↾ (0..^𝑁))) → ((𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹))) ↔ (((coe1𝑥) ↾ (0..^𝑁)) finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹)))))
181165ffund 6721 . . . . . . . . 9 ((𝜑𝑥𝑆) → Fun (coe1𝑥))
182 fzofi 13944 . . . . . . . . . 10 (0..^𝑁) ∈ Fin
183182a1i 11 . . . . . . . . 9 ((𝜑𝑥𝑆) → (0..^𝑁) ∈ Fin)
184 fvexd 6906 . . . . . . . . 9 ((𝜑𝑥𝑆) → (0g‘(Scalar‘𝑃)) ∈ V)
185181, 183, 184resfifsupp 9396 . . . . . . . 8 ((𝜑𝑥𝑆) → ((coe1𝑥) ↾ (0..^𝑁)) finSupp (0g‘(Scalar‘𝑃)))
186 ringcmn 20171 . . . . . . . . . . . 12 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
1876, 93, 1863syl 18 . . . . . . . . . . 11 (𝜑𝑃 ∈ CMnd)
188187adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑆) → 𝑃 ∈ CMnd)
189 nn0ex 12483 . . . . . . . . . . 11 0 ∈ V
190189a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ℕ0 ∈ V)
19123ad2antrr 723 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → 𝑃 ∈ LMod)
192168ffvelcdmda 7086 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → ((coe1𝑥)‘𝑖) ∈ (Base‘(Scalar‘𝑃)))
1936ad2antrr 723 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → 𝑅 ∈ Ring)
194193, 93, 943syl 18 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
195 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
196193, 100syl 17 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → (var1𝑅) ∈ (Base‘𝑃))
19791, 92, 194, 195, 196mulgnn0cld 19012 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
19833, 47, 48, 49lmodvscl 20633 . . . . . . . . . . . 12 ((𝑃 ∈ LMod ∧ ((coe1𝑥)‘𝑖) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃)) → (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ (Base‘𝑃))
199191, 192, 197, 198syl3anc 1370 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ (Base‘𝑃))
200 eqid 2731 . . . . . . . . . . 11 (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) = (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))
201199, 200fmptd 7115 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))):ℕ0⟶(Base‘𝑃))
202 nfv 1916 . . . . . . . . . . . 12 𝑖(𝜑𝑥𝑆)
203202, 199, 200fnmptd 6691 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) Fn ℕ0)
204 fveq2 6891 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((coe1𝑥)‘𝑖) = ((coe1𝑥)‘𝑗))
205 oveq1 7419 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)))
206204, 205oveq12d 7430 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))) = (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
207 simplr 766 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑗 ∈ ℕ0)
208 ovexd 7447 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ V)
209200, 206, 207, 208fvmptd3 7021 . . . . . . . . . . . 12 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗) = (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
210162ad2antrr 723 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑥 ∈ (Base‘𝑃))
211 icossxr 13414 . . . . . . . . . . . . . . . . 17 (-∞[,)𝑁) ⊆ ℝ*
212211, 76sselid 3980 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑆) → (𝐷𝑥) ∈ ℝ*)
213212ad2antrr 723 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (𝐷𝑥) ∈ ℝ*)
21455ad3antrrr 727 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑁 ∈ ℝ*)
215207nn0red 12538 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑗 ∈ ℝ)
216215rexrd 11269 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑗 ∈ ℝ*)
21780ad2antrr 723 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (𝐷𝑥) < 𝑁)
218 simpr 484 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑁𝑗)
219213, 214, 216, 217, 218xrltletrd 13145 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (𝐷𝑥) < 𝑗)
22024, 1, 33, 9, 163deg1lt 25851 . . . . . . . . . . . . . 14 ((𝑥 ∈ (Base‘𝑃) ∧ 𝑗 ∈ ℕ0 ∧ (𝐷𝑥) < 𝑗) → ((coe1𝑥)‘𝑗) = (0g𝑅))
221210, 207, 219, 220syl3anc 1370 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((coe1𝑥)‘𝑗) = (0g𝑅))
222221oveq1d 7427 . . . . . . . . . . . 12 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = ((0g𝑅)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
223148ad3antrrr 727 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
224223oveq1d 7427 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((0g𝑅)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
22523ad3antrrr 727 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑃 ∈ LMod)
22695ad3antrrr 727 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (mulGrp‘𝑃) ∈ Mnd)
227101ad3antrrr 727 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (var1𝑅) ∈ (Base‘𝑃))
22891, 92, 226, 207, 227mulgnn0cld 19012 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
229 eqid 2731 . . . . . . . . . . . . . . 15 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
23033, 47, 48, 229, 10lmod0vs 20650 . . . . . . . . . . . . . 14 ((𝑃 ∈ LMod ∧ (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃)) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
231225, 228, 230syl2anc 583 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
232224, 231eqtrd 2771 . . . . . . . . . . . 12 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((0g𝑅)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
233209, 222, 2323eqtrd 2775 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗) = (0g𝑃))
2343nn0zd 12589 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
235234adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑁 ∈ ℤ)
236203, 233, 235suppssnn0 32285 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) supp (0g𝑃)) ⊆ (0..^𝑁))
237190mptexd 7228 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∈ V)
238203fnfund 6650 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → Fun (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))))
239 fvexd 6906 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (0g𝑃) ∈ V)
240 suppssfifsupp 9382 . . . . . . . . . . 11 ((((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∈ V ∧ Fun (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∧ (0g𝑃) ∈ V) ∧ ((0..^𝑁) ∈ Fin ∧ ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) supp (0g𝑃)) ⊆ (0..^𝑁))) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) finSupp (0g𝑃))
241237, 238, 239, 183, 236, 240syl32anc 1377 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) finSupp (0g𝑃))
24233, 10, 188, 190, 201, 236, 241gsumres 19823 . . . . . . . . 9 ((𝜑𝑥𝑆) → (𝑃 Σg ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁))) = (𝑃 Σg (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))))
243 fvexd 6906 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → (coe1𝑥) ∈ V)
244 ovexd 7447 . . . . . . . . . . . . . 14 (𝜑 → (0..^𝑁) ∈ V)
245155, 244fexd 7231 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
246245adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → 𝐹 ∈ V)
247 offres 7974 . . . . . . . . . . . 12 (((coe1𝑥) ∈ V ∧ 𝐹 ∈ V) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) ↾ (0..^𝑁)) = (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)(𝐹 ↾ (0..^𝑁))))
248243, 246, 247syl2anc 583 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) ↾ (0..^𝑁)) = (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)(𝐹 ↾ (0..^𝑁))))
249165ffnd 6718 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑆) → (coe1𝑥) Fn ℕ0)
250155ffnd 6718 . . . . . . . . . . . . . . . 16 (𝜑𝐹 Fn (0..^𝑁))
251250adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑆) → 𝐹 Fn (0..^𝑁))
252 sseqin2 4215 . . . . . . . . . . . . . . . 16 ((0..^𝑁) ⊆ ℕ0 ↔ (ℕ0 ∩ (0..^𝑁)) = (0..^𝑁))
253169, 252mpbi 229 . . . . . . . . . . . . . . 15 (ℕ0 ∩ (0..^𝑁)) = (0..^𝑁)
254 eqidd 2732 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) → ((coe1𝑥)‘𝑗) = ((coe1𝑥)‘𝑗))
255 oveq1 7419 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)))
256 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
257 ovexd 7447 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ V)
2588, 255, 256, 257fvmptd3 7021 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹𝑗) = (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)))
259249, 251, 190, 161, 253, 254, 258ofval 7685 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹)‘𝑗) = (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
260169, 256sselid 3980 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℕ0)
261 ovexd 7447 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ V)
262200, 206, 260, 261fvmptd3 7021 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗) = (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
263259, 262eqtr4d 2774 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗))
264263ralrimiva 3145 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → ∀𝑗 ∈ (0..^𝑁)(((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗))
265249, 251, 190, 161, 253offn 7687 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → ((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) Fn (0..^𝑁))
266 ssidd 4005 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → (0..^𝑁) ⊆ (0..^𝑁))
267 fvreseq0 7039 . . . . . . . . . . . . 13 (((((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) Fn (0..^𝑁) ∧ (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) Fn ℕ0) ∧ ((0..^𝑁) ⊆ (0..^𝑁) ∧ (0..^𝑁) ⊆ ℕ0)) → ((((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) ↾ (0..^𝑁)) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁)) ↔ ∀𝑗 ∈ (0..^𝑁)(((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗)))
268265, 203, 266, 170, 267syl22anc 836 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → ((((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) ↾ (0..^𝑁)) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁)) ↔ ∀𝑗 ∈ (0..^𝑁)(((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗)))
269264, 268mpbird 257 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) ↾ (0..^𝑁)) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁)))
270 fnresdm 6669 . . . . . . . . . . . . . 14 (𝐹 Fn (0..^𝑁) → (𝐹 ↾ (0..^𝑁)) = 𝐹)
271250, 270syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ↾ (0..^𝑁)) = 𝐹)
272271adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → (𝐹 ↾ (0..^𝑁)) = 𝐹)
273272oveq2d 7428 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)(𝐹 ↾ (0..^𝑁))) = (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹))
274248, 269, 2733eqtr3rd 2780 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁)))
275274oveq2d 7428 . . . . . . . . 9 ((𝜑𝑥𝑆) → (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹)) = (𝑃 Σg ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁))))
2766adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑆) → 𝑅 ∈ Ring)
2771, 99, 33, 48, 90, 92, 163ply1coe 22041 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 = (𝑃 Σg (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))))
278276, 162, 277syl2anc 583 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝑥 = (𝑃 Σg (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))))
279242, 275, 2783eqtr4rd 2782 . . . . . . . 8 ((𝜑𝑥𝑆) → 𝑥 = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹)))
280185, 279jca 511 . . . . . . 7 ((𝜑𝑥𝑆) → (((coe1𝑥) ↾ (0..^𝑁)) finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹))))
281174, 180, 280rspcedvd 3614 . . . . . 6 ((𝜑𝑥𝑆) → ∃𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))(𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹))))
282103, 8fmptd 7115 . . . . . . . 8 (𝜑𝐹:(0..^𝑁)⟶(Base‘𝑃))
283157, 33, 49, 47, 229, 48, 282, 23, 244ellspd 21577 . . . . . . 7 (𝜑 → (𝑥 ∈ ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))(𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹)))))
284283adantr 480 . . . . . 6 ((𝜑𝑥𝑆) → (𝑥 ∈ ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))(𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹)))))
285281, 284mpbird 257 . . . . 5 ((𝜑𝑥𝑆) → 𝑥 ∈ ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
286 imadmrn 6069 . . . . . . . 8 (𝐹 “ dom 𝐹) = ran 𝐹
287155fdmd 6728 . . . . . . . . 9 (𝜑 → dom 𝐹 = (0..^𝑁))
288287imaeq2d 6059 . . . . . . . 8 (𝜑 → (𝐹 “ dom 𝐹) = (𝐹 “ (0..^𝑁)))
289286, 288eqtr3id 2785 . . . . . . 7 (𝜑 → ran 𝐹 = (𝐹 “ (0..^𝑁)))
290289fveq2d 6895 . . . . . 6 (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
291290adantr 480 . . . . 5 ((𝜑𝑥𝑆) → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
292285, 291eleqtrrd 2835 . . . 4 ((𝜑𝑥𝑆) → 𝑥 ∈ ((LSpan‘𝑃)‘ran 𝐹))
293159, 292eqelssd 4003 . . 3 (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = 𝑆)
294 eqid 2731 . . . . 5 (LSpan‘𝐸) = (LSpan‘𝐸)
29541, 157, 294, 27lsslsp 20771 . . . 4 ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃) ∧ ran 𝐹𝑆) → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝐸)‘ran 𝐹))
29623, 26, 156, 295syl3anc 1370 . . 3 (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝐸)‘ran 𝐹))
297293, 296, 433eqtr3d 2779 . 2 (𝜑 → ((LSpan‘𝐸)‘ran 𝐹) = (Base‘𝐸))
298 eqid 2731 . . 3 (Base‘𝐸) = (Base‘𝐸)
29924fvexi 6905 . . . . . . 7 𝐷 ∈ V
300 cnvexg 7919 . . . . . . 7 (𝐷 ∈ V → 𝐷 ∈ V)
301 imaexg 7910 . . . . . . 7 (𝐷 ∈ V → (𝐷 “ (-∞[,)𝑁)) ∈ V)
302299, 300, 301mp2b 10 . . . . . 6 (𝐷 “ (-∞[,)𝑁)) ∈ V
30325, 302eqeltri 2828 . . . . 5 𝑆 ∈ V
30441, 47resssca 17293 . . . . 5 (𝑆 ∈ V → (Scalar‘𝑃) = (Scalar‘𝐸))
305303, 304ax-mp 5 . . . 4 (Scalar‘𝑃) = (Scalar‘𝐸)
306305fveq2i 6894 . . 3 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝐸))
307 eqid 2731 . . 3 (Scalar‘𝐸) = (Scalar‘𝐸)
30841, 48ressvsca 17294 . . . 4 (𝑆 ∈ V → ( ·𝑠𝑃) = ( ·𝑠𝐸))
309303, 308ax-mp 5 . . 3 ( ·𝑠𝑃) = ( ·𝑠𝐸)
310 eqid 2731 . . 3 (0g𝐸) = (0g𝐸)
311305fveq2i 6894 . . 3 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝐸))
312 eqid 2731 . . 3 (LBasis‘𝐸) = (LBasis‘𝐸)
31341, 27lsslvec 20865 . . . . 5 ((𝑃 ∈ LVec ∧ 𝑆 ∈ (LSubSp‘𝑃)) → 𝐸 ∈ LVec)
31422, 26, 313syl2anc 583 . . . 4 (𝜑𝐸 ∈ LVec)
315314lveclmodd 20863 . . 3 (𝜑𝐸 ∈ LMod)
31614, 5eqeltrrd 2833 . . . . 5 (𝜑 → (Scalar‘𝑃) ∈ DivRing)
317 drngnzr 20521 . . . . 5 ((Scalar‘𝑃) ∈ DivRing → (Scalar‘𝑃) ∈ NzRing)
318316, 317syl 17 . . . 4 (𝜑 → (Scalar‘𝑃) ∈ NzRing)
319305, 318eqeltrrid 2837 . . 3 (𝜑 → (Scalar‘𝐸) ∈ NzRing)
320121ralrimiva 3145 . . . 4 (𝜑 → ∀𝑛 ∈ (0..^𝑁)(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝐸))
321 drngnzr 20521 . . . . . . . . . 10 (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
3225, 321syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ NzRing)
323322ad2antrr 723 . . . . . . . 8 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑅 ∈ NzRing)
32498adantr 480 . . . . . . . 8 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
325 elfzonn0 13682 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
326325adantl 481 . . . . . . . 8 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
3271, 99, 92, 323, 324, 326ply1moneq 32940 . . . . . . 7 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) ↔ 𝑛 = 𝑖))
328327biimpd 228 . . . . . 6 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) → 𝑛 = 𝑖))
329328anasss 466 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (0..^𝑁) ∧ 𝑖 ∈ (0..^𝑁))) → ((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) → 𝑛 = 𝑖))
330329ralrimivva 3199 . . . 4 (𝜑 → ∀𝑛 ∈ (0..^𝑁)∀𝑖 ∈ (0..^𝑁)((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) → 𝑛 = 𝑖))
331 oveq1 7419 . . . . 5 (𝑛 = 𝑖 → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))
3328, 331f1mpt 7263 . . . 4 (𝐹:(0..^𝑁)–1-1→(Base‘𝐸) ↔ (∀𝑛 ∈ (0..^𝑁)(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝐸) ∧ ∀𝑛 ∈ (0..^𝑁)∀𝑖 ∈ (0..^𝑁)((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) → 𝑛 = 𝑖)))
333320, 330, 332sylanbrc 582 . . 3 (𝜑𝐹:(0..^𝑁)–1-1→(Base‘𝐸))
334298, 306, 307, 309, 310, 311, 312, 294, 315, 319, 244, 333islbs5 32771 . 2 (𝜑 → (ran 𝐹 ∈ (LBasis‘𝐸) ↔ (∀𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))((𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑎 = ((0..^𝑁) × {(0g‘(Scalar‘𝑃))})) ∧ ((LSpan‘𝐸)‘ran 𝐹) = (Base‘𝐸))))
335154, 297, 334mpbir2and 710 1 (𝜑 → ran 𝐹 ∈ (LBasis‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  wrex 3069  Vcvv 3473  cin 3947  wss 3948  {csn 4628   class class class wbr 5148  cmpt 5231   × cxp 5674  ccnv 5675  dom cdm 5676  ran crn 5677  cres 5678  cima 5679  Fun wfun 6537   Fn wfn 6538  wf 6539  1-1wf1 6540  cfv 6543  (class class class)co 7412  f cof 7672   supp csupp 8150  m cmap 8824  Fincfn 8943   finSupp cfsupp 9365  0cc0 11114  -∞cmnf 11251  *cxr 11252   < clt 11253  cle 11254  0cn0 12477  cz 12563  [,)cico 13331  ..^cfzo 13632  Basecbs 17149  s cress 17178  Scalarcsca 17205   ·𝑠 cvsca 17206  0gc0g 17390   Σg cgsu 17391  Mndcmnd 18660  SubMndcsubmnd 18705  .gcmg 18987  SubGrpcsubg 19037  CMndccmn 19690  mulGrpcmgp 20029  Ringcrg 20128  NzRingcnzr 20404  DivRingcdr 20501  LModclmod 20615  LSubSpclss 20687  LSpanclspn 20727  LBasisclbs 20830  LVecclvec 20858  var1cv1 21920  Poly1cpl1 21921  coe1cco1 21922   deg1 cdg1 25805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191  ax-pre-sup 11192  ax-addf 11193  ax-mulf 11194
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  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 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-tp 4633  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7674  df-ofr 7675  df-om 7860  df-1st 7979  df-2nd 7980  df-supp 8151  df-tpos 8215  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-pm 8827  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9366  df-sup 9441  df-oi 9509  df-card 9938  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-ico 13335  df-fz 13490  df-fzo 13633  df-seq 13972  df-hash 14296  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-starv 17217  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-unif 17225  df-hom 17226  df-cco 17227  df-0g 17392  df-gsum 17393  df-prds 17398  df-pws 17400  df-mre 17535  df-mrc 17536  df-acs 17538  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-mhm 18706  df-submnd 18707  df-grp 18859  df-minusg 18860  df-sbg 18861  df-mulg 18988  df-subg 19040  df-ghm 19129  df-cntz 19223  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-srg 20082  df-ring 20130  df-cring 20131  df-oppr 20226  df-dvdsr 20249  df-unit 20250  df-nzr 20405  df-subrng 20435  df-subrg 20460  df-drng 20503  df-lmod 20617  df-lss 20688  df-lsp 20728  df-lmhm 20778  df-lbs 20831  df-lvec 20859  df-sra 20931  df-rgmod 20932  df-cnfld 21146  df-dsmm 21507  df-frlm 21522  df-uvc 21558  df-lindf 21581  df-linds 21582  df-psr 21682  df-mvr 21683  df-mpl 21684  df-opsr 21686  df-psr1 21924  df-vr1 21925  df-ply1 21926  df-coe1 21927  df-mdeg 25806  df-deg1 25807
This theorem is referenced by:  ply1degltdim  32997
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