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Theorem ply1degltdimlem 33674
Description: Lemma for ply1degltdim 33675. (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 2736 . . . . . 6 (Base‘𝑅) = (Base‘𝑅)
3 ply1degltdim.n . . . . . . 7 (𝜑𝑁 ∈ ℕ0)
43ad3antrrr 730 . . . . . 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 20738 . . . . . . 7 (𝜑𝑅 ∈ Ring)
76ad3antrrr 730 . . . . . 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 2736 . . . . . 6 (0g𝑅) = (0g𝑅)
10 eqid 2736 . . . . . 6 (0g𝑃) = (0g𝑃)
11 elmapi 8890 . . . . . . . . 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 22255 . . . . . . . . . . . 12 (𝑅 ∈ DivRing → 𝑅 = (Scalar‘𝑃))
145, 13syl 17 . . . . . . . . . . 11 (𝜑𝑅 = (Scalar‘𝑃))
1514fveq2d 6909 . . . . . . . . . 10 (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
1615adantr 480 . . . . . . . . 9 ((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
1716feq3d 6722 . . . . . . . 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 726 . . . . . 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 7467 . . . . . . . 8 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → (0..^𝑁) ∈ V)
221, 5ply1lvec 33586 . . . . . . . . . . . 12 (𝜑𝑃 ∈ LVec)
2322lveclmodd 21107 . . . . . . . . . . 11 (𝜑𝑃 ∈ LMod)
24 ply1degltdim.d . . . . . . . . . . . 12 𝐷 = (deg1𝑅)
25 ply1degltdim.s . . . . . . . . . . . 12 𝑆 = (𝐷 “ (-∞[,)𝑁))
261, 24, 25, 3, 6ply1degltlss 33618 . . . . . . . . . . 11 (𝜑𝑆 ∈ (LSubSp‘𝑃))
27 eqid 2736 . . . . . . . . . . . 12 (LSubSp‘𝑃) = (LSubSp‘𝑃)
2827lsssubg 20956 . . . . . . . . . . 11 ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃)) → 𝑆 ∈ (SubGrp‘𝑃))
2923, 26, 28syl2anc 584 . . . . . . . . . 10 (𝜑𝑆 ∈ (SubGrp‘𝑃))
30 subgsubm 19167 . . . . . . . . . 10 (𝑆 ∈ (SubGrp‘𝑃) → 𝑆 ∈ (SubMnd‘𝑃))
3129, 30syl 17 . . . . . . . . 9 (𝜑𝑆 ∈ (SubMnd‘𝑃))
3231ad3antrrr 730 . . . . . . . 8 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑆 ∈ (SubMnd‘𝑃))
33 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘𝑃)
3424, 1, 33deg1xrf 26121 . . . . . . . . . . . . . 14 𝐷:(Base‘𝑃)⟶ℝ*
35 ffn 6735 . . . . . . . . . . . . . 14 (𝐷:(Base‘𝑃)⟶ℝ*𝐷 Fn (Base‘𝑃))
3634, 35mp1i 13 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷 Fn (Base‘𝑃))
37 eqid 2736 . . . . . . . . . . . . . 14 (Scalar‘𝑃) = (Scalar‘𝑃)
38 eqid 2736 . . . . . . . . . . . . . 14 ( ·𝑠𝑃) = ( ·𝑠𝑃)
39 eqid 2736 . . . . . . . . . . . . . 14 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
4023ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑃 ∈ LMod)
41 simplr 768 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑘 ∈ (Base‘(Scalar‘𝑃)))
4233, 27lssss 20935 . . . . . . . . . . . . . . . . . . 19 (𝑆 ∈ (LSubSp‘𝑃) → 𝑆 ⊆ (Base‘𝑃))
4326, 42syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑𝑆 ⊆ (Base‘𝑃))
44 ply1degltdim.e . . . . . . . . . . . . . . . . . . 19 𝐸 = (𝑃s 𝑆)
4544, 33ressbas2 17284 . . . . . . . . . . . . . . . . . 18 (𝑆 ⊆ (Base‘𝑃) → 𝑆 = (Base‘𝐸))
4643, 45syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑆 = (Base‘𝐸))
4746, 43eqsstrrd 4018 . . . . . . . . . . . . . . . 16 (𝜑 → (Base‘𝐸) ⊆ (Base‘𝑃))
4847sselda 3982 . . . . . . . . . . . . . . 15 ((𝜑𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝑃))
4948adantlr 715 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝑃))
5033, 37, 38, 39, 40, 41, 49lmodvscld 20878 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·𝑠𝑃)𝑥) ∈ (Base‘𝑃))
51 mnfxr 11319 . . . . . . . . . . . . . . 15 -∞ ∈ ℝ*
5251a1i 11 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → -∞ ∈ ℝ*)
533nn0red 12590 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℝ)
5453rexrd 11312 . . . . . . . . . . . . . . 15 (𝜑𝑁 ∈ ℝ*)
5554ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑁 ∈ ℝ*)
5634a1i 11 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷:(Base‘𝑃)⟶ℝ*)
5756, 50ffvelcdmd 7104 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·𝑠𝑃)𝑥)) ∈ ℝ*)
5857mnfled 13179 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → -∞ ≤ (𝐷‘(𝑘( ·𝑠𝑃)𝑥)))
5956, 49ffvelcdmd 7104 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷𝑥) ∈ ℝ*)
606ad2antrr 726 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑅 ∈ Ring)
6115ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
6241, 61eleqtrrd 2843 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑘 ∈ (Base‘𝑅))
631, 24, 60, 33, 2, 38, 62, 49deg1vscale 26144 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·𝑠𝑃)𝑥)) ≤ (𝐷𝑥))
64 simpll 766 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝜑)
65 simpr 484 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝐸))
6646ad2antrr 726 . . . . . . . . . . . . . . . . 17 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑆 = (Base‘𝐸))
6765, 66eleqtrrd 2843 . . . . . . . . . . . . . . . 16 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥𝑆)
6851a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑆) → -∞ ∈ ℝ*)
6954adantr 480 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑆) → 𝑁 ∈ ℝ*)
7034, 35mp1i 13 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑆) → 𝐷 Fn (Base‘𝑃))
71 simpr 484 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑥𝑆) → 𝑥𝑆)
7271, 25eleqtrdi 2850 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑥𝑆) → 𝑥 ∈ (𝐷 “ (-∞[,)𝑁)))
73 elpreima 7077 . . . . . . . . . . . . . . . . . . 19 (𝐷 Fn (Base‘𝑃) → (𝑥 ∈ (𝐷 “ (-∞[,)𝑁)) ↔ (𝑥 ∈ (Base‘𝑃) ∧ (𝐷𝑥) ∈ (-∞[,)𝑁))))
7473simplbda 499 . . . . . . . . . . . . . . . . . 18 ((𝐷 Fn (Base‘𝑃) ∧ 𝑥 ∈ (𝐷 “ (-∞[,)𝑁))) → (𝐷𝑥) ∈ (-∞[,)𝑁))
7570, 72, 74syl2anc 584 . . . . . . . . . . . . . . . . 17 ((𝜑𝑥𝑆) → (𝐷𝑥) ∈ (-∞[,)𝑁))
76 elico1 13431 . . . . . . . . . . . . . . . . . . 19 ((-∞ ∈ ℝ*𝑁 ∈ ℝ*) → ((𝐷𝑥) ∈ (-∞[,)𝑁) ↔ ((𝐷𝑥) ∈ ℝ* ∧ -∞ ≤ (𝐷𝑥) ∧ (𝐷𝑥) < 𝑁)))
7776biimpa 476 . . . . . . . . . . . . . . . . . 18 (((-∞ ∈ ℝ*𝑁 ∈ ℝ*) ∧ (𝐷𝑥) ∈ (-∞[,)𝑁)) → ((𝐷𝑥) ∈ ℝ* ∧ -∞ ≤ (𝐷𝑥) ∧ (𝐷𝑥) < 𝑁))
7877simp3d 1144 . . . . . . . . . . . . . . . . 17 (((-∞ ∈ ℝ*𝑁 ∈ ℝ*) ∧ (𝐷𝑥) ∈ (-∞[,)𝑁)) → (𝐷𝑥) < 𝑁)
7968, 69, 75, 78syl21anc 837 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑆) → (𝐷𝑥) < 𝑁)
8064, 67, 79syl2anc 584 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷𝑥) < 𝑁)
8157, 59, 55, 63, 80xrlelttrd 13203 . . . . . . . . . . . . . 14 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·𝑠𝑃)𝑥)) < 𝑁)
8252, 55, 57, 58, 81elicod 13438 . . . . . . . . . . . . 13 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·𝑠𝑃)𝑥)) ∈ (-∞[,)𝑁))
8336, 50, 82elpreimad 7078 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·𝑠𝑃)𝑥) ∈ (𝐷 “ (-∞[,)𝑁)))
8483, 25eleqtrrdi 2851 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·𝑠𝑃)𝑥) ∈ 𝑆)
8584anasss 466 . . . . . . . . . 10 ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑥 ∈ (Base‘𝐸))) → (𝑘( ·𝑠𝑃)𝑥) ∈ 𝑆)
8685ad5ant15 758 . . . . . . . . 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)
8712ad2antrr 726 . . . . . . . . 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)))
8834, 35mp1i 13 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝐷 Fn (Base‘𝑃))
89 eqid 2736 . . . . . . . . . . . . . . . 16 (mulGrp‘𝑃) = (mulGrp‘𝑃)
9089, 33mgpbas 20143 . . . . . . . . . . . . . . 15 (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
91 eqid 2736 . . . . . . . . . . . . . . 15 (.g‘(mulGrp‘𝑃)) = (.g‘(mulGrp‘𝑃))
921ply1ring 22250 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → 𝑃 ∈ Ring)
9389ringmgp 20237 . . . . . . . . . . . . . . . . 17 (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
946, 92, 933syl 18 . . . . . . . . . . . . . . . 16 (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
9594adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → (mulGrp‘𝑃) ∈ Mnd)
96 elfzonn0 13748 . . . . . . . . . . . . . . . 16 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ0)
9796adantl 481 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
98 eqid 2736 . . . . . . . . . . . . . . . . . 18 (var1𝑅) = (var1𝑅)
9998, 1, 33vr1cl 22220 . . . . . . . . . . . . . . . . 17 (𝑅 ∈ Ring → (var1𝑅) ∈ (Base‘𝑃))
1006, 99syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → (var1𝑅) ∈ (Base‘𝑃))
101100adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → (var1𝑅) ∈ (Base‘𝑃))
10290, 91, 95, 97, 101mulgnn0cld 19114 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
10351a1i 11 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → -∞ ∈ ℝ*)
10454adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ*)
10524, 1, 33deg1xrcl 26122 . . . . . . . . . . . . . . . 16 ((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ ℝ*)
106102, 105syl 17 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ ℝ*)
107106mnfled 13179 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → -∞ ≤ (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))))
10896nn0red 12590 . . . . . . . . . . . . . . . . . 18 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℝ)
109108rexrd 11312 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℝ*)
110109adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ*)
11124, 1, 98, 89, 91deg1pwle 26160 . . . . . . . . . . . . . . . . 17 ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ0) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ≤ 𝑛)
1126, 96, 111syl2an 596 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ≤ 𝑛)
113 elfzolt2 13709 . . . . . . . . . . . . . . . . 17 (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
114113adantl 481 . . . . . . . . . . . . . . . 16 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
115106, 110, 104, 112, 114xrlelttrd 13203 . . . . . . . . . . . . . . 15 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) < 𝑁)
116103, 104, 106, 107, 115elicod 13438 . . . . . . . . . . . . . 14 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ (-∞[,)𝑁))
11788, 102, 116elpreimad 7078 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (𝐷 “ (-∞[,)𝑁)))
118117, 25eleqtrrdi 2851 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ 𝑆)
11946adantr 480 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ (0..^𝑁)) → 𝑆 = (Base‘𝐸))
120118, 119eleqtrd 2842 . . . . . . . . . . 11 ((𝜑𝑛 ∈ (0..^𝑁)) → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝐸))
121120, 8fmptd 7133 . . . . . . . . . 10 (𝜑𝐹:(0..^𝑁)⟶(Base‘𝐸))
122121ad3antrrr 730 . . . . . . . . 9 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝐹:(0..^𝑁)⟶(Base‘𝐸))
123 inidm 4226 . . . . . . . . 9 ((0..^𝑁) ∩ (0..^𝑁)) = (0..^𝑁)
12486, 87, 122, 21, 21, 123off 7716 . . . . . . . 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)
12521, 32, 124, 44gsumsubm 18849 . . . . . . 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)))
126 ringmnd 20241 . . . . . . . . . 10 (𝑃 ∈ Ring → 𝑃 ∈ Mnd)
1276, 92, 1263syl 18 . . . . . . . . 9 (𝜑𝑃 ∈ Mnd)
12834, 35mp1i 13 . . . . . . . . . . 11 (𝜑𝐷 Fn (Base‘𝑃))
12933, 10mndidcl 18763 . . . . . . . . . . . 12 (𝑃 ∈ Mnd → (0g𝑃) ∈ (Base‘𝑃))
130127, 129syl 17 . . . . . . . . . . 11 (𝜑 → (0g𝑃) ∈ (Base‘𝑃))
13151a1i 11 . . . . . . . . . . . 12 (𝜑 → -∞ ∈ ℝ*)
13224, 1, 33deg1xrcl 26122 . . . . . . . . . . . . 13 ((0g𝑃) ∈ (Base‘𝑃) → (𝐷‘(0g𝑃)) ∈ ℝ*)
133130, 132syl 17 . . . . . . . . . . . 12 (𝜑 → (𝐷‘(0g𝑃)) ∈ ℝ*)
134133mnfled 13179 . . . . . . . . . . . 12 (𝜑 → -∞ ≤ (𝐷‘(0g𝑃)))
13524, 1, 10deg1z 26127 . . . . . . . . . . . . . 14 (𝑅 ∈ Ring → (𝐷‘(0g𝑃)) = -∞)
1366, 135syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐷‘(0g𝑃)) = -∞)
13753mnfltd 13167 . . . . . . . . . . . . 13 (𝜑 → -∞ < 𝑁)
138136, 137eqbrtrd 5164 . . . . . . . . . . . 12 (𝜑 → (𝐷‘(0g𝑃)) < 𝑁)
139131, 54, 133, 134, 138elicod 13438 . . . . . . . . . . 11 (𝜑 → (𝐷‘(0g𝑃)) ∈ (-∞[,)𝑁))
140128, 130, 139elpreimad 7078 . . . . . . . . . 10 (𝜑 → (0g𝑃) ∈ (𝐷 “ (-∞[,)𝑁)))
141140, 25eleqtrrdi 2851 . . . . . . . . 9 (𝜑 → (0g𝑃) ∈ 𝑆)
14244, 33, 10ress0g 18776 . . . . . . . . 9 ((𝑃 ∈ Mnd ∧ (0g𝑃) ∈ 𝑆𝑆 ⊆ (Base‘𝑃)) → (0g𝑃) = (0g𝐸))
143127, 141, 43, 142syl3anc 1372 . . . . . . . 8 (𝜑 → (0g𝑃) = (0g𝐸))
144143ad3antrrr 730 . . . . . . 7 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → (0g𝑃) = (0g𝐸))
14520, 125, 1443eqtr4d 2786 . . . . . 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))
1461, 2, 4, 7, 8, 9, 10, 19, 145ply1gsumz 33620 . . . . 5 ((((𝜑𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))) ∧ 𝑎 finSupp (0g‘(Scalar‘𝑃))) ∧ (𝐸 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (0g𝐸)) → 𝑎 = ((0..^𝑁) × {(0g𝑅)}))
14714fveq2d 6909 . . . . . . . 8 (𝜑 → (0g𝑅) = (0g‘(Scalar‘𝑃)))
148147sneqd 4637 . . . . . . 7 (𝜑 → {(0g𝑅)} = {(0g‘(Scalar‘𝑃))})
149148xpeq2d 5714 . . . . . 6 (𝜑 → ((0..^𝑁) × {(0g𝑅)}) = ((0..^𝑁) × {(0g‘(Scalar‘𝑃))}))
150149ad3antrrr 730 . . . . 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))}))
151146, 150eqtrd 2776 . . . 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))}))
152151expl 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))})))
153152ralrimiva 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))})))
154118, 8fmptd 7133 . . . . . 6 (𝜑𝐹:(0..^𝑁)⟶𝑆)
155154frnd 6743 . . . . 5 (𝜑 → ran 𝐹𝑆)
156 eqid 2736 . . . . . 6 (LSpan‘𝑃) = (LSpan‘𝑃)
15727, 156lspssp 20987 . . . . 5 ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃) ∧ ran 𝐹𝑆) → ((LSpan‘𝑃)‘ran 𝐹) ⊆ 𝑆)
15823, 26, 155, 157syl3anc 1372 . . . 4 (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) ⊆ 𝑆)
159 breq1 5145 . . . . . . . 8 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → (𝑎 finSupp (0g‘(Scalar‘𝑃)) ↔ ((coe1𝑥) ↾ (0..^𝑁)) finSupp (0g‘(Scalar‘𝑃))))
160 oveq1 7439 . . . . . . . . . 10 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → (𝑎f ( ·𝑠𝑃)𝐹) = (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹))
161160oveq2d 7448 . . . . . . . . 9 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹)) = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹)))
162161eqeq2d 2747 . . . . . . . 8 (𝑎 = ((coe1𝑥) ↾ (0..^𝑁)) → (𝑥 = (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹)) ↔ 𝑥 = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹))))
163159, 162anbi12d 632 . . . . . . 7 (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)))))
164 fvexd 6920 . . . . . . . 8 ((𝜑𝑥𝑆) → (Base‘(Scalar‘𝑃)) ∈ V)
165 ovexd 7467 . . . . . . . 8 ((𝜑𝑥𝑆) → (0..^𝑁) ∈ V)
16643sselda 3982 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑥 ∈ (Base‘𝑃))
167 eqid 2736 . . . . . . . . . . . 12 (coe1𝑥) = (coe1𝑥)
168167, 33, 1, 2coe1f 22214 . . . . . . . . . . 11 (𝑥 ∈ (Base‘𝑃) → (coe1𝑥):ℕ0⟶(Base‘𝑅))
169166, 168syl 17 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (coe1𝑥):ℕ0⟶(Base‘𝑅))
17015adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
171170feq3d 6722 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ((coe1𝑥):ℕ0⟶(Base‘𝑅) ↔ (coe1𝑥):ℕ0⟶(Base‘(Scalar‘𝑃))))
172169, 171mpbid 232 . . . . . . . . 9 ((𝜑𝑥𝑆) → (coe1𝑥):ℕ0⟶(Base‘(Scalar‘𝑃)))
173 fzo0ssnn0 13786 . . . . . . . . . 10 (0..^𝑁) ⊆ ℕ0
174173a1i 11 . . . . . . . . 9 ((𝜑𝑥𝑆) → (0..^𝑁) ⊆ ℕ0)
175172, 174fssresd 6774 . . . . . . . 8 ((𝜑𝑥𝑆) → ((coe1𝑥) ↾ (0..^𝑁)):(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
176164, 165, 175elmapdd 8882 . . . . . . 7 ((𝜑𝑥𝑆) → ((coe1𝑥) ↾ (0..^𝑁)) ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁)))
177169ffund 6739 . . . . . . . . 9 ((𝜑𝑥𝑆) → Fun (coe1𝑥))
178 fzofi 14016 . . . . . . . . . 10 (0..^𝑁) ∈ Fin
179178a1i 11 . . . . . . . . 9 ((𝜑𝑥𝑆) → (0..^𝑁) ∈ Fin)
180 fvexd 6920 . . . . . . . . 9 ((𝜑𝑥𝑆) → (0g‘(Scalar‘𝑃)) ∈ V)
181177, 179, 180resfifsupp 9438 . . . . . . . 8 ((𝜑𝑥𝑆) → ((coe1𝑥) ↾ (0..^𝑁)) finSupp (0g‘(Scalar‘𝑃)))
182 ringcmn 20280 . . . . . . . . . . . 12 (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
1836, 92, 1823syl 18 . . . . . . . . . . 11 (𝜑𝑃 ∈ CMnd)
184183adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑆) → 𝑃 ∈ CMnd)
185 nn0ex 12534 . . . . . . . . . . 11 0 ∈ V
186185a1i 11 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ℕ0 ∈ V)
18723ad2antrr 726 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → 𝑃 ∈ LMod)
188172ffvelcdmda 7103 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → ((coe1𝑥)‘𝑖) ∈ (Base‘(Scalar‘𝑃)))
1896ad2antrr 726 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → 𝑅 ∈ Ring)
190189, 92, 933syl 18 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → (mulGrp‘𝑃) ∈ Mnd)
191 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → 𝑖 ∈ ℕ0)
192189, 99syl 17 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → (var1𝑅) ∈ (Base‘𝑃))
19390, 91, 190, 191, 192mulgnn0cld 19114 . . . . . . . . . . . 12 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
19433, 37, 38, 39, 187, 188, 193lmodvscld 20878 . . . . . . . . . . 11 (((𝜑𝑥𝑆) ∧ 𝑖 ∈ ℕ0) → (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ (Base‘𝑃))
195 eqid 2736 . . . . . . . . . . 11 (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) = (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))
196194, 195fmptd 7133 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))):ℕ0⟶(Base‘𝑃))
197 nfv 1913 . . . . . . . . . . . 12 𝑖(𝜑𝑥𝑆)
198197, 194, 195fnmptd 6708 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) Fn ℕ0)
199 fveq2 6905 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → ((coe1𝑥)‘𝑖) = ((coe1𝑥)‘𝑗))
200 oveq1 7439 . . . . . . . . . . . . . 14 (𝑖 = 𝑗 → (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)))
201199, 200oveq12d 7450 . . . . . . . . . . . . 13 (𝑖 = 𝑗 → (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))) = (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
202 simplr 768 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑗 ∈ ℕ0)
203 ovexd 7467 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ V)
204195, 201, 202, 203fvmptd3 7038 . . . . . . . . . . . 12 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗) = (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
205166ad2antrr 726 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑥 ∈ (Base‘𝑃))
206 icossxr 13473 . . . . . . . . . . . . . . . . 17 (-∞[,)𝑁) ⊆ ℝ*
207206, 75sselid 3980 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑆) → (𝐷𝑥) ∈ ℝ*)
208207ad2antrr 726 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (𝐷𝑥) ∈ ℝ*)
20954ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑁 ∈ ℝ*)
210202nn0red 12590 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑗 ∈ ℝ)
211210rexrd 11312 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑗 ∈ ℝ*)
21279ad2antrr 726 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (𝐷𝑥) < 𝑁)
213 simpr 484 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑁𝑗)
214208, 209, 211, 212, 213xrltletrd 13204 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (𝐷𝑥) < 𝑗)
21524, 1, 33, 9, 167deg1lt 26137 . . . . . . . . . . . . . 14 ((𝑥 ∈ (Base‘𝑃) ∧ 𝑗 ∈ ℕ0 ∧ (𝐷𝑥) < 𝑗) → ((coe1𝑥)‘𝑗) = (0g𝑅))
216205, 202, 214, 215syl3anc 1372 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((coe1𝑥)‘𝑗) = (0g𝑅))
217216oveq1d 7447 . . . . . . . . . . . 12 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = ((0g𝑅)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
218147ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (0g𝑅) = (0g‘(Scalar‘𝑃)))
219218oveq1d 7447 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((0g𝑅)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
22023ad3antrrr 730 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → 𝑃 ∈ LMod)
22194ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (mulGrp‘𝑃) ∈ Mnd)
222100ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (var1𝑅) ∈ (Base‘𝑃))
22390, 91, 221, 202, 222mulgnn0cld 19114 . . . . . . . . . . . . . 14 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃))
224 eqid 2736 . . . . . . . . . . . . . . 15 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝑃))
22533, 37, 38, 224, 10lmod0vs 20894 . . . . . . . . . . . . . 14 ((𝑃 ∈ LMod ∧ (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝑃)) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
226220, 223, 225syl2anc 584 . . . . . . . . . . . . 13 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((0g‘(Scalar‘𝑃))( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
227219, 226eqtrd 2776 . . . . . . . . . . . 12 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((0g𝑅)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) = (0g𝑃))
228204, 217, 2273eqtrd 2780 . . . . . . . . . . 11 ((((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) ∧ 𝑁𝑗) → ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗) = (0g𝑃))
2293nn0zd 12641 . . . . . . . . . . . 12 (𝜑𝑁 ∈ ℤ)
230229adantr 480 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → 𝑁 ∈ ℤ)
231198, 228, 230suppssnn0 32810 . . . . . . . . . 10 ((𝜑𝑥𝑆) → ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) supp (0g𝑃)) ⊆ (0..^𝑁))
232186mptexd 7245 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ∈ V)
233198fnfund 6668 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → Fun (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))))
234 fvexd 6920 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (0g𝑃) ∈ V)
235 suppssfifsupp 9421 . . . . . . . . . . 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))
236232, 233, 234, 179, 231, 235syl32anc 1379 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) finSupp (0g𝑃))
23733, 10, 184, 186, 196, 231, 236gsumres 19932 . . . . . . . . 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))))))
238 fvexd 6920 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → (coe1𝑥) ∈ V)
239 ovexd 7467 . . . . . . . . . . . . . 14 (𝜑 → (0..^𝑁) ∈ V)
240154, 239fexd 7248 . . . . . . . . . . . . 13 (𝜑𝐹 ∈ V)
241240adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → 𝐹 ∈ V)
242 offres 8009 . . . . . . . . . . . 12 (((coe1𝑥) ∈ V ∧ 𝐹 ∈ V) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) ↾ (0..^𝑁)) = (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)(𝐹 ↾ (0..^𝑁))))
243238, 241, 242syl2anc 584 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) ↾ (0..^𝑁)) = (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)(𝐹 ↾ (0..^𝑁))))
244169ffnd 6736 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑆) → (coe1𝑥) Fn ℕ0)
245154ffnd 6736 . . . . . . . . . . . . . . . 16 (𝜑𝐹 Fn (0..^𝑁))
246245adantr 480 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑆) → 𝐹 Fn (0..^𝑁))
247 sseqin2 4222 . . . . . . . . . . . . . . . 16 ((0..^𝑁) ⊆ ℕ0 ↔ (ℕ0 ∩ (0..^𝑁)) = (0..^𝑁))
248173, 247mpbi 230 . . . . . . . . . . . . . . 15 (ℕ0 ∩ (0..^𝑁)) = (0..^𝑁)
249 eqidd 2737 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ ℕ0) → ((coe1𝑥)‘𝑗) = ((coe1𝑥)‘𝑗))
250 oveq1 7439 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑗 → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)))
251 simpr 484 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
252 ovexd 7467 . . . . . . . . . . . . . . . 16 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ V)
2538, 250, 251, 252fvmptd3 7038 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹𝑗) = (𝑗(.g‘(mulGrp‘𝑃))(var1𝑅)))
254244, 246, 186, 165, 248, 249, 253ofval 7709 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹)‘𝑗) = (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
255173, 251sselid 3980 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℕ0)
256 ovexd 7467 . . . . . . . . . . . . . . 15 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))) ∈ V)
257195, 201, 255, 256fvmptd3 7038 . . . . . . . . . . . . . 14 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗) = (((coe1𝑥)‘𝑗)( ·𝑠𝑃)(𝑗(.g‘(mulGrp‘𝑃))(var1𝑅))))
258254, 257eqtr4d 2779 . . . . . . . . . . . . 13 (((𝜑𝑥𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗))
259258ralrimiva 3145 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → ∀𝑗 ∈ (0..^𝑁)(((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))‘𝑗))
260244, 246, 186, 165, 248offn 7711 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → ((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) Fn (0..^𝑁))
261 ssidd 4006 . . . . . . . . . . . . 13 ((𝜑𝑥𝑆) → (0..^𝑁) ⊆ (0..^𝑁))
262 fvreseq0 7057 . . . . . . . . . . . . 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)))
263260, 198, 261, 174, 262syl22anc 838 . . . . . . . . . . . 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)))
264259, 263mpbird 257 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (((coe1𝑥) ∘f ( ·𝑠𝑃)𝐹) ↾ (0..^𝑁)) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁)))
265 fnresdm 6686 . . . . . . . . . . . . . 14 (𝐹 Fn (0..^𝑁) → (𝐹 ↾ (0..^𝑁)) = 𝐹)
266245, 265syl 17 . . . . . . . . . . . . 13 (𝜑 → (𝐹 ↾ (0..^𝑁)) = 𝐹)
267266adantr 480 . . . . . . . . . . . 12 ((𝜑𝑥𝑆) → (𝐹 ↾ (0..^𝑁)) = 𝐹)
268267oveq2d 7448 . . . . . . . . . . 11 ((𝜑𝑥𝑆) → (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)(𝐹 ↾ (0..^𝑁))) = (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹))
269243, 264, 2683eqtr3rd 2785 . . . . . . . . . 10 ((𝜑𝑥𝑆) → (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹) = ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁)))
270269oveq2d 7448 . . . . . . . . 9 ((𝜑𝑥𝑆) → (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹)) = (𝑃 Σg ((𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))) ↾ (0..^𝑁))))
2716adantr 480 . . . . . . . . . 10 ((𝜑𝑥𝑆) → 𝑅 ∈ Ring)
2721, 98, 33, 38, 89, 91, 167ply1coe 22303 . . . . . . . . . 10 ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 = (𝑃 Σg (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))))
273271, 166, 272syl2anc 584 . . . . . . . . 9 ((𝜑𝑥𝑆) → 𝑥 = (𝑃 Σg (𝑖 ∈ ℕ0 ↦ (((coe1𝑥)‘𝑖)( ·𝑠𝑃)(𝑖(.g‘(mulGrp‘𝑃))(var1𝑅))))))
274237, 270, 2733eqtr4rd 2787 . . . . . . . 8 ((𝜑𝑥𝑆) → 𝑥 = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹)))
275181, 274jca 511 . . . . . . 7 ((𝜑𝑥𝑆) → (((coe1𝑥) ↾ (0..^𝑁)) finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (((coe1𝑥) ↾ (0..^𝑁)) ∘f ( ·𝑠𝑃)𝐹))))
276163, 176, 275rspcedvdw 3624 . . . . . 6 ((𝜑𝑥𝑆) → ∃𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑m (0..^𝑁))(𝑎 finSupp (0g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σg (𝑎f ( ·𝑠𝑃)𝐹))))
277102, 8fmptd 7133 . . . . . . . 8 (𝜑𝐹:(0..^𝑁)⟶(Base‘𝑃))
278156, 33, 39, 37, 224, 38, 277, 23, 239ellspd 21823 . . . . . . 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)))))
279278adantr 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)))))
280276, 279mpbird 257 . . . . 5 ((𝜑𝑥𝑆) → 𝑥 ∈ ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
281 imadmrn 6087 . . . . . . . 8 (𝐹 “ dom 𝐹) = ran 𝐹
282154fdmd 6745 . . . . . . . . 9 (𝜑 → dom 𝐹 = (0..^𝑁))
283282imaeq2d 6077 . . . . . . . 8 (𝜑 → (𝐹 “ dom 𝐹) = (𝐹 “ (0..^𝑁)))
284281, 283eqtr3id 2790 . . . . . . 7 (𝜑 → ran 𝐹 = (𝐹 “ (0..^𝑁)))
285284fveq2d 6909 . . . . . 6 (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
286285adantr 480 . . . . 5 ((𝜑𝑥𝑆) → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
287280, 286eleqtrrd 2843 . . . 4 ((𝜑𝑥𝑆) → 𝑥 ∈ ((LSpan‘𝑃)‘ran 𝐹))
288158, 287eqelssd 4004 . . 3 (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = 𝑆)
289 eqid 2736 . . . . . 6 (LSpan‘𝐸) = (LSpan‘𝐸)
29044, 156, 289, 27lsslsp 21014 . . . . 5 ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃) ∧ ran 𝐹𝑆) → ((LSpan‘𝐸)‘ran 𝐹) = ((LSpan‘𝑃)‘ran 𝐹))
291290eqcomd 2742 . . . 4 ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃) ∧ ran 𝐹𝑆) → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝐸)‘ran 𝐹))
29223, 26, 155, 291syl3anc 1372 . . 3 (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝐸)‘ran 𝐹))
293288, 292, 463eqtr3d 2784 . 2 (𝜑 → ((LSpan‘𝐸)‘ran 𝐹) = (Base‘𝐸))
294 eqid 2736 . . 3 (Base‘𝐸) = (Base‘𝐸)
29524fvexi 6919 . . . . . . 7 𝐷 ∈ V
296 cnvexg 7947 . . . . . . 7 (𝐷 ∈ V → 𝐷 ∈ V)
297 imaexg 7936 . . . . . . 7 (𝐷 ∈ V → (𝐷 “ (-∞[,)𝑁)) ∈ V)
298295, 296, 297mp2b 10 . . . . . 6 (𝐷 “ (-∞[,)𝑁)) ∈ V
29925, 298eqeltri 2836 . . . . 5 𝑆 ∈ V
30044, 37resssca 17388 . . . . 5 (𝑆 ∈ V → (Scalar‘𝑃) = (Scalar‘𝐸))
301299, 300ax-mp 5 . . . 4 (Scalar‘𝑃) = (Scalar‘𝐸)
302301fveq2i 6908 . . 3 (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝐸))
303 eqid 2736 . . 3 (Scalar‘𝐸) = (Scalar‘𝐸)
30444, 38ressvsca 17389 . . . 4 (𝑆 ∈ V → ( ·𝑠𝑃) = ( ·𝑠𝐸))
305299, 304ax-mp 5 . . 3 ( ·𝑠𝑃) = ( ·𝑠𝐸)
306 eqid 2736 . . 3 (0g𝐸) = (0g𝐸)
307301fveq2i 6908 . . 3 (0g‘(Scalar‘𝑃)) = (0g‘(Scalar‘𝐸))
308 eqid 2736 . . 3 (LBasis‘𝐸) = (LBasis‘𝐸)
30944, 27lsslvec 21109 . . . . 5 ((𝑃 ∈ LVec ∧ 𝑆 ∈ (LSubSp‘𝑃)) → 𝐸 ∈ LVec)
31022, 26, 309syl2anc 584 . . . 4 (𝜑𝐸 ∈ LVec)
311310lveclmodd 21107 . . 3 (𝜑𝐸 ∈ LMod)
31214, 5eqeltrrd 2841 . . . . 5 (𝜑 → (Scalar‘𝑃) ∈ DivRing)
313 drngnzr 20749 . . . . 5 ((Scalar‘𝑃) ∈ DivRing → (Scalar‘𝑃) ∈ NzRing)
314312, 313syl 17 . . . 4 (𝜑 → (Scalar‘𝑃) ∈ NzRing)
315301, 314eqeltrrid 2845 . . 3 (𝜑 → (Scalar‘𝐸) ∈ NzRing)
316120ralrimiva 3145 . . . 4 (𝜑 → ∀𝑛 ∈ (0..^𝑁)(𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) ∈ (Base‘𝐸))
317 drngnzr 20749 . . . . . . . . . 10 (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
3185, 317syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ NzRing)
319318ad2antrr 726 . . . . . . . 8 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑅 ∈ NzRing)
32097adantr 480 . . . . . . . 8 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ0)
321 elfzonn0 13748 . . . . . . . . 9 (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ0)
322321adantl 481 . . . . . . . 8 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ0)
3231, 98, 91, 319, 320, 322ply1moneq 33612 . . . . . . 7 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) ↔ 𝑛 = 𝑖))
324323biimpd 229 . . . . . 6 (((𝜑𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) → 𝑛 = 𝑖))
325324anasss 466 . . . . 5 ((𝜑 ∧ (𝑛 ∈ (0..^𝑁) ∧ 𝑖 ∈ (0..^𝑁))) → ((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) → 𝑛 = 𝑖))
326325ralrimivva 3201 . . . 4 (𝜑 → ∀𝑛 ∈ (0..^𝑁)∀𝑖 ∈ (0..^𝑁)((𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)) → 𝑛 = 𝑖))
327 oveq1 7439 . . . . 5 (𝑛 = 𝑖 → (𝑛(.g‘(mulGrp‘𝑃))(var1𝑅)) = (𝑖(.g‘(mulGrp‘𝑃))(var1𝑅)))
3288, 327f1mpt 7282 . . . 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)))
329316, 326, 328sylanbrc 583 . . 3 (𝜑𝐹:(0..^𝑁)–1-1→(Base‘𝐸))
330294, 302, 303, 305, 306, 307, 308, 289, 311, 315, 239, 329islbs5 33409 . 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))))
331153, 293, 330mpbir2and 713 1 (𝜑 → ran 𝐹 ∈ (LBasis‘𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wcel 2107  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950  {csn 4625   class class class wbr 5142  cmpt 5224   × cxp 5682  ccnv 5683  dom cdm 5684  ran crn 5685  cres 5686  cima 5687  Fun wfun 6554   Fn wfn 6555  wf 6556  1-1wf1 6557  cfv 6560  (class class class)co 7432  f cof 7696   supp csupp 8186  m cmap 8867  Fincfn 8986   finSupp cfsupp 9402  0cc0 11156  -∞cmnf 11294  *cxr 11295   < clt 11296  cle 11297  0cn0 12528  cz 12615  [,)cico 13390  ..^cfzo 13695  Basecbs 17248  s cress 17275  Scalarcsca 17301   ·𝑠 cvsca 17302  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18748  SubMndcsubmnd 18796  .gcmg 19086  SubGrpcsubg 19139  CMndccmn 19799  mulGrpcmgp 20138  Ringcrg 20231  NzRingcnzr 20513  DivRingcdr 20730  LModclmod 20859  LSubSpclss 20930  LSpanclspn 20970  LBasisclbs 21074  LVecclvec 21102  var1cv1 22178  Poly1cpl1 22179  coe1cco1 22180  deg1cdg1 26094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-addf 11235
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-ofr 7699  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-z 12616  df-dec 12736  df-uz 12880  df-ico 13394  df-fz 13549  df-fzo 13696  df-seq 14044  df-hash 14371  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-srg 20185  df-ring 20233  df-cring 20234  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-nzr 20514  df-subrng 20547  df-subrg 20571  df-drng 20732  df-lmod 20861  df-lss 20931  df-lsp 20971  df-lmhm 21022  df-lbs 21075  df-lvec 21103  df-sra 21173  df-rgmod 21174  df-cnfld 21366  df-dsmm 21753  df-frlm 21768  df-uvc 21804  df-lindf 21827  df-linds 21828  df-psr 21930  df-mvr 21931  df-mpl 21932  df-opsr 21934  df-psr1 22182  df-vr1 22183  df-ply1 22184  df-coe1 22185  df-mdeg 26095  df-deg1 26096
This theorem is referenced by:  ply1degltdim  33675
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