ply1degltdimlem - Mathbox for Thierry Arnoux

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Theorem ply1degltdimlem 32695

Description: Lemma for ply1degltdim 32696. (Contributed by Thierry Arnoux, 20-Feb-2025.)

**Hypotheses**
Ref	Expression
ply1degltdim.p	⊢ 𝑃 = (Poly₁‘𝑅)
ply1degltdim.d	⊢ 𝐷 = ( deg₁ ‘𝑅)
ply1degltdim.s	⊢ 𝑆 = (^◡𝐷 “ (-∞[,)𝑁))
ply1degltdim.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
ply1degltdim.r	⊢ (𝜑 → 𝑅 ∈ DivRing)
ply1degltdim.e	⊢ 𝐸 = (𝑃 ↾_s 𝑆)
ply1degltdimlem.f	⊢ 𝐹 = (𝑛 ∈ (0..^𝑁) ↦ (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))

**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.

Step	Hyp	Ref	Expression
1		ply1degltdim.p	. . . . . 6 ⊢ 𝑃 = (Poly₁‘𝑅)
2		eqid 2732	. . . . . 6 ⊢ (Base‘𝑅) = (Base‘𝑅)
3		ply1degltdim.n	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	3	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑁 ∈ ℕ₀)
5		ply1degltdim.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ DivRing)
6	5	drngringd 20315	. . . . . . 7 ⊢ (𝜑 → 𝑅 ∈ Ring)
7	6	ad3antrrr 728	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑅 ∈ Ring)
8		ply1degltdimlem.f	. . . . . 6 ⊢ 𝐹 = (𝑛 ∈ (0..^𝑁) ↦ (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
9		eqid 2732	. . . . . 6 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
10		eqid 2732	. . . . . 6 ⊢ (0_g‘𝑃) = (0_g‘𝑃)
11		elmapi 8839	. . . . . . . . 9 ⊢ (𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁)) → 𝑎:(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
12	11	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) → 𝑎:(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
13	1	ply1sca 21766	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ DivRing → 𝑅 = (Scalar‘𝑃))
14	5, 13	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 = (Scalar‘𝑃))
15	14	fveq2d 6892	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
16	15	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
17	16	feq3d 6701	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) → (𝑎:(0..^𝑁)⟶(Base‘𝑅) ↔ 𝑎:(0..^𝑁)⟶(Base‘(Scalar‘𝑃))))
18	12, 17	mpbird 256	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) → 𝑎:(0..^𝑁)⟶(Base‘𝑅))
19	18	ad2antrr 724	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑎:(0..^𝑁)⟶(Base‘𝑅))
20		simpr 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸))
21		ovexd 7440	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → (0..^𝑁) ∈ V)
22	1, 5	ply1lvec 32626	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑃 ∈ LVec)
23	22	lveclmodd 20710	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ LMod)
24		ply1degltdim.d	. . . . . . . . . . . 12 ⊢ 𝐷 = ( deg₁ ‘𝑅)
25		ply1degltdim.s	. . . . . . . . . . . 12 ⊢ 𝑆 = (^◡𝐷 “ (-∞[,)𝑁))
26	1, 24, 25, 3, 6	ply1degltlss 32655	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ (LSubSp‘𝑃))
27		eqid 2732	. . . . . . . . . . . 12 ⊢ (LSubSp‘𝑃) = (LSubSp‘𝑃)
28	27	lsssubg 20560	. . . . . . . . . . 11 ⊢ ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃)) → 𝑆 ∈ (SubGrp‘𝑃))
29	23, 26, 28	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ (SubGrp‘𝑃))
30		subgsubm 19022	. . . . . . . . . 10 ⊢ (𝑆 ∈ (SubGrp‘𝑃) → 𝑆 ∈ (SubMnd‘𝑃))
31	29, 30	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (SubMnd‘𝑃))
32	31	ad3antrrr 728	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑆 ∈ (SubMnd‘𝑃))
33		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘𝑃)
34	24, 1, 33	deg1xrf 25590	. . . . . . . . . . . . . 14 ⊢ 𝐷:(Base‘𝑃)⟶ℝ^*
35		ffn 6714	. . . . . . . . . . . . . 14 ⊢ (𝐷:(Base‘𝑃)⟶ℝ^* → 𝐷 Fn (Base‘𝑃))
36	34, 35	mp1i 13	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷 Fn (Base‘𝑃))
37	23	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑃 ∈ LMod)
38		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑘 ∈ (Base‘(Scalar‘𝑃)))
39	33, 27	lssss 20539	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ∈ (LSubSp‘𝑃) → 𝑆 ⊆ (Base‘𝑃))
40	26, 39	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑆 ⊆ (Base‘𝑃))
41		ply1degltdim.e	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐸 = (𝑃 ↾_s 𝑆)
42	41, 33	ressbas2 17178	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ⊆ (Base‘𝑃) → 𝑆 = (Base‘𝐸))
43	40, 42	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑆 = (Base‘𝐸))
44	43, 40	eqsstrrd 4020	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (Base‘𝐸) ⊆ (Base‘𝑃))
45	44	sselda 3981	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝑃))
46	45	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝑃))
47		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (Scalar‘𝑃) = (Scalar‘𝑃)
48		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝑃)
49		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝑃))
50	33, 47, 48, 49	lmodvscl 20481	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑥 ∈ (Base‘𝑃)) → (𝑘( ·_𝑠 ‘𝑃)𝑥) ∈ (Base‘𝑃))
51	37, 38, 46, 50	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·_𝑠 ‘𝑃)𝑥) ∈ (Base‘𝑃))
52		mnfxr 11267	. . . . . . . . . . . . . . 15 ⊢ -∞ ∈ ℝ^*
53	52	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → -∞ ∈ ℝ^*)
54	3	nn0red 12529	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℝ)
55	54	rexrd 11260	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
56	55	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑁 ∈ ℝ^*)
57	34	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝐷:(Base‘𝑃)⟶ℝ^*)
58	57, 51	ffvelcdmd 7084	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·_𝑠 ‘𝑃)𝑥)) ∈ ℝ^*)
59	58	mnfled 13111	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → -∞ ≤ (𝐷‘(𝑘( ·_𝑠 ‘𝑃)𝑥)))
60	57, 46	ffvelcdmd 7084	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘𝑥) ∈ ℝ^*)
61	6	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑅 ∈ Ring)
62	15	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
63	38, 62	eleqtrrd 2836	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑘 ∈ (Base‘𝑅))
64	1, 24, 61, 33, 2, 48, 63, 46	deg1vscale 25613	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·_𝑠 ‘𝑃)𝑥)) ≤ (𝐷‘𝑥))
65		simpll 765	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝜑)
66		simpr 485	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ (Base‘𝐸))
67	43	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑆 = (Base‘𝐸))
68	66, 67	eleqtrrd 2836	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → 𝑥 ∈ 𝑆)
69	52	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → -∞ ∈ ℝ^*)
70	55	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑁 ∈ ℝ^*)
71	34, 35	mp1i 13	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐷 Fn (Base‘𝑃))
72		simpr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ 𝑆)
73	72, 25	eleqtrdi 2843	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (^◡𝐷 “ (-∞[,)𝑁)))
74		elpreima 7056	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐷 Fn (Base‘𝑃) → (𝑥 ∈ (^◡𝐷 “ (-∞[,)𝑁)) ↔ (𝑥 ∈ (Base‘𝑃) ∧ (𝐷‘𝑥) ∈ (-∞[,)𝑁))))
75	74	simplbda 500	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐷 Fn (Base‘𝑃) ∧ 𝑥 ∈ (^◡𝐷 “ (-∞[,)𝑁))) → (𝐷‘𝑥) ∈ (-∞[,)𝑁))
76	71, 73, 75	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐷‘𝑥) ∈ (-∞[,)𝑁))
77		elico1 13363	. . . . . . . . . . . . . . . . . . 19 ⊢ ((-∞ ∈ ℝ^* ∧ 𝑁 ∈ ℝ^) → ((𝐷‘𝑥) ∈ (-∞[,)𝑁) ↔ ((𝐷‘𝑥) ∈ ℝ^ ∧ -∞ ≤ (𝐷‘𝑥) ∧ (𝐷‘𝑥) < 𝑁)))
78	77	biimpa 477	. . . . . . . . . . . . . . . . . 18 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑁 ∈ ℝ^) ∧ (𝐷‘𝑥) ∈ (-∞[,)𝑁)) → ((𝐷‘𝑥) ∈ ℝ^ ∧ -∞ ≤ (𝐷‘𝑥) ∧ (𝐷‘𝑥) < 𝑁))
79	78	simp3d 1144	. . . . . . . . . . . . . . . . 17 ⊢ (((-∞ ∈ ℝ^* ∧ 𝑁 ∈ ℝ^*) ∧ (𝐷‘𝑥) ∈ (-∞[,)𝑁)) → (𝐷‘𝑥) < 𝑁)
80	69, 70, 76, 79	syl21anc 836	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐷‘𝑥) < 𝑁)
81	65, 68, 80	syl2anc 584	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘𝑥) < 𝑁)
82	58, 60, 56, 64, 81	xrlelttrd 13135	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·_𝑠 ‘𝑃)𝑥)) < 𝑁)
83	53, 56, 58, 59, 82	elicod 13370	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝐷‘(𝑘( ·_𝑠 ‘𝑃)𝑥)) ∈ (-∞[,)𝑁))
84	36, 51, 83	elpreimad 7057	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·_𝑠 ‘𝑃)𝑥) ∈ (^◡𝐷 “ (-∞[,)𝑁)))
85	84, 25	eleqtrrdi 2844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (Base‘(Scalar‘𝑃))) ∧ 𝑥 ∈ (Base‘𝐸)) → (𝑘( ·_𝑠 ‘𝑃)𝑥) ∈ 𝑆)
86	85	anasss 467	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑥 ∈ (Base‘𝐸))) → (𝑘( ·_𝑠 ‘𝑃)𝑥) ∈ 𝑆)
87	86	ad5ant15 757	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) ∧ (𝑘 ∈ (Base‘(Scalar‘𝑃)) ∧ 𝑥 ∈ (Base‘𝐸))) → (𝑘( ·_𝑠 ‘𝑃)𝑥) ∈ 𝑆)
88	12	ad2antrr 724	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑎:(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
89	34, 35	mp1i 13	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → 𝐷 Fn (Base‘𝑃))
90		eqid 2732	. . . . . . . . . . . . . . . 16 ⊢ (mulGrp‘𝑃) = (mulGrp‘𝑃)
91	90, 33	mgpbas 19987	. . . . . . . . . . . . . . 15 ⊢ (Base‘𝑃) = (Base‘(mulGrp‘𝑃))
92		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (._g‘(mulGrp‘𝑃)) = (._g‘(mulGrp‘𝑃))
93	1	ply1ring 21761	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → 𝑃 ∈ Ring)
94	90	ringmgp 20055	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ Ring → (mulGrp‘𝑃) ∈ Mnd)
95	6, 93, 94	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (mulGrp‘𝑃) ∈ Mnd)
96	95	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (mulGrp‘𝑃) ∈ Mnd)
97		elfzonn0 13673	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℕ₀)
98	97	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ₀)
99		eqid 2732	. . . . . . . . . . . . . . . . . 18 ⊢ (var₁‘𝑅) = (var₁‘𝑅)
100	99, 1, 33	vr1cl 21732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑅 ∈ Ring → (var₁‘𝑅) ∈ (Base‘𝑃))
101	6, 100	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (var₁‘𝑅) ∈ (Base‘𝑃))
102	101	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (var₁‘𝑅) ∈ (Base‘𝑃))
103	91, 92, 96, 98, 102	mulgnn0cld 18969	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃))
104	52	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → -∞ ∈ ℝ^*)
105	55	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → 𝑁 ∈ ℝ^*)
106	24, 1, 33	deg1xrcl 25591	. . . . . . . . . . . . . . . 16 ⊢ ((𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃) → (𝐷‘(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ ℝ^*)
107	103, 106	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ ℝ^*)
108	107	mnfled 13111	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → -∞ ≤ (𝐷‘(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
109	97	nn0red 12529	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℝ)
110	109	rexrd 11260	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 ∈ ℝ^*)
111	110	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 ∈ ℝ^*)
112	24, 1, 99, 90, 92	deg1pwle 25628	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ ℕ₀) → (𝐷‘(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ≤ 𝑛)
113	6, 97, 112	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ≤ 𝑛)
114		elfzolt2 13637	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ (0..^𝑁) → 𝑛 < 𝑁)
115	114	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → 𝑛 < 𝑁)
116	107, 111, 105, 113, 115	xrlelttrd 13135	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) < 𝑁)
117	104, 105, 107, 108, 116	elicod 13370	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (𝐷‘(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ (-∞[,)𝑁))
118	89, 103, 117	elpreimad 7057	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (^◡𝐷 “ (-∞[,)𝑁)))
119	118, 25	eleqtrrdi 2844	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ 𝑆)
120	43	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → 𝑆 = (Base‘𝐸))
121	119, 120	eleqtrd 2835	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) → (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝐸))
122	121, 8	fmptd 7110	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(0..^𝑁)⟶(Base‘𝐸))
123	122	ad3antrrr 728	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝐹:(0..^𝑁)⟶(Base‘𝐸))
124		inidm 4217	. . . . . . . . 9 ⊢ ((0..^𝑁) ∩ (0..^𝑁)) = (0..^𝑁)
125	87, 88, 123, 21, 21, 124	off 7684	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹):(0..^𝑁)⟶𝑆)
126	21, 32, 125, 41	gsumsubm 18712	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)))
127		ringmnd 20059	. . . . . . . . . 10 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ Mnd)
128	6, 93, 127	3syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ Mnd)
129	34, 35	mp1i 13	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 Fn (Base‘𝑃))
130	33, 10	mndidcl 18636	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Mnd → (0_g‘𝑃) ∈ (Base‘𝑃))
131	128, 130	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘𝑃) ∈ (Base‘𝑃))
132	52	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ ∈ ℝ^*)
133	24, 1, 33	deg1xrcl 25591	. . . . . . . . . . . . 13 ⊢ ((0_g‘𝑃) ∈ (Base‘𝑃) → (𝐷‘(0_g‘𝑃)) ∈ ℝ^*)
134	131, 133	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘(0_g‘𝑃)) ∈ ℝ^*)
135	134	mnfled 13111	. . . . . . . . . . . 12 ⊢ (𝜑 → -∞ ≤ (𝐷‘(0_g‘𝑃)))
136	24, 1, 10	deg1z 25596	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Ring → (𝐷‘(0_g‘𝑃)) = -∞)
137	6, 136	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐷‘(0_g‘𝑃)) = -∞)
138	54	mnfltd 13100	. . . . . . . . . . . . 13 ⊢ (𝜑 → -∞ < 𝑁)
139	137, 138	eqbrtrd 5169	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐷‘(0_g‘𝑃)) < 𝑁)
140	132, 55, 134, 135, 139	elicod 13370	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷‘(0_g‘𝑃)) ∈ (-∞[,)𝑁))
141	129, 131, 140	elpreimad 7057	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝑃) ∈ (^◡𝐷 “ (-∞[,)𝑁)))
142	141, 25	eleqtrrdi 2844	. . . . . . . . 9 ⊢ (𝜑 → (0_g‘𝑃) ∈ 𝑆)
143	41, 33, 10	ress0g 18649	. . . . . . . . 9 ⊢ ((𝑃 ∈ Mnd ∧ (0_g‘𝑃) ∈ 𝑆 ∧ 𝑆 ⊆ (Base‘𝑃)) → (0_g‘𝑃) = (0_g‘𝐸))
144	128, 142, 40, 143	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝑃) = (0_g‘𝐸))
145	144	ad3antrrr 728	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → (0_g‘𝑃) = (0_g‘𝐸))
146	20, 126, 145	3eqtr4d 2782	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝑃))
147	1, 2, 4, 7, 8, 9, 10, 19, 146	ply1gsumz 32657	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑎 = ((0..^𝑁) × {(0_g‘𝑅)}))
148	14	fveq2d 6892	. . . . . . . 8 ⊢ (𝜑 → (0_g‘𝑅) = (0_g‘(Scalar‘𝑃)))
149	148	sneqd 4639	. . . . . . 7 ⊢ (𝜑 → {(0_g‘𝑅)} = {(0_g‘(Scalar‘𝑃))})
150	149	xpeq2d 5705	. . . . . 6 ⊢ (𝜑 → ((0..^𝑁) × {(0_g‘𝑅)}) = ((0..^𝑁) × {(0_g‘(Scalar‘𝑃))}))
151	150	ad3antrrr 728	. . . . 5 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → ((0..^𝑁) × {(0_g‘𝑅)}) = ((0..^𝑁) × {(0_g‘(Scalar‘𝑃))}))
152	147, 151	eqtrd 2772	. . . 4 ⊢ ((((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) ∧ 𝑎 finSupp (0_g‘(Scalar‘𝑃))) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑎 = ((0..^𝑁) × {(0_g‘(Scalar‘𝑃))}))
153	152	expl 458	. . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))) → ((𝑎 finSupp (0_g‘(Scalar‘𝑃)) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑎 = ((0..^𝑁) × {(0_g‘(Scalar‘𝑃))})))
154	153	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))((𝑎 finSupp (0_g‘(Scalar‘𝑃)) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑎 = ((0..^𝑁) × {(0_g‘(Scalar‘𝑃))})))
155	119, 8	fmptd 7110	. . . . . 6 ⊢ (𝜑 → 𝐹:(0..^𝑁)⟶𝑆)
156	155	frnd 6722	. . . . 5 ⊢ (𝜑 → ran 𝐹 ⊆ 𝑆)
157		eqid 2732	. . . . . 6 ⊢ (LSpan‘𝑃) = (LSpan‘𝑃)
158	27, 157	lspssp 20591	. . . . 5 ⊢ ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃) ∧ ran 𝐹 ⊆ 𝑆) → ((LSpan‘𝑃)‘ran 𝐹) ⊆ 𝑆)
159	23, 26, 156, 158	syl3anc 1371	. . . 4 ⊢ (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) ⊆ 𝑆)
160		fvexd 6903	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Base‘(Scalar‘𝑃)) ∈ V)
161		ovexd 7440	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (0..^𝑁) ∈ V)
162	40	sselda 3981	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑃))
163		eqid 2732	. . . . . . . . . . . 12 ⊢ (coe₁‘𝑥) = (coe₁‘𝑥)
164	163, 33, 1, 2	coe1f 21726	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (Base‘𝑃) → (coe₁‘𝑥):ℕ₀⟶(Base‘𝑅))
165	162, 164	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (coe₁‘𝑥):ℕ₀⟶(Base‘𝑅))
166	15	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (Base‘𝑅) = (Base‘(Scalar‘𝑃)))
167	166	feq3d 6701	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((coe₁‘𝑥):ℕ₀⟶(Base‘𝑅) ↔ (coe₁‘𝑥):ℕ₀⟶(Base‘(Scalar‘𝑃))))
168	165, 167	mpbid 231	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (coe₁‘𝑥):ℕ₀⟶(Base‘(Scalar‘𝑃)))
169		fzo0ssnn0 13709	. . . . . . . . . 10 ⊢ (0..^𝑁) ⊆ ℕ₀
170	169	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (0..^𝑁) ⊆ ℕ₀)
171	168, 170	fssresd 6755	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((coe₁‘𝑥) ↾ (0..^𝑁)):(0..^𝑁)⟶(Base‘(Scalar‘𝑃)))
172		elmapg 8829	. . . . . . . . 9 ⊢ (((Base‘(Scalar‘𝑃)) ∈ V ∧ (0..^𝑁) ∈ V) → (((coe₁‘𝑥) ↾ (0..^𝑁)) ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁)) ↔ ((coe₁‘𝑥) ↾ (0..^𝑁)):(0..^𝑁)⟶(Base‘(Scalar‘𝑃))))
173	172	biimpar 478	. . . . . . . 8 ⊢ ((((Base‘(Scalar‘𝑃)) ∈ V ∧ (0..^𝑁) ∈ V) ∧ ((coe₁‘𝑥) ↾ (0..^𝑁)):(0..^𝑁)⟶(Base‘(Scalar‘𝑃))) → ((coe₁‘𝑥) ↾ (0..^𝑁)) ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁)))
174	160, 161, 171, 173	syl21anc 836	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((coe₁‘𝑥) ↾ (0..^𝑁)) ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁)))
175		breq1 5150	. . . . . . . . 9 ⊢ (𝑎 = ((coe₁‘𝑥) ↾ (0..^𝑁)) → (𝑎 finSupp (0_g‘(Scalar‘𝑃)) ↔ ((coe₁‘𝑥) ↾ (0..^𝑁)) finSupp (0_g‘(Scalar‘𝑃))))
176		oveq1 7412	. . . . . . . . . . 11 ⊢ (𝑎 = ((coe₁‘𝑥) ↾ (0..^𝑁)) → (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹) = (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹))
177	176	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑎 = ((coe₁‘𝑥) ↾ (0..^𝑁)) → (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (𝑃 Σ_g (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹)))
178	177	eqeq2d 2743	. . . . . . . . 9 ⊢ (𝑎 = ((coe₁‘𝑥) ↾ (0..^𝑁)) → (𝑥 = (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) ↔ 𝑥 = (𝑃 Σ_g (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹))))
179	175, 178	anbi12d 631	. . . . . . . 8 ⊢ (𝑎 = ((coe₁‘𝑥) ↾ (0..^𝑁)) → ((𝑎 finSupp (0_g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹))) ↔ (((coe₁‘𝑥) ↾ (0..^𝑁)) finSupp (0_g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σ_g (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹)))))
180	179	adantl 482	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑎 = ((coe₁‘𝑥) ↾ (0..^𝑁))) → ((𝑎 finSupp (0_g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹))) ↔ (((coe₁‘𝑥) ↾ (0..^𝑁)) finSupp (0_g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σ_g (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹)))))
181	165	ffund 6718	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → Fun (coe₁‘𝑥))
182		fzofi 13935	. . . . . . . . . 10 ⊢ (0..^𝑁) ∈ Fin
183	182	a1i 11	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (0..^𝑁) ∈ Fin)
184		fvexd 6903	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (0_g‘(Scalar‘𝑃)) ∈ V)
185	181, 183, 184	resfifsupp 9388	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((coe₁‘𝑥) ↾ (0..^𝑁)) finSupp (0_g‘(Scalar‘𝑃)))
186		ringcmn 20092	. . . . . . . . . . . 12 ⊢ (𝑃 ∈ Ring → 𝑃 ∈ CMnd)
187	6, 93, 186	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ CMnd)
188	187	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑃 ∈ CMnd)
189		nn0ex 12474	. . . . . . . . . . 11 ⊢ ℕ₀ ∈ V
190	189	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ℕ₀ ∈ V)
191	23	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑖 ∈ ℕ₀) → 𝑃 ∈ LMod)
192	168	ffvelcdmda 7083	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑖 ∈ ℕ₀) → ((coe₁‘𝑥)‘𝑖) ∈ (Base‘(Scalar‘𝑃)))
193	6	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑖 ∈ ℕ₀) → 𝑅 ∈ Ring)
194	193, 93, 94	3syl 18	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑖 ∈ ℕ₀) → (mulGrp‘𝑃) ∈ Mnd)
195		simpr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑖 ∈ ℕ₀) → 𝑖 ∈ ℕ₀)
196	193, 100	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑖 ∈ ℕ₀) → (var₁‘𝑅) ∈ (Base‘𝑃))
197	91, 92, 194, 195, 196	mulgnn0cld 18969	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑖 ∈ ℕ₀) → (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃))
198	33, 47, 48, 49	lmodvscl 20481	. . . . . . . . . . . 12 ⊢ ((𝑃 ∈ LMod ∧ ((coe₁‘𝑥)‘𝑖) ∈ (Base‘(Scalar‘𝑃)) ∧ (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃)) → (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ (Base‘𝑃))
199	191, 192, 197, 198	syl3anc 1371	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑖 ∈ ℕ₀) → (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ (Base‘𝑃))
200		eqid 2732	. . . . . . . . . . 11 ⊢ (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) = (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
201	199, 200	fmptd 7110	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))):ℕ₀⟶(Base‘𝑃))
202		nfv 1917	. . . . . . . . . . . 12 ⊢ Ⅎ𝑖(𝜑 ∧ 𝑥 ∈ 𝑆)
203	202, 199, 200	fnmptd 6688	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) Fn ℕ₀)
204		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → ((coe₁‘𝑥)‘𝑖) = ((coe₁‘𝑥)‘𝑗))
205		oveq1 7412	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑗 → (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
206	204, 205	oveq12d 7423	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑗 → (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) = (((coe₁‘𝑥)‘𝑗)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
207		simplr 767	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → 𝑗 ∈ ℕ₀)
208		ovexd 7440	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (((coe₁‘𝑥)‘𝑗)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ V)
209	200, 206, 207, 208	fvmptd3 7018	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝑗) = (((coe₁‘𝑥)‘𝑗)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
210	162	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → 𝑥 ∈ (Base‘𝑃))
211		icossxr 13405	. . . . . . . . . . . . . . . . 17 ⊢ (-∞[,)𝑁) ⊆ ℝ^*
212	211, 76	sselid 3979	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐷‘𝑥) ∈ ℝ^*)
213	212	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (𝐷‘𝑥) ∈ ℝ^*)
214	55	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → 𝑁 ∈ ℝ^*)
215	207	nn0red 12529	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → 𝑗 ∈ ℝ)
216	215	rexrd 11260	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → 𝑗 ∈ ℝ^*)
217	80	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (𝐷‘𝑥) < 𝑁)
218		simpr 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → 𝑁 ≤ 𝑗)
219	213, 214, 216, 217, 218	xrltletrd 13136	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (𝐷‘𝑥) < 𝑗)
220	24, 1, 33, 9, 163	deg1lt 25606	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ (Base‘𝑃) ∧ 𝑗 ∈ ℕ₀ ∧ (𝐷‘𝑥) < 𝑗) → ((coe₁‘𝑥)‘𝑗) = (0_g‘𝑅))
221	210, 207, 219, 220	syl3anc 1371	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → ((coe₁‘𝑥)‘𝑗) = (0_g‘𝑅))
222	221	oveq1d 7420	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (((coe₁‘𝑥)‘𝑗)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) = ((0_g‘𝑅)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
223	148	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (0_g‘𝑅) = (0_g‘(Scalar‘𝑃)))
224	223	oveq1d 7420	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → ((0_g‘𝑅)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) = ((0_g‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
225	23	ad3antrrr 728	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → 𝑃 ∈ LMod)
226	95	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (mulGrp‘𝑃) ∈ Mnd)
227	101	ad3antrrr 728	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (var₁‘𝑅) ∈ (Base‘𝑃))
228	91, 92, 226, 207, 227	mulgnn0cld 18969	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → (𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃))
229		eqid 2732	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(Scalar‘𝑃)) = (0_g‘(Scalar‘𝑃))
230	33, 47, 48, 229, 10	lmod0vs 20497	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ LMod ∧ (𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝑃)) → ((0_g‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) = (0_g‘𝑃))
231	225, 228, 230	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → ((0_g‘(Scalar‘𝑃))( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) = (0_g‘𝑃))
232	224, 231	eqtrd 2772	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → ((0_g‘𝑅)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) = (0_g‘𝑃))
233	209, 222, 232	3eqtrd 2776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) ∧ 𝑁 ≤ 𝑗) → ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝑗) = (0_g‘𝑃))
234	3	nn0zd 12580	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℤ)
235	234	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑁 ∈ ℤ)
236	203, 233, 235	suppssnn0 32004	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) supp (0_g‘𝑃)) ⊆ (0..^𝑁))
237	190	mptexd 7222	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ V)
238	203	fnfund 6647	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → Fun (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))))
239		fvexd 6903	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (0_g‘𝑃) ∈ V)
240		suppssfifsupp 9374	. . . . . . . . . . 11 ⊢ ((((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∈ V ∧ Fun (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ∧ (0_g‘𝑃) ∈ V) ∧ ((0..^𝑁) ∈ Fin ∧ ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) supp (0_g‘𝑃)) ⊆ (0..^𝑁))) → (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) finSupp (0_g‘𝑃))
241	237, 238, 239, 183, 236, 240	syl32anc 1378	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) finSupp (0_g‘𝑃))
242	33, 10, 188, 190, 201, 236, 241	gsumres 19775	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑃 Σ_g ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ↾ (0..^𝑁))) = (𝑃 Σ_g (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))
243		fvexd 6903	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (coe₁‘𝑥) ∈ V)
244		ovexd 7440	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^𝑁) ∈ V)
245	155, 244	fexd 7225	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ V)
246	245	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐹 ∈ V)
247		offres 7966	. . . . . . . . . . . 12 ⊢ (((coe₁‘𝑥) ∈ V ∧ 𝐹 ∈ V) → (((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹) ↾ (0..^𝑁)) = (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)(𝐹 ↾ (0..^𝑁))))
248	243, 246, 247	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹) ↾ (0..^𝑁)) = (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)(𝐹 ↾ (0..^𝑁))))
249	165	ffnd 6715	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (coe₁‘𝑥) Fn ℕ₀)
250	155	ffnd 6715	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐹 Fn (0..^𝑁))
251	250	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝐹 Fn (0..^𝑁))
252		sseqin2 4214	. . . . . . . . . . . . . . . 16 ⊢ ((0..^𝑁) ⊆ ℕ₀ ↔ (ℕ₀ ∩ (0..^𝑁)) = (0..^𝑁))
253	169, 252	mpbi 229	. . . . . . . . . . . . . . 15 ⊢ (ℕ₀ ∩ (0..^𝑁)) = (0..^𝑁)
254		eqidd 2733	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ ℕ₀) → ((coe₁‘𝑥)‘𝑗) = ((coe₁‘𝑥)‘𝑗))
255		oveq1 7412	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 = 𝑗 → (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
256		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ (0..^𝑁))
257		ovexd 7440	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ V)
258	8, 255, 256, 257	fvmptd3 7018	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (𝐹‘𝑗) = (𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
259	249, 251, 190, 161, 253, 254, 258	ofval 7677	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹)‘𝑗) = (((coe₁‘𝑥)‘𝑗)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
260	169, 256	sselid 3979	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → 𝑗 ∈ ℕ₀)
261		ovexd 7440	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe₁‘𝑥)‘𝑗)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))) ∈ V)
262	200, 206, 260, 261	fvmptd3 7018	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝑗) = (((coe₁‘𝑥)‘𝑗)( ·_𝑠 ‘𝑃)(𝑗(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))
263	259, 262	eqtr4d 2775	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑆) ∧ 𝑗 ∈ (0..^𝑁)) → (((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝑗))
264	263	ralrimiva 3146	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∀𝑗 ∈ (0..^𝑁)(((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝑗))
265	249, 251, 190, 161, 253	offn 7679	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹) Fn (0..^𝑁))
266		ssidd 4004	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (0..^𝑁) ⊆ (0..^𝑁))
267		fvreseq0 7036	. . . . . . . . . . . . 13 ⊢ (((((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹) Fn (0..^𝑁) ∧ (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) Fn ℕ₀) ∧ ((0..^𝑁) ⊆ (0..^𝑁) ∧ (0..^𝑁) ⊆ ℕ₀)) → ((((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹) ↾ (0..^𝑁)) = ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ↾ (0..^𝑁)) ↔ ∀𝑗 ∈ (0..^𝑁)(((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝑗)))
268	265, 203, 266, 170, 267	syl22anc 837	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹) ↾ (0..^𝑁)) = ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ↾ (0..^𝑁)) ↔ ∀𝑗 ∈ (0..^𝑁)(((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹)‘𝑗) = ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))‘𝑗)))
269	264, 268	mpbird 256	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((coe₁‘𝑥) ∘_f ( ·_𝑠 ‘𝑃)𝐹) ↾ (0..^𝑁)) = ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ↾ (0..^𝑁)))
270		fnresdm 6666	. . . . . . . . . . . . . 14 ⊢ (𝐹 Fn (0..^𝑁) → (𝐹 ↾ (0..^𝑁)) = 𝐹)
271	250, 270	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)) = 𝐹)
272	271	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝐹 ↾ (0..^𝑁)) = 𝐹)
273	272	oveq2d 7421	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)(𝐹 ↾ (0..^𝑁))) = (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹))
274	248, 269, 273	3eqtr3rd 2781	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹) = ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ↾ (0..^𝑁)))
275	274	oveq2d 7421	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑃 Σ_g (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (𝑃 Σ_g ((𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))) ↾ (0..^𝑁))))
276	6	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑅 ∈ Ring)
277	1, 99, 33, 48, 90, 92, 163	ply1coe 21811	. . . . . . . . . 10 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ (Base‘𝑃)) → 𝑥 = (𝑃 Σ_g (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))
278	276, 162, 277	syl2anc 584	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 = (𝑃 Σ_g (𝑖 ∈ ℕ₀ ↦ (((coe₁‘𝑥)‘𝑖)( ·_𝑠 ‘𝑃)(𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅))))))
279	242, 275, 278	3eqtr4rd 2783	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 = (𝑃 Σ_g (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹)))
280	185, 279	jca 512	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (((coe₁‘𝑥) ↾ (0..^𝑁)) finSupp (0_g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σ_g (((coe₁‘𝑥) ↾ (0..^𝑁)) ∘_f ( ·_𝑠 ‘𝑃)𝐹))))
281	174, 180, 280	rspcedvd 3614	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ∃𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))(𝑎 finSupp (0_g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹))))
282	103, 8	fmptd 7110	. . . . . . . 8 ⊢ (𝜑 → 𝐹:(0..^𝑁)⟶(Base‘𝑃))
283	157, 33, 49, 47, 229, 48, 282, 23, 244	ellspd 21348	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))(𝑎 finSupp (0_g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)))))
284	283	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (𝑥 ∈ ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))) ↔ ∃𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))(𝑎 finSupp (0_g‘(Scalar‘𝑃)) ∧ 𝑥 = (𝑃 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)))))
285	281, 284	mpbird 256	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
286		imadmrn 6067	. . . . . . . 8 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹
287	155	fdmd 6725	. . . . . . . . 9 ⊢ (𝜑 → dom 𝐹 = (0..^𝑁))
288	287	imaeq2d 6057	. . . . . . . 8 ⊢ (𝜑 → (𝐹 “ dom 𝐹) = (𝐹 “ (0..^𝑁)))
289	286, 288	eqtr3id 2786	. . . . . . 7 ⊢ (𝜑 → ran 𝐹 = (𝐹 “ (0..^𝑁)))
290	289	fveq2d 6892	. . . . . 6 ⊢ (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
291	290	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝑃)‘(𝐹 “ (0..^𝑁))))
292	285, 291	eleqtrrd 2836	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ ((LSpan‘𝑃)‘ran 𝐹))
293	159, 292	eqelssd 4002	. . 3 ⊢ (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = 𝑆)
294		eqid 2732	. . . . 5 ⊢ (LSpan‘𝐸) = (LSpan‘𝐸)
295	41, 157, 294, 27	lsslsp 20618	. . . 4 ⊢ ((𝑃 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑃) ∧ ran 𝐹 ⊆ 𝑆) → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝐸)‘ran 𝐹))
296	23, 26, 156, 295	syl3anc 1371	. . 3 ⊢ (𝜑 → ((LSpan‘𝑃)‘ran 𝐹) = ((LSpan‘𝐸)‘ran 𝐹))
297	293, 296, 43	3eqtr3d 2780	. 2 ⊢ (𝜑 → ((LSpan‘𝐸)‘ran 𝐹) = (Base‘𝐸))
298		eqid 2732	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
299	24	fvexi 6902	. . . . . . 7 ⊢ 𝐷 ∈ V
300		cnvexg 7911	. . . . . . 7 ⊢ (𝐷 ∈ V → ^◡𝐷 ∈ V)
301		imaexg 7902	. . . . . . 7 ⊢ (^◡𝐷 ∈ V → (^◡𝐷 “ (-∞[,)𝑁)) ∈ V)
302	299, 300, 301	mp2b 10	. . . . . 6 ⊢ (^◡𝐷 “ (-∞[,)𝑁)) ∈ V
303	25, 302	eqeltri 2829	. . . . 5 ⊢ 𝑆 ∈ V
304	41, 47	resssca 17284	. . . . 5 ⊢ (𝑆 ∈ V → (Scalar‘𝑃) = (Scalar‘𝐸))
305	303, 304	ax-mp 5	. . . 4 ⊢ (Scalar‘𝑃) = (Scalar‘𝐸)
306	305	fveq2i 6891	. . 3 ⊢ (Base‘(Scalar‘𝑃)) = (Base‘(Scalar‘𝐸))
307		eqid 2732	. . 3 ⊢ (Scalar‘𝐸) = (Scalar‘𝐸)
308	41, 48	ressvsca 17285	. . . 4 ⊢ (𝑆 ∈ V → ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝐸))
309	303, 308	ax-mp 5	. . 3 ⊢ ( ·_𝑠 ‘𝑃) = ( ·_𝑠 ‘𝐸)
310		eqid 2732	. . 3 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
311	305	fveq2i 6891	. . 3 ⊢ (0_g‘(Scalar‘𝑃)) = (0_g‘(Scalar‘𝐸))
312		eqid 2732	. . 3 ⊢ (LBasis‘𝐸) = (LBasis‘𝐸)
313	41, 27	lsslvec 20711	. . . . 5 ⊢ ((𝑃 ∈ LVec ∧ 𝑆 ∈ (LSubSp‘𝑃)) → 𝐸 ∈ LVec)
314	22, 26, 313	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐸 ∈ LVec)
315	314	lveclmodd 20710	. . 3 ⊢ (𝜑 → 𝐸 ∈ LMod)
316	14, 5	eqeltrrd 2834	. . . . 5 ⊢ (𝜑 → (Scalar‘𝑃) ∈ DivRing)
317		drngnzr 20327	. . . . 5 ⊢ ((Scalar‘𝑃) ∈ DivRing → (Scalar‘𝑃) ∈ NzRing)
318	316, 317	syl 17	. . . 4 ⊢ (𝜑 → (Scalar‘𝑃) ∈ NzRing)
319	305, 318	eqeltrrid 2838	. . 3 ⊢ (𝜑 → (Scalar‘𝐸) ∈ NzRing)
320	121	ralrimiva 3146	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ (0..^𝑁)(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝐸))
321		drngnzr 20327	. . . . . . . . . 10 ⊢ (𝑅 ∈ DivRing → 𝑅 ∈ NzRing)
322	5, 321	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ∈ NzRing)
323	322	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑅 ∈ NzRing)
324	98	adantr 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑛 ∈ ℕ₀)
325		elfzonn0 13673	. . . . . . . . 9 ⊢ (𝑖 ∈ (0..^𝑁) → 𝑖 ∈ ℕ₀)
326	325	adantl 482	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → 𝑖 ∈ ℕ₀)
327	1, 99, 92, 323, 324, 326	ply1moneq 32653	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ↔ 𝑛 = 𝑖))
328	327	biimpd 228	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (0..^𝑁)) ∧ 𝑖 ∈ (0..^𝑁)) → ((𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) → 𝑛 = 𝑖))
329	328	anasss 467	. . . . 5 ⊢ ((𝜑 ∧ (𝑛 ∈ (0..^𝑁) ∧ 𝑖 ∈ (0..^𝑁))) → ((𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) → 𝑛 = 𝑖))
330	329	ralrimivva 3200	. . . 4 ⊢ (𝜑 → ∀𝑛 ∈ (0..^𝑁)∀𝑖 ∈ (0..^𝑁)((𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) → 𝑛 = 𝑖))
331		oveq1 7412	. . . . 5 ⊢ (𝑛 = 𝑖 → (𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)))
332	8, 331	f1mpt 7256	. . . 4 ⊢ (𝐹:(0..^𝑁)–1-1→(Base‘𝐸) ↔ (∀𝑛 ∈ (0..^𝑁)(𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) ∈ (Base‘𝐸) ∧ ∀𝑛 ∈ (0..^𝑁)∀𝑖 ∈ (0..^𝑁)((𝑛(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) = (𝑖(._g‘(mulGrp‘𝑃))(var₁‘𝑅)) → 𝑛 = 𝑖)))
333	320, 330, 332	sylanbrc 583	. . 3 ⊢ (𝜑 → 𝐹:(0..^𝑁)–1-1→(Base‘𝐸))
334	298, 306, 307, 309, 310, 311, 312, 294, 315, 319, 244, 333	islbs5 32484	. 2 ⊢ (𝜑 → (ran 𝐹 ∈ (LBasis‘𝐸) ↔ (∀𝑎 ∈ ((Base‘(Scalar‘𝑃)) ↑_m (0..^𝑁))((𝑎 finSupp (0_g‘(Scalar‘𝑃)) ∧ (𝐸 Σ_g (𝑎 ∘_f ( ·_𝑠 ‘𝑃)𝐹)) = (0_g‘𝐸)) → 𝑎 = ((0..^𝑁) × {(0_g‘(Scalar‘𝑃))})) ∧ ((LSpan‘𝐸)‘ran 𝐹) = (Base‘𝐸))))
335	154, 297, 334	mpbir2and 711	1 ⊢ (𝜑 → ran 𝐹 ∈ (LBasis‘𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ∩ cin 3946 ⊆ wss 3947 {csn 4627 class class class wbr 5147 ↦ cmpt 5230 × cxp 5673 ^◡ccnv 5674 dom cdm 5675 ran crn 5676 ↾ cres 5677 “ cima 5678 Fun wfun 6534 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 ‘cfv 6540 (class class class)co 7405 ∘_f cof 7664 supp csupp 8142 ↑_m cmap 8816 Fincfn 8935 finSupp cfsupp 9357 0cc0 11106 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℕ₀cn0 12468 ℤcz 12554 [,)cico 13322 ..^cfzo 13623 Basecbs 17140 ↾_s cress 17169 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 Σ_g cgsu 17382 Mndcmnd 18621 SubMndcsubmnd 18666 ._gcmg 18944 SubGrpcsubg 18994 CMndccmn 19642 mulGrpcmgp 19981 Ringcrg 20049 NzRingcnzr 20283 DivRingcdr 20307 LModclmod 20463 LSubSpclss 20534 LSpanclspn 20574 LBasisclbs 20677 LVecclvec 20705 var₁cv1 21691 Poly₁cpl1 21692 coe₁cco1 21693 deg₁ cdg1 25560

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-tpos 8207 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-sup 9433 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-ico 13326 df-fz 13481 df-fzo 13624 df-seq 13963 df-hash 14287 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-0g 17383 df-gsum 17384 df-prds 17389 df-pws 17391 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-ghm 19084 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-srg 20003 df-ring 20051 df-cring 20052 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-nzr 20284 df-drng 20309 df-subrg 20353 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lmhm 20625 df-lbs 20678 df-lvec 20706 df-sra 20777 df-rgmod 20778 df-cnfld 20937 df-dsmm 21278 df-frlm 21293 df-uvc 21329 df-lindf 21352 df-linds 21353 df-psr 21453 df-mvr 21454 df-mpl 21455 df-opsr 21457 df-psr1 21695 df-vr1 21696 df-ply1 21697 df-coe1 21698 df-mdeg 25561 df-deg1 25562

This theorem is referenced by: ply1degltdim 32696

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