aks6d1c6lem1 - Mathbox for metakunt

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Theorem aks6d1c6lem1 42211

Description: Lemma for claim 6, deduce exact degree of the polynomial. (Contributed by metakunt, 7-May-2025.)

**Hypotheses**
Ref	Expression
aks6d1c6.1	⊢ ∼ = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓 ∈ (Base‘(Poly₁‘𝐾)) ∧ ∀𝑦 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅)(𝑒(._g‘(mulGrp‘𝐾))(((eval₁‘𝐾)‘𝑓)‘𝑦)) = (((eval₁‘𝐾)‘𝑓)‘(𝑒(._g‘(mulGrp‘𝐾))𝑦)))}
aks6d1c6.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c6.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c6.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c6.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c6.6	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c6.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c6.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c6.9	⊢ (𝜑 → 𝐴 < 𝑃)
aks6d1c6.10	⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c6.11	⊢ (𝜑 → 𝐴 ∈ ℕ₀)
aks6d1c6.12	⊢ 𝐸 = (𝑘 ∈ ℕ₀, 𝑙 ∈ ℕ₀ ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c6.13	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c6.14	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c6.15	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(._g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c6.16	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c6.17	⊢ 𝐻 = (ℎ ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ (((eval₁‘𝐾)‘(𝐺‘ℎ))‘𝑀))
aks6d1c6.18	⊢ 𝐷 = (♯‘(𝐿 “ (𝐸 “ (ℕ₀ × ℕ₀))))
aks6d1c6.19	⊢ 𝑆 = {𝑠 ∈ (ℕ₀ ↑_m (0...𝐴)) ∣ Σ𝑡 ∈ (0...𝐴)(𝑠‘𝑡) ≤ (𝐷 − 1)}
aks6d1c6lem1.1	⊢ (𝜑 → 𝑈 ∈ (ℕ₀ ↑_m (0...𝐴)))

**Assertion**
Ref	Expression
aks6d1c6lem1	⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐺‘𝑈)) = Σ𝑡 ∈ (0...𝐴)(𝑈‘𝑡))

Distinct variable groups: 𝐴,𝑔,𝑖 𝑡,𝐴,𝑖 𝑔,𝐾,𝑖 𝑡,𝐾 𝑈,𝑔,𝑖 𝑡,𝑈 𝜑,𝑔,𝑖 𝜑,𝑡 Allowed substitution hints: 𝜑(𝑥,𝑦,𝑒,𝑓,ℎ,𝑘,𝑠,𝑎,𝑙) 𝐴(𝑥,𝑦,𝑒,𝑓,ℎ,𝑘,𝑠,𝑎,𝑙) 𝐷(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝑃(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) ∼ (𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝑅(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝑆(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝑈(𝑥,𝑦,𝑒,𝑓,ℎ,𝑘,𝑠,𝑎,𝑙) 𝐸(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝐺(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝐻(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝐾(𝑥,𝑦,𝑒,𝑓,ℎ,𝑘,𝑠,𝑎,𝑙) 𝐿(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝑀(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙) 𝑁(𝑥,𝑦,𝑡,𝑒,𝑓,𝑔,ℎ,𝑖,𝑘,𝑠,𝑎,𝑙)

**Proof of Theorem aks6d1c6lem1**
Step	Hyp	Ref	Expression
1		aks6d1c6.10	. . . . 5 ⊢ 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
2	1	a1i 11	. . . 4 ⊢ (𝜑 → 𝐺 = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))
3	2	fveq1d 6824	. . 3 ⊢ (𝜑 → (𝐺‘𝑈) = ((𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘𝑈))
4	3	fveq2d 6826	. 2 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐺‘𝑈)) = ((deg₁‘𝐾)‘((𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘𝑈)))
5		eqidd 2732	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) = (𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))
6		simplr 768	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 = 𝑈) ∧ 𝑖 ∈ (0...𝐴)) → 𝑔 = 𝑈)
7	6	fveq1d 6824	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 = 𝑈) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) = (𝑈‘𝑖))
8	7	oveq1d 7361	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑔 = 𝑈) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))
9	8	mpteq2dva 5182	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 = 𝑈) → (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))
10	9	oveq2d 7362	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 = 𝑈) → ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) = ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
11		aks6d1c6lem1.1	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ (ℕ₀ ↑_m (0...𝐴)))
12		ovexd 7381	. . . . 5 ⊢ (𝜑 → ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ V)
13	5, 10, 11, 12	fvmptd 6936	. . . 4 ⊢ (𝜑 → ((𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘𝑈) = ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
14	13	fveq2d 6826	. . 3 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘𝑈)) = ((deg₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))))
15		aks6d1c6.3	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Field)
16		fldidom 20686	. . . . . . 7 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ IDomn)
18		fzfid 13880	. . . . . 6 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
19		eqid 2731	. . . . . . . . . 10 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
20		eqid 2731	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
21	19, 20	mgpbas 20063	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
22		eqid 2731	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐾))) = (._g‘(mulGrp‘(Poly₁‘𝐾)))
23	15	fldcrngd 20657	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ CRing)
24		crngring 20163	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Ring)
26		eqid 2731	. . . . . . . . . . . . 13 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
27	26	ply1ring 22160	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (Poly₁‘𝐾) ∈ Ring)
28	25, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
29	19	ringmgp 20157	. . . . . . . . . . 11 ⊢ ((Poly₁‘𝐾) ∈ Ring → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
30	28, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
31	30	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
32		nn0ex 12387	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
33	32	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ₀ ∈ V)
34		ovexd 7381	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...𝐴) ∈ V)
35	33, 34	elmapd 8764	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 ∈ (ℕ₀ ↑_m (0...𝐴)) ↔ 𝑈:(0...𝐴)⟶ℕ₀))
36	11, 35	mpbid 232	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈:(0...𝐴)⟶ℕ₀)
37	36	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑈:(0...𝐴)⟶ℕ₀)
38		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ (0...𝐴))
39	37, 38	ffvelcdmd 7018	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑈‘𝑖) ∈ ℕ₀)
40		2fveq3 6827	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑖 → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))
41	40	oveq2d 7362	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑖 → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) = ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))
42	41	eleq1d 2816	. . . . . . . . . 10 ⊢ (𝑡 = 𝑖 → (((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾)) ↔ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly₁‘𝐾))))
43		ringmnd 20161	. . . . . . . . . . . . . . 15 ⊢ ((Poly₁‘𝐾) ∈ Ring → (Poly₁‘𝐾) ∈ Mnd)
44	28, 43	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Mnd)
45	44	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (Poly₁‘𝐾) ∈ Mnd)
46	25	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝐾 ∈ Ring)
47		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
48	47, 26, 20	vr1cl 22130	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)))
49	46, 48	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)))
50		eqid 2731	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
51	50	zrhrhm 21448	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
52	25, 51	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
53		zringbas 21390	. . . . . . . . . . . . . . . . . 18 ⊢ ℤ = (Base‘ℤ_ring)
54		eqid 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐾) = (Base‘𝐾)
55	53, 54	rhmf 20402	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
56	52, 55	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
57	56	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
58		elfzelz 13424	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (0...𝐴) → 𝑡 ∈ ℤ)
59	58	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝑡 ∈ ℤ)
60	57, 59	ffvelcdmd 7018	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾))
61		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
62	26, 61, 54, 20	ply1sclcl 22200	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾)) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾)))
63	46, 60, 62	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾)))
64		eqid 2731	. . . . . . . . . . . . . 14 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
65	20, 64	mndcl 18650	. . . . . . . . . . . . 13 ⊢ (((Poly₁‘𝐾) ∈ Mnd ∧ (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)) ∧ ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾))) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾)))
66	45, 49, 63, 65	syl3anc 1373	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾)))
67	66	ralrimiva 3124	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑡 ∈ (0...𝐴)((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾)))
68	67	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ∀𝑡 ∈ (0...𝐴)((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾)))
69	42, 68, 38	rspcdva 3573	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ∈ (Base‘(Poly₁‘𝐾)))
70	21, 22, 31, 39, 69	mulgnn0cld 19008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)))
71	26	ply1idom 26057	. . . . . . . . . . 11 ⊢ (𝐾 ∈ IDomn → (Poly₁‘𝐾) ∈ IDomn)
72	17, 71	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ IDomn)
73	72	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (Poly₁‘𝐾) ∈ IDomn)
74	41	neeq1d 2987	. . . . . . . . . 10 ⊢ (𝑡 = 𝑖 → (((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≠ (0_g‘(Poly₁‘𝐾)) ↔ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ≠ (0_g‘(Poly₁‘𝐾))))
75		eqid 2731	. . . . . . . . . . . . . . . 16 ⊢ (deg₁‘𝐾) = (deg₁‘𝐾)
76	75, 26, 20	deg1xrcl 26014	. . . . . . . . . . . . . . . . . . 19 ⊢ (((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ ℝ^*)
77	63, 76	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ ℝ^*)
78		0xr 11159	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ^*
79	78	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 0 ∈ ℝ^*)
80		1xr 11171	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 ∈ ℝ^*)
82	75, 26, 54, 61	deg1sclle 26044	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≤ 0)
83	46, 60, 82	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≤ 0)
84		0lt1 11639	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 < 1
85	84	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 0 < 1)
86	77, 79, 81, 83, 85	xrlelttrd 13059	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) < 1)
87	21, 22	mulg1 18994	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)) → (1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) = (var₁‘𝐾))
88	49, 87	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) = (var₁‘𝐾))
89	88	eqcomd 2737	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (var₁‘𝐾) = (1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
90	89	fveq2d 6826	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(var₁‘𝐾)) = ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
91		isfld 20655	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
92		drngnzr 20663	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
93	92	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing) → 𝐾 ∈ NzRing)
94	91, 93	sylbi 217	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐾 ∈ Field → 𝐾 ∈ NzRing)
95	15, 94	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 ∈ NzRing)
96	95	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝐾 ∈ NzRing)
97		1nn0 12397	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ₀
98	97	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 ∈ ℕ₀)
99	75, 26, 47, 19, 22	deg1pw 26053	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ NzRing ∧ 1 ∈ ℕ₀) → ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = 1)
100	96, 98, 99	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = 1)
101	90, 100	eqtr2d 2767	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 = ((deg₁‘𝐾)‘(var₁‘𝐾)))
102	86, 101	breqtrd 5115	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) < ((deg₁‘𝐾)‘(var₁‘𝐾)))
103	26, 75, 46, 20, 64, 49, 63, 102	deg1add 26035	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) = ((deg₁‘𝐾)‘(var₁‘𝐾)))
104	90, 100	eqtrd 2766	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(var₁‘𝐾)) = 1)
105	103, 104	eqtrd 2766	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) = 1)
106	105, 98	eqeltrd 2831	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) ∈ ℕ₀)
107		eqid 2731	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(Poly₁‘𝐾)) = (0_g‘(Poly₁‘𝐾))
108	75, 26, 107, 20	deg1nn0clb 26022	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Ring ∧ ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾))) → (((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≠ (0_g‘(Poly₁‘𝐾)) ↔ ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) ∈ ℕ₀))
109	46, 66, 108	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≠ (0_g‘(Poly₁‘𝐾)) ↔ ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) ∈ ℕ₀))
110	106, 109	mpbird 257	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≠ (0_g‘(Poly₁‘𝐾)))
111	110	ralrimiva 3124	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑡 ∈ (0...𝐴)((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≠ (0_g‘(Poly₁‘𝐾)))
112	111	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ∀𝑡 ∈ (0...𝐴)((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≠ (0_g‘(Poly₁‘𝐾)))
113	74, 112, 38	rspcdva 3573	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) ≠ (0_g‘(Poly₁‘𝐾)))
114	73, 69, 113, 39, 22	idomnnzpownz 42173	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ≠ (0_g‘(Poly₁‘𝐾)))
115	70, 114	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)) ∧ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ≠ (0_g‘(Poly₁‘𝐾))))
116	115	ralrimiva 3124	. . . . . 6 ⊢ (𝜑 → ∀𝑖 ∈ (0...𝐴)(((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ∈ (Base‘(Poly₁‘𝐾)) ∧ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) ≠ (0_g‘(Poly₁‘𝐾))))
117	17, 18, 116	deg1gprod 42181	. . . . 5 ⊢ (𝜑 → (((deg₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) = Σ𝑡 ∈ (0...𝐴)((deg₁‘𝐾)‘((𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘𝑡)) ∧ 0 ≤ ((deg₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))))
118	117	simpld 494	. . . 4 ⊢ (𝜑 → ((deg₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) = Σ𝑡 ∈ (0...𝐴)((deg₁‘𝐾)‘((𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘𝑡)))
119		eqidd 2732	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))
120		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → 𝑖 = 𝑡)
121	120	fveq2d 6826	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → (𝑈‘𝑖) = (𝑈‘𝑡))
122	120	fveq2d 6826	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → ((ℤRHom‘𝐾)‘𝑖) = ((ℤRHom‘𝐾)‘𝑡))
123	122	fveq2d 6826	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))
124	123	oveq2d 7362	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))) = ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))))
125	121, 124	oveq12d 7364	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝑈‘𝑡)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))))
126		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝑡 ∈ (0...𝐴))
127		ovexd 7381	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((𝑈‘𝑡)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) ∈ V)
128	119, 125, 126, 127	fvmptd 6936	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘𝑡) = ((𝑈‘𝑡)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))))
129	128	fveq2d 6826	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘𝑡)) = ((deg₁‘𝐾)‘((𝑈‘𝑡)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))))))
130	17	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝐾 ∈ IDomn)
131	36	ffvelcdmda 7017	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (𝑈‘𝑡) ∈ ℕ₀)
132	130, 66, 110, 131, 22, 75	deg1pow 42182	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((𝑈‘𝑡)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))))) = ((𝑈‘𝑡) · ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))))))
133	105	oveq2d 7362	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((𝑈‘𝑡) · ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))))) = ((𝑈‘𝑡) · 1))
134	131	nn0cnd 12444	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (𝑈‘𝑡) ∈ ℂ)
135	134	mulridd 11129	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((𝑈‘𝑡) · 1) = (𝑈‘𝑡))
136	133, 135	eqtrd 2766	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((𝑈‘𝑡) · ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))))) = (𝑈‘𝑡))
137	132, 136	eqtrd 2766	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((𝑈‘𝑡)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))))) = (𝑈‘𝑡))
138	129, 137	eqtrd 2766	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘𝑡)) = (𝑈‘𝑡))
139	138	sumeq2dv 15609	. . . 4 ⊢ (𝜑 → Σ𝑡 ∈ (0...𝐴)((deg₁‘𝐾)‘((𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))‘𝑡)) = Σ𝑡 ∈ (0...𝐴)(𝑈‘𝑡))
140	118, 139	eqtrd 2766	. . 3 ⊢ (𝜑 → ((deg₁‘𝐾)‘((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑈‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖))))))) = Σ𝑡 ∈ (0...𝐴)(𝑈‘𝑡))
141	14, 140	eqtrd 2766	. 2 ⊢ (𝜑 → ((deg₁‘𝐾)‘((𝑔 ∈ (ℕ₀ ↑_m (0...𝐴)) ↦ ((mulGrp‘(Poly₁‘𝐾)) Σ_g (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(._g‘(mulGrp‘(Poly₁‘𝐾)))((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))‘𝑈)) = Σ𝑡 ∈ (0...𝐴)(𝑈‘𝑡))
142	4, 141	eqtrd 2766	1 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐺‘𝑈)) = Σ𝑡 ∈ (0...𝐴)(𝑈‘𝑡))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 {crab 3395 Vcvv 3436 class class class wbr 5089 {copab 5151 ↦ cmpt 5170 × cxp 5612 “ cima 5617 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 ↑_m cmap 8750 0cc0 11006 1c1 11007 · cmul 11011 ℝ^*cxr 11145 < clt 11146 ≤ cle 11147 − cmin 11344 / cdiv 11774 ℕcn 12125 ℕ₀cn0 12381 ℤcz 12468 ...cfz 13407 ↑cexp 13968 ♯chash 14237 Σcsu 15593 ∥ cdvds 16163 gcd cgcd 16405 ℙcprime 16582 Basecbs 17120 +_gcplusg 17161 0_gc0g 17343 Σ_g cgsu 17344 Mndcmnd 18642 ._gcmg 18980 mulGrpcmgp 20058 Ringcrg 20151 CRingccrg 20152 RingHom crh 20387 RingIso crs 20388 NzRingcnzr 20427 IDomncidom 20608 DivRingcdr 20644 Fieldcfield 20645 ℤ_ringczring 21383 ℤRHomczrh 21436 chrcchr 21438 ℤ/nℤczn 21439 algSccascl 21789 var₁cv1 22088 Poly₁cpl1 22089 eval₁ce1 22229 deg₁cdg1 25986 PrimRoots cprimroots 42132

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-ofr 7611 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-tpos 8156 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-pm 8753 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-sup 9326 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-rp 12891 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-0g 17345 df-gsum 17346 df-prds 17351 df-pws 17353 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-mulg 18981 df-subg 19036 df-ghm 19125 df-cntz 19229 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-cring 20154 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-rhm 20390 df-nzr 20428 df-subrng 20461 df-subrg 20485 df-rlreg 20609 df-domn 20610 df-idom 20611 df-drng 20646 df-field 20647 df-lmod 20795 df-lss 20865 df-cnfld 21292 df-zring 21384 df-zrh 21440 df-ascl 21792 df-psr 21846 df-mvr 21847 df-mpl 21848 df-opsr 21850 df-psr1 22092 df-vr1 22093 df-ply1 22094 df-coe1 22095 df-mdeg 25987 df-deg1 25988

This theorem is referenced by: aks6d1c6lem3 42213

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