aks6d1c6lem1 - Mathbox for metakunt

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Theorem aks6d1c6lem1 42609

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 6842	. . 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 6844	. 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 2737	. . . . 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 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 = 𝑈) ∧ 𝑖 ∈ (0...𝐴)) → 𝑔 = 𝑈)
7	6	fveq1d 6842	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 = 𝑈) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) = (𝑈‘𝑖))
8	7	oveq1d 7382	. . . . . . 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 5178	. . . . . 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 7383	. . . . 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 7402	. . . . 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 6955	. . . 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 6844	. . 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 20748	. . . . . . 7 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
17	15, 16	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ IDomn)
18		fzfid 13935	. . . . . 6 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
19		eqid 2736	. . . . . . . . . 10 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
20		eqid 2736	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
21	19, 20	mgpbas 20126	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
22		eqid 2736	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐾))) = (._g‘(mulGrp‘(Poly₁‘𝐾)))
23	15	fldcrngd 20719	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ CRing)
24		crngring 20226	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
25	23, 24	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Ring)
26		eqid 2736	. . . . . . . . . . . . 13 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
27	26	ply1ring 22211	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (Poly₁‘𝐾) ∈ Ring)
28	25, 27	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
29	19	ringmgp 20220	. . . . . . . . . . 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 12443	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
33	32	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ₀ ∈ V)
34		ovexd 7402	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...𝐴) ∈ V)
35	33, 34	elmapd 8787	. . . . . . . . . . . 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 7037	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑈‘𝑖) ∈ ℕ₀)
40		2fveq3 6845	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑖 → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))
41	40	oveq2d 7383	. . . . . . . . . . 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 2821	. . . . . . . . . 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 20224	. . . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . 15 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
48	47, 26, 20	vr1cl 22181	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
51	50	zrhrhm 21491	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
52	25, 51	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
53		zringbas 21433	. . . . . . . . . . . . . . . . . 18 ⊢ ℤ = (Base‘ℤ_ring)
54		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐾) = (Base‘𝐾)
55	53, 54	rhmf 20464	. . . . . . . . . . . . . . . . 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 13478	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (0...𝐴) → 𝑡 ∈ ℤ)
59	58	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝑡 ∈ ℤ)
60	57, 59	ffvelcdmd 7037	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾))
61		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
62	26, 61, 54, 20	ply1sclcl 22251	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾)) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾)))
63	46, 60, 62	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾)))
64		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
65	20, 64	mndcl 18710	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾)))
67	66	ralrimiva 3129	. . . . . . . . . . 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 3565	. . . . . . . . 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 19071	. . . . . . . 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 26090	. . . . . . . . . . 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 2991	. . . . . . . . . 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 2736	. . . . . . . . . . . . . . . 16 ⊢ (deg₁‘𝐾) = (deg₁‘𝐾)
76	75, 26, 20	deg1xrcl 26047	. . . . . . . . . . . . . . . . . . 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 11192	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ^*
79	78	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 0 ∈ ℝ^*)
80		1xr 11204	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 ∈ ℝ^*)
82	75, 26, 54, 61	deg1sclle 26077	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≤ 0)
83	46, 60, 82	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≤ 0)
84		0lt1 11672	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 < 1
85	84	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 0 < 1)
86	77, 79, 81, 83, 85	xrlelttrd 13111	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) < 1)
87	21, 22	mulg1 19057	. . . . . . . . . . . . . . . . . . . . 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 2742	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (var₁‘𝐾) = (1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
90	89	fveq2d 6844	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(var₁‘𝐾)) = ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
91		isfld 20717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
92		drngnzr 20725	. . . . . . . . . . . . . . . . . . . . . . 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 12453	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ₀
98	97	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 ∈ ℕ₀)
99	75, 26, 47, 19, 22	deg1pw 26086	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ NzRing ∧ 1 ∈ ℕ₀) → ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = 1)
100	96, 98, 99	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = 1)
101	90, 100	eqtr2d 2772	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 = ((deg₁‘𝐾)‘(var₁‘𝐾)))
102	86, 101	breqtrd 5111	. . . . . . . . . . . . . . . 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 26068	. . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(var₁‘𝐾)) = 1)
105	103, 104	eqtrd 2771	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) = 1)
106	105, 98	eqeltrd 2836	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) ∈ ℕ₀)
107		eqid 2736	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(Poly₁‘𝐾)) = (0_g‘(Poly₁‘𝐾))
108	75, 26, 107, 20	deg1nn0clb 26055	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 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 3129	. . . . . . . . . . 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 3565	. . . . . . . . 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 42571	. . . . . . . 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 3129	. . . . . 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 42579	. . . . 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 2737	. . . . . . . 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 6844	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → (𝑈‘𝑖) = (𝑈‘𝑡))
122	120	fveq2d 6844	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → ((ℤRHom‘𝐾)‘𝑖) = ((ℤRHom‘𝐾)‘𝑡))
123	122	fveq2d 6844	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))
124	123	oveq2d 7383	. . . . . . . . 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 7385	. . . . . . . 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 7402	. . . . . . . 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 6955	. . . . . . 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 6844	. . . . . 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 7036	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (𝑈‘𝑡) ∈ ℕ₀)
132	130, 66, 110, 131, 22, 75	deg1pow 42580	. . . . . . 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 7383	. . . . . . . 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 12500	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (𝑈‘𝑡) ∈ ℂ)
135	134	mulridd 11162	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((𝑈‘𝑡) · 1) = (𝑈‘𝑡))
136	133, 135	eqtrd 2771	. . . . . . 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 2771	. . . . . 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 2771	. . . . 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 15664	. . . 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 2771	. . 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 2771	. 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 2771	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 1087 = wceq 1542 ∈ wcel 2114 ≠ wne 2932 ∀wral 3051 {crab 3389 Vcvv 3429 class class class wbr 5085 {copab 5147 ↦ cmpt 5166 × cxp 5629 “ cima 5634 ⟶wf 6494 ‘cfv 6498 (class class class)co 7367 ∈ cmpo 7369 ↑_m cmap 8773 0cc0 11038 1c1 11039 · cmul 11043 ℝ^*cxr 11178 < clt 11179 ≤ cle 11180 − cmin 11377 / cdiv 11807 ℕcn 12174 ℕ₀cn0 12437 ℤcz 12524 ...cfz 13461 ↑cexp 14023 ♯chash 14292 Σcsu 15648 ∥ cdvds 16221 gcd cgcd 16463 ℙcprime 16640 Basecbs 17179 +_gcplusg 17220 0_gc0g 17402 Σ_g cgsu 17403 Mndcmnd 18702 ._gcmg 19043 mulGrpcmgp 20121 Ringcrg 20214 CRingccrg 20215 RingHom crh 20449 RingIso crs 20450 NzRingcnzr 20489 IDomncidom 20670 DivRingcdr 20706 Fieldcfield 20707 ℤ_ringczring 21426 ℤRHomczrh 21479 chrcchr 21481 ℤ/nℤczn 21482 algSccascl 21832 var₁cv1 22139 Poly₁cpl1 22140 eval₁ce1 22279 deg₁cdg1 26019 PrimRoots cprimroots 42530

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-ofr 7632 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-er 8643 df-map 8775 df-pm 8776 df-ixp 8846 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-oi 9425 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-z 12525 df-dec 12645 df-uz 12789 df-rp 12943 df-fz 13462 df-fzo 13609 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-hom 17244 df-cco 17245 df-0g 17404 df-gsum 17405 df-prds 17410 df-pws 17412 df-mre 17548 df-mrc 17549 df-acs 17551 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-ghm 19188 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-rhm 20452 df-nzr 20490 df-subrng 20523 df-subrg 20547 df-rlreg 20671 df-domn 20672 df-idom 20673 df-drng 20708 df-field 20709 df-lmod 20857 df-lss 20927 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-ascl 21835 df-psr 21889 df-mvr 21890 df-mpl 21891 df-opsr 21893 df-psr1 22143 df-vr1 22144 df-ply1 22145 df-coe1 22146 df-mdeg 26020 df-deg1 26021

This theorem is referenced by: aks6d1c6lem3 42611

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