aks6d1c6lem1 - Mathbox for metakunt

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Theorem aks6d1c6lem1 43357

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 6892	. . 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 6894	. 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 2761	. . . . 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 781	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑔 = 𝑈) ∧ 𝑖 ∈ (0...𝐴)) → 𝑔 = 𝑈)
7	6	fveq1d 6892	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑔 = 𝑈) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔‘𝑖) = (𝑈‘𝑖))
8	7	oveq1d 7439	. . . . . . 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 5197	. . . . . 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 7440	. . . . 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 7459	. . . . 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 7006	. . . 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 6894	. . 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 21038	. . . . . . 7 ⊢ (𝐾 ∈ Field → 𝐾 ∈ IDomn)
17	15, 16	syl 18	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ IDomn)
18		fzfid 14123	. . . . . 6 ⊢ (𝜑 → (0...𝐴) ∈ Fin)
19		eqid 2760	. . . . . . . . . 10 ⊢ (mulGrp‘(Poly₁‘𝐾)) = (mulGrp‘(Poly₁‘𝐾))
20		eqid 2760	. . . . . . . . . 10 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(Poly₁‘𝐾))
21	19, 20	mgpbas 20374	. . . . . . . . 9 ⊢ (Base‘(Poly₁‘𝐾)) = (Base‘(mulGrp‘(Poly₁‘𝐾)))
22		eqid 2760	. . . . . . . . 9 ⊢ (._g‘(mulGrp‘(Poly₁‘𝐾))) = (._g‘(mulGrp‘(Poly₁‘𝐾)))
23	15	fldcrngd 21004	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐾 ∈ CRing)
24		crngring 20481	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ CRing → 𝐾 ∈ Ring)
25	23, 24	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐾 ∈ Ring)
26		eqid 2760	. . . . . . . . . . . . 13 ⊢ (Poly₁‘𝐾) = (Poly₁‘𝐾)
27	26	ply1ring 22574	. . . . . . . . . . . 12 ⊢ (𝐾 ∈ Ring → (Poly₁‘𝐾) ∈ Ring)
28	25, 27	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Ring)
29	19	ringmgp 20474	. . . . . . . . . . 11 ⊢ ((Poly₁‘𝐾) ∈ Ring → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
30	28, 29	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
31	30	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (mulGrp‘(Poly₁‘𝐾)) ∈ Mnd)
32		nn0ex 12619	. . . . . . . . . . . . . 14 ⊢ ℕ₀ ∈ V
33	32	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℕ₀ ∈ V)
34		ovexd 7459	. . . . . . . . . . . . 13 ⊢ (𝜑 → (0...𝐴) ∈ V)
35	33, 34	elmapd 8868	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑈 ∈ (ℕ₀ ↑_m (0...𝐴)) ↔ 𝑈:(0...𝐴)⟶ℕ₀))
36	11, 35	mpbid 235	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈:(0...𝐴)⟶ℕ₀)
37	36	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑈:(0...𝐴)⟶ℕ₀)
38		simpr 490	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → 𝑖 ∈ (0...𝐴))
39	37, 38	ffvelcdmd 7090	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (𝑈‘𝑖) ∈ ℕ₀)
40		2fveq3 6895	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑖 → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))
41	40	oveq2d 7440	. . . . . . . . . . 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 2845	. . . . . . . . . 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 20479	. . . . . . . . . . . . . . 15 ⊢ ((Poly₁‘𝐾) ∈ Ring → (Poly₁‘𝐾) ∈ Mnd)
44	28, 43	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ Mnd)
45	44	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (Poly₁‘𝐾) ∈ Mnd)
46	25	adantr 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝐾 ∈ Ring)
47		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (var₁‘𝐾) = (var₁‘𝐾)
48	47, 26, 20	vr1cl 22544	. . . . . . . . . . . . . 14 ⊢ (𝐾 ∈ Ring → (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)))
49	46, 48	syl 18	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)))
50		eqid 2760	. . . . . . . . . . . . . . . . . . 19 ⊢ (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
51	50	zrhrhm 21826	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
52	25, 51	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾))
53		zringbas 21768	. . . . . . . . . . . . . . . . . 18 ⊢ ℤ = (Base‘ℤ_ring)
54		eqid 2760	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘𝐾) = (Base‘𝐾)
55	53, 54	rhmf 20724	. . . . . . . . . . . . . . . . 17 ⊢ ((ℤRHom‘𝐾) ∈ (ℤ_ring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
56	52, 55	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
57	56	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
58		elfzelz 13663	. . . . . . . . . . . . . . . 16 ⊢ (𝑡 ∈ (0...𝐴) → 𝑡 ∈ ℤ)
59	58	adantl 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝑡 ∈ ℤ)
60	57, 59	ffvelcdmd 7090	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾))
61		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (algSc‘(Poly₁‘𝐾)) = (algSc‘(Poly₁‘𝐾))
62	26, 61, 54, 20	ply1sclcl 22614	. . . . . . . . . . . . . 14 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾)) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾)))
63	46, 60, 62	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾)))
64		eqid 2760	. . . . . . . . . . . . . 14 ⊢ (+_g‘(Poly₁‘𝐾)) = (+_g‘(Poly₁‘𝐾))
65	20, 64	mndcl 18940	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾)))
67	66	ralrimiva 3154	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑡 ∈ (0...𝐴)((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ (Base‘(Poly₁‘𝐾)))
68	67	adantr 486	. . . . . . . . . 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 3577	. . . . . . . . 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 19314	. . . . . . . 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 26452	. . . . . . . . . . 11 ⊢ (𝐾 ∈ IDomn → (Poly₁‘𝐾) ∈ IDomn)
72	17, 71	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (Poly₁‘𝐾) ∈ IDomn)
73	72	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (0...𝐴)) → (Poly₁‘𝐾) ∈ IDomn)
74	41	neeq1d 3014	. . . . . . . . . 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 2760	. . . . . . . . . . . . . . . 16 ⊢ (deg₁‘𝐾) = (deg₁‘𝐾)
76	75, 26, 20	deg1xrcl 26409	. . . . . . . . . . . . . . . . . . 19 ⊢ (((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)) ∈ (Base‘(Poly₁‘𝐾)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ ℝ^*)
77	63, 76	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ∈ ℝ^*)
78		0xr 11363	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ^*
79	78	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 0 ∈ ℝ^*)
80		1xr 11375	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
81	80	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 ∈ ℝ^*)
82	75, 26, 54, 61	deg1sclle 26439	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑡) ∈ (Base‘𝐾)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≤ 0)
83	46, 60, 82	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≤ 0)
84		0lt1 11845	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 < 1
85	84	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 0 < 1)
86	77, 79, 81, 83, 85	xrlelttrd 13296	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) < 1)
87	21, 22	mulg1 19300	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((var₁‘𝐾) ∈ (Base‘(Poly₁‘𝐾)) → (1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) = (var₁‘𝐾))
88	49, 87	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)) = (var₁‘𝐾))
89	88	eqcomd 2766	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (var₁‘𝐾) = (1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾)))
90	89	fveq2d 6894	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(var₁‘𝐾)) = ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))))
91		isfld 21002	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐾 ∈ Field ↔ (𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing))
92		drngnzr 21011	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐾 ∈ DivRing → 𝐾 ∈ NzRing)
93	92	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐾 ∈ DivRing ∧ 𝐾 ∈ CRing) → 𝐾 ∈ NzRing)
94	91, 93	sylbi 220	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐾 ∈ Field → 𝐾 ∈ NzRing)
95	15, 94	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐾 ∈ NzRing)
96	95	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝐾 ∈ NzRing)
97		1nn0 12629	. . . . . . . . . . . . . . . . . . . 20 ⊢ 1 ∈ ℕ₀
98	97	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 ∈ ℕ₀)
99	75, 26, 47, 19, 22	deg1pw 26448	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐾 ∈ NzRing ∧ 1 ∈ ℕ₀) → ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = 1)
100	96, 98, 99	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(1(._g‘(mulGrp‘(Poly₁‘𝐾)))(var₁‘𝐾))) = 1)
101	90, 100	eqtr2d 2796	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 1 = ((deg₁‘𝐾)‘(var₁‘𝐾)))
102	86, 101	breqtrd 5130	. . . . . . . . . . . . . . . 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 26430	. . . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘(var₁‘𝐾)) = 1)
105	103, 104	eqtrd 2795	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) = 1)
106	105, 98	eqeltrd 2860	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((deg₁‘𝐾)‘((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))) ∈ ℕ₀)
107		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (0_g‘(Poly₁‘𝐾)) = (0_g‘(Poly₁‘𝐾))
108	75, 26, 107, 20	deg1nn0clb 26417	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 260	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≠ (0_g‘(Poly₁‘𝐾)))
111	110	ralrimiva 3154	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑡 ∈ (0...𝐴)((var₁‘𝐾)(+_g‘(Poly₁‘𝐾))((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡))) ≠ (0_g‘(Poly₁‘𝐾)))
112	111	adantr 486	. . . . . . . . . 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 3577	. . . . . . . . 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 43319	. . . . . . . 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 521	. . . . . . 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 3154	. . . . . 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 43327	. . . . 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 500	. . . 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 2761	. . . . . . . 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 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → 𝑖 = 𝑡)
121	120	fveq2d 6894	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → (𝑈‘𝑖) = (𝑈‘𝑡))
122	120	fveq2d 6894	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → ((ℤRHom‘𝐾)‘𝑖) = ((ℤRHom‘𝐾)‘𝑡))
123	122	fveq2d 6894	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑡 ∈ (0...𝐴)) ∧ 𝑖 = 𝑡) → ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)) = ((algSc‘(Poly₁‘𝐾))‘((ℤRHom‘𝐾)‘𝑡)))
124	123	oveq2d 7440	. . . . . . . . 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 7442	. . . . . . . 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 490	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝑡 ∈ (0...𝐴))
127		ovexd 7459	. . . . . . . 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 7006	. . . . . . 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 6894	. . . . . 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 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → 𝐾 ∈ IDomn)
131	36	ffvelcdmda 7089	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (𝑈‘𝑡) ∈ ℕ₀)
132	130, 66, 110, 131, 22, 75	deg1pow 43328	. . . . . . 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 7440	. . . . . . . 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 12676	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → (𝑈‘𝑡) ∈ ℂ)
135	134	mulridd 11333	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (0...𝐴)) → ((𝑈‘𝑡) · 1) = (𝑈‘𝑡))
136	133, 135	eqtrd 2795	. . . . . . 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 2795	. . . . . 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 2795	. . . . 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 15878	. . . 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 2795	. . 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 2795	. 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 2795	1 ⊢ (𝜑 → ((deg₁‘𝐾)‘(𝐺‘𝑈)) = Σ𝑡 ∈ (0...𝐴)(𝑈‘𝑡))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 ∀wral 3076 {crab 3412 Vcvv 3450 class class class wbr 5102 {copab 5166 ↦ cmpt 5185 × cxp 5649 “ cima 5654 ⟶wf 6539 ‘cfv 6543 (class class class)co 7424 ∈ cmpo 7426 ↑_m cmap 8855 0cc0 11207 1c1 11208 · cmul 11212 ℝ^*cxr 11349 < clt 11350 ≤ cle 11351 − cmin 11548 / cdiv 11980 ℕcn 12342 ℕ₀cn0 12613 ℤcz 12700 ...cfz 13646 ↑cexp 14211 ♯chash 14481 Σcsu 15862 ∥ cdvds 16431 gcd cgcd 16673 ℙcprime 16855 Basecbs 17396 +_gcplusg 17437 0_gc0g 17619 Σ_g cgsu 17620 Mndcmnd 18932 ._gcmg 19286 mulGrpcmgp 20369 Ringcrg 20468 CRingccrg 20469 RingHom crh 20708 RingIso crs 20709 NzRingcnzr 20771 IDomncidom 20954 DivRingcdr 20989 Fieldcfield 20990 ℤ_ringczring 21761 ℤRHomczrh 21814 chrcchr 21816 ℤ/nℤczn 21817 algSccascl 22169 var₁cv1 22503 Poly₁cpl1 22504 eval₁ce1 22641 deg₁cdg1 26381 PrimRoots cprimroots 43278

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5327 ax-pr 5391 ax-un 7755 ax-inf2 9649 ax-cnex 11263 ax-resscn 11264 ax-1cn 11265 ax-icn 11266 ax-addcl 11267 ax-addrcl 11268 ax-mulcl 11269 ax-mulrcl 11270 ax-mulcom 11271 ax-addass 11272 ax-mulass 11273 ax-distr 11274 ax-i2m1 11275 ax-1ne0 11276 ax-1rid 11277 ax-rnegex 11278 ax-rrecex 11279 ax-cnre 11280 ax-pre-lttri 11281 ax-pre-lttrn 11282 ax-pre-ltadd 11283 ax-pre-mulgt0 11284 ax-pre-sup 11285 ax-addf 11286 ax-mulf 11287

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3848 df-ss 3897 df-pss 4013 df-dif 4046 df-in 4090 df-un 4150 df-nul 4278 df-if 4482 df-pw 4558 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4835 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7381 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7697 df-ofr 7698 df-om 7882 df-1st 8005 df-2nd 8006 df-supp 8185 df-tpos 8251 df-frecs 8307 df-wrecs 8338 df-recs 8387 df-rdg 8426 df-1o 8484 df-2o 8485 df-er 8725 df-map 8857 df-pm 8858 df-ixp 8934 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-fsupp 9362 df-sup 9441 df-oi 9511 df-card 10027 df-pnf 11352 df-mnf 11353 df-xr 11354 df-ltxr 11355 df-le 11356 df-sub 11550 df-neg 11551 df-div 11981 df-nn 12343 df-2 12412 df-3 12413 df-4 12414 df-5 12415 df-6 12416 df-7 12417 df-8 12418 df-9 12419 df-n0 12614 df-z 12701 df-dec 12822 df-uz 12973 df-rp 13128 df-fz 13647 df-fzo 13796 df-seq 14152 df-exp 14212 df-hash 14482 df-cj 15275 df-re 15276 df-im 15277 df-sqrt 15411 df-abs 15412 df-clim 15664 df-sum 15863 df-struct 17334 df-sets 17351 df-slot 17369 df-ndx 17381 df-base 17397 df-ress 17418 df-plusg 17450 df-mulr 17451 df-starv 17452 df-sca 17453 df-vsca 17454 df-ip 17455 df-tset 17456 df-ple 17457 df-ds 17459 df-unif 17460 df-hom 17461 df-cco 17462 df-0g 17621 df-gsum 17622 df-prds 17627 df-pws 17629 df-mre 17765 df-mrc 17766 df-acs 17768 df-mgm 18825 df-sgrp 18917 df-mnd 18933 df-mhm 18987 df-submnd 18988 df-grp 19156 df-minusg 19157 df-sbg 19158 df-mulg 19287 df-subg 19342 df-ghm 19437 df-cntz 19540 df-cmn 20005 df-abl 20006 df-mgp 20370 df-rng 20384 df-ur 20417 df-ring 20470 df-cring 20471 df-oppr 20576 df-dvdsr 20596 df-unit 20597 df-invr 20627 df-rhm 20711 df-nzr 20772 df-subrng 20807 df-subrg 20831 df-rlreg 20955 df-domn 20956 df-idom 20957 df-drng 20991 df-field 20992 df-lmod 21146 df-lss 21216 df-cnfld 21688 df-zring 21762 df-zrh 21818 df-ascl 22172 df-psr 22226 df-mvr 22227 df-mpl 22228 df-opsr 22230 df-psr1 22507 df-vr1 22508 df-ply1 22509 df-coe1 22510 df-mdeg 26382 df-deg1 26383

This theorem is used by: aks6d1c6lem3 43359

W3C validator