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Theorem aks6d1c1 42098
Description: Claim 1 of Theorem 6.1 https://www3.nd.edu/%7eandyp/notes/AKS.pdf. (Contributed by metakunt, 30-Apr-2025.)
Hypotheses
Ref Expression
aks6d1c1.1 = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ((𝑂𝑓)‘𝑦)) = ((𝑂𝑓)‘(𝑒 𝑦)))}
aks6d1c1.2 𝑆 = (Poly1𝐾)
aks6d1c1.3 𝐵 = (Base‘𝑆)
aks6d1c1.4 𝑋 = (var1𝐾)
aks6d1c1.5 𝑊 = (mulGrp‘𝑆)
aks6d1c1.6 𝑉 = (mulGrp‘𝐾)
aks6d1c1.7 = (.g𝑉)
aks6d1c1.8 𝐶 = (algSc‘𝑆)
aks6d1c1.9 𝐷 = (.g𝑊)
aks6d1c1.10 𝑃 = (chr‘𝐾)
aks6d1c1.11 𝑂 = (eval1𝐾)
aks6d1c1.12 + = (+g𝑆)
aks6d1c1.13 (𝜑𝐾 ∈ Field)
aks6d1c1.14 (𝜑𝑃 ∈ ℙ)
aks6d1c1.15 (𝜑𝑅 ∈ ℕ)
aks6d1c1.16 (𝜑𝑁 ∈ ℕ)
aks6d1c1.17 (𝜑𝑃𝑁)
aks6d1c1.18 (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c1.19 (𝜑𝐹:(0...𝐴)⟶ℕ0)
aks6d1c1.20 𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c1.21 (𝜑𝐴 ∈ ℕ0)
aks6d1c1.22 (𝜑𝑈 ∈ ℕ0)
aks6d1c1.23 (𝜑𝐿 ∈ ℕ0)
aks6d1c1.24 𝐸 = ((𝑃𝑈) · ((𝑁 / 𝑃)↑𝐿))
aks6d1c1.25 (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c1.26 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingIso 𝐾))
Assertion
Ref Expression
aks6d1c1 (𝜑𝐸 (𝐺𝐹))
Distinct variable groups:   + ,𝑎   + ,𝑒,𝑓,𝑦   + ,𝑔,𝑖   ,𝑒,𝑓,𝑦   𝑥, ,𝑦   ,𝑎   𝑦,   𝐴,𝑎   𝐴,𝑔,𝑖   𝑦,𝐴,𝑖   𝑥,𝐴   𝐵,𝑒,𝑓   𝐶,𝑎   𝐶,𝑒,𝑓,𝑦   𝐶,𝑔,𝑖   𝐷,𝑒,𝑓,𝑦   𝐷,𝑔,𝑖   𝑒,𝐸,𝑓,𝑦   𝑒,𝐹,𝑓,𝑦   𝑔,𝐹,𝑖   𝐾,𝑎   𝑒,𝐾,𝑓,𝑦   𝑔,𝐾,𝑖   𝑥,𝐾   𝑒,𝐿,𝑓,𝑦   𝑁,𝑎   𝑒,𝑁,𝑓,𝑦   𝑥,𝑁   𝑒,𝑂,𝑓,𝑦   𝑃,𝑒,𝑓,𝑦   𝑥,𝑃   𝑅,𝑒,𝑓,𝑦   𝑥,𝑅   𝑈,𝑒,𝑓,𝑦   𝑒,𝑉,𝑓,𝑦   𝑥,𝑉   𝑒,𝑊,𝑓,𝑦   𝑔,𝑊,𝑖   𝑋,𝑎   𝑒,𝑋,𝑓,𝑦   𝑔,𝑋,𝑖   𝜑,𝑎   𝜑,𝑔,𝑖   𝜑,𝑦,𝑥
Allowed substitution hints:   𝜑(𝑒,𝑓)   𝐴(𝑒,𝑓)   𝐵(𝑥,𝑦,𝑔,𝑖,𝑎)   𝐶(𝑥)   𝐷(𝑥,𝑎)   𝑃(𝑔,𝑖,𝑎)   + (𝑥)   (𝑥,𝑒,𝑓,𝑔,𝑖)   𝑅(𝑔,𝑖,𝑎)   𝑆(𝑥,𝑦,𝑒,𝑓,𝑔,𝑖,𝑎)   𝑈(𝑥,𝑔,𝑖,𝑎)   𝐸(𝑥,𝑔,𝑖,𝑎)   (𝑔,𝑖,𝑎)   𝐹(𝑥,𝑎)   𝐺(𝑥,𝑦,𝑒,𝑓,𝑔,𝑖,𝑎)   𝐿(𝑥,𝑔,𝑖,𝑎)   𝑁(𝑔,𝑖)   𝑂(𝑥,𝑔,𝑖,𝑎)   𝑉(𝑔,𝑖,𝑎)   𝑊(𝑥,𝑎)   𝑋(𝑥)

Proof of Theorem aks6d1c1
Dummy variables 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 aks6d1c1.21 . . . . 5 (𝜑𝐴 ∈ ℕ0)
21nn0zd 12637 . . . 4 (𝜑𝐴 ∈ ℤ)
31nn0ge0d 12588 . . . 4 (𝜑 → 0 ≤ 𝐴)
41nn0red 12586 . . . . 5 (𝜑𝐴 ∈ ℝ)
54leidd 11827 . . . 4 (𝜑𝐴𝐴)
62, 3, 53jca 1127 . . 3 (𝜑 → (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴𝐴𝐴))
7 oveq2 7439 . . . . . . . 8 ( = 0 → (0...) = (0...0))
87mpteq1d 5243 . . . . . . 7 ( = 0 → (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
98oveq2d 7447 . . . . . 6 ( = 0 → (𝑊 Σg (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
109breq2d 5160 . . . . 5 ( = 0 → (𝐸 (𝑊 Σg (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) ↔ 𝐸 (𝑊 Σg (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))))
11 oveq2 7439 . . . . . . . 8 ( = 𝑗 → (0...) = (0...𝑗))
1211mpteq1d 5243 . . . . . . 7 ( = 𝑗 → (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
1312oveq2d 7447 . . . . . 6 ( = 𝑗 → (𝑊 Σg (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
1413breq2d 5160 . . . . 5 ( = 𝑗 → (𝐸 (𝑊 Σg (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) ↔ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))))
15 oveq2 7439 . . . . . . . 8 ( = (𝑗 + 1) → (0...) = (0...(𝑗 + 1)))
1615mpteq1d 5243 . . . . . . 7 ( = (𝑗 + 1) → (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
1716oveq2d 7447 . . . . . 6 ( = (𝑗 + 1) → (𝑊 Σg (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
1817breq2d 5160 . . . . 5 ( = (𝑗 + 1) → (𝐸 (𝑊 Σg (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) ↔ 𝐸 (𝑊 Σg (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))))
19 oveq2 7439 . . . . . . . 8 ( = 𝐴 → (0...) = (0...𝐴))
2019mpteq1d 5243 . . . . . . 7 ( = 𝐴 → (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
2120oveq2d 7447 . . . . . 6 ( = 𝐴 → (𝑊 Σg (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
2221breq2d 5160 . . . . 5 ( = 𝐴 → (𝐸 (𝑊 Σg (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) ↔ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))))
23 aks6d1c1.1 . . . . . . . 8 = {⟨𝑒, 𝑓⟩ ∣ (𝑒 ∈ ℕ ∧ 𝑓𝐵 ∧ ∀𝑦 ∈ (𝑉 PrimRoots 𝑅)(𝑒 ((𝑂𝑓)‘𝑦)) = ((𝑂𝑓)‘(𝑒 𝑦)))}
24 aks6d1c1.2 . . . . . . . 8 𝑆 = (Poly1𝐾)
25 aks6d1c1.3 . . . . . . . 8 𝐵 = (Base‘𝑆)
26 aks6d1c1.4 . . . . . . . 8 𝑋 = (var1𝐾)
27 aks6d1c1.5 . . . . . . . 8 𝑊 = (mulGrp‘𝑆)
28 aks6d1c1.6 . . . . . . . 8 𝑉 = (mulGrp‘𝐾)
29 aks6d1c1.7 . . . . . . . 8 = (.g𝑉)
30 aks6d1c1.8 . . . . . . . 8 𝐶 = (algSc‘𝑆)
31 aks6d1c1.9 . . . . . . . 8 𝐷 = (.g𝑊)
32 aks6d1c1.10 . . . . . . . 8 𝑃 = (chr‘𝐾)
33 aks6d1c1.11 . . . . . . . 8 𝑂 = (eval1𝐾)
34 aks6d1c1.12 . . . . . . . 8 + = (+g𝑆)
35 aks6d1c1.13 . . . . . . . 8 (𝜑𝐾 ∈ Field)
36 aks6d1c1.14 . . . . . . . 8 (𝜑𝑃 ∈ ℙ)
37 aks6d1c1.15 . . . . . . . 8 (𝜑𝑅 ∈ ℕ)
38 aks6d1c1.16 . . . . . . . 8 (𝜑𝑁 ∈ ℕ)
39 aks6d1c1.17 . . . . . . . 8 (𝜑𝑃𝑁)
40 aks6d1c1.18 . . . . . . . 8 (𝜑 → (𝑁 gcd 𝑅) = 1)
41 aks6d1c1.24 . . . . . . . . . . . 12 𝐸 = ((𝑃𝑈) · ((𝑁 / 𝑃)↑𝐿))
42 prmnn 16708 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4336, 42syl 17 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ)
44 aks6d1c1.22 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ ℕ0)
4543, 44nnexpcld 14281 . . . . . . . . . . . . 13 (𝜑 → (𝑃𝑈) ∈ ℕ)
4643nnzd 12638 . . . . . . . . . . . . . . . . . 18 (𝜑𝑃 ∈ ℤ)
4743nnne0d 12314 . . . . . . . . . . . . . . . . . 18 (𝜑𝑃 ≠ 0)
4838nnzd 12638 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℤ)
49 dvdsval2 16290 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
5046, 47, 48, 49syl3anc 1370 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
5139, 50mpbid 232 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
5238nnred 12279 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℝ)
5343nnred 12279 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 ∈ ℝ)
5438nngt0d 12313 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝑁)
5543nngt0d 12313 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝑃)
5652, 53, 54, 55divgt0d 12201 . . . . . . . . . . . . . . . 16 (𝜑 → 0 < (𝑁 / 𝑃))
5751, 56jca 511 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
58 elnnz 12621 . . . . . . . . . . . . . . 15 ((𝑁 / 𝑃) ∈ ℕ ↔ ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
5957, 58sylibr 234 . . . . . . . . . . . . . 14 (𝜑 → (𝑁 / 𝑃) ∈ ℕ)
60 aks6d1c1.23 . . . . . . . . . . . . . 14 (𝜑𝐿 ∈ ℕ0)
6159, 60nnexpcld 14281 . . . . . . . . . . . . 13 (𝜑 → ((𝑁 / 𝑃)↑𝐿) ∈ ℕ)
6245, 61nnmulcld 12317 . . . . . . . . . . . 12 (𝜑 → ((𝑃𝑈) · ((𝑁 / 𝑃)↑𝐿)) ∈ ℕ)
6341, 62eqeltrid 2843 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℕ)
6423, 24, 25, 26, 28, 29, 32, 33, 35, 36, 37, 38, 39, 40, 63aks6d1c1p7 42095 . . . . . . . . . 10 (𝜑𝐸 𝑋)
6535fldcrngd 20759 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ CRing)
6624ply1crng 22216 . . . . . . . . . . . . . 14 (𝐾 ∈ CRing → 𝑆 ∈ CRing)
6765, 66syl 17 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ CRing)
68 crngring 20263 . . . . . . . . . . . . . 14 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
69 ringcmn 20296 . . . . . . . . . . . . . 14 (𝑆 ∈ Ring → 𝑆 ∈ CMnd)
7068, 69syl 17 . . . . . . . . . . . . 13 (𝑆 ∈ CRing → 𝑆 ∈ CMnd)
7167, 70syl 17 . . . . . . . . . . . 12 (𝜑𝑆 ∈ CMnd)
72 cmnmnd 19830 . . . . . . . . . . . 12 (𝑆 ∈ CMnd → 𝑆 ∈ Mnd)
7371, 72syl 17 . . . . . . . . . . 11 (𝜑𝑆 ∈ Mnd)
74 crngring 20263 . . . . . . . . . . . . 13 (𝐾 ∈ CRing → 𝐾 ∈ Ring)
7565, 74syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Ring)
76 eqid 2735 . . . . . . . . . . . . 13 (Base‘𝑆) = (Base‘𝑆)
7726, 24, 76vr1cl 22235 . . . . . . . . . . . 12 (𝐾 ∈ Ring → 𝑋 ∈ (Base‘𝑆))
7875, 77syl 17 . . . . . . . . . . 11 (𝜑𝑋 ∈ (Base‘𝑆))
79 eqid 2735 . . . . . . . . . . . 12 (0g𝑆) = (0g𝑆)
8076, 34, 79mndrid 18781 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑋 ∈ (Base‘𝑆)) → (𝑋 + (0g𝑆)) = 𝑋)
8173, 78, 80syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝑋 + (0g𝑆)) = 𝑋)
8264, 81breqtrrd 5176 . . . . . . . . 9 (𝜑𝐸 (𝑋 + (0g𝑆)))
83 eqid 2735 . . . . . . . . . . . . . 14 (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
84 eqid 2735 . . . . . . . . . . . . . 14 (0g𝐾) = (0g𝐾)
8583, 84zrh0 21542 . . . . . . . . . . . . 13 (𝐾 ∈ Ring → ((ℤRHom‘𝐾)‘0) = (0g𝐾))
8675, 85syl 17 . . . . . . . . . . . 12 (𝜑 → ((ℤRHom‘𝐾)‘0) = (0g𝐾))
8786fveq2d 6911 . . . . . . . . . . 11 (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘0)) = (𝐶‘(0g𝐾)))
8824, 30, 84, 79, 75ply1ascl0 22272 . . . . . . . . . . 11 (𝜑 → (𝐶‘(0g𝐾)) = (0g𝑆))
8987, 88eqtrd 2775 . . . . . . . . . 10 (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘0)) = (0g𝑆))
9089oveq2d 7447 . . . . . . . . 9 (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))) = (𝑋 + (0g𝑆)))
9182, 90breqtrrd 5176 . . . . . . . 8 (𝜑𝐸 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))))
92 aks6d1c1.19 . . . . . . . . 9 (𝜑𝐹:(0...𝐴)⟶ℕ0)
93 0zd 12623 . . . . . . . . . 10 (𝜑 → 0 ∈ ℤ)
94 0red 11262 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
9594leidd 11827 . . . . . . . . . 10 (𝜑 → 0 ≤ 0)
9693, 2, 93, 95, 3elfzd 13552 . . . . . . . . 9 (𝜑 → 0 ∈ (0...𝐴))
9792, 96ffvelcdmd 7105 . . . . . . . 8 (𝜑 → (𝐹‘0) ∈ ℕ0)
9823, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 91, 97aks6d1c1p6 42096 . . . . . . 7 (𝜑𝐸 ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))))
9927crngmgp 20259 . . . . . . . . . 10 (𝑆 ∈ CRing → 𝑊 ∈ CMnd)
10067, 99syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ CMnd)
101100cmnmndd 19837 . . . . . . . 8 (𝜑𝑊 ∈ Mnd)
102 0z 12622 . . . . . . . . 9 0 ∈ ℤ
103102a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℤ)
104 eqid 2735 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
105 0le0 12365 . . . . . . . . . . . 12 0 ≤ 0
106105a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ≤ 0)
107103, 2, 103, 106, 3elfzd 13552 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝐴))
10892, 107ffvelcdmd 7105 . . . . . . . . 9 (𝜑 → (𝐹‘0) ∈ ℕ0)
10983zrhrhm 21540 . . . . . . . . . . . . . . 15 (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
11075, 109syl 17 . . . . . . . . . . . . . 14 (𝜑 → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
111 zringbas 21482 . . . . . . . . . . . . . . 15 ℤ = (Base‘ℤring)
112 eqid 2735 . . . . . . . . . . . . . . 15 (Base‘𝐾) = (Base‘𝐾)
113111, 112rhmf 20502 . . . . . . . . . . . . . 14 ((ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
114110, 113syl 17 . . . . . . . . . . . . 13 (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
115114, 93ffvelcdmd 7105 . . . . . . . . . . . 12 (𝜑 → ((ℤRHom‘𝐾)‘0) ∈ (Base‘𝐾))
11624, 30, 112, 76ply1sclcl 22305 . . . . . . . . . . . 12 ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘0) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘0)) ∈ (Base‘𝑆))
11775, 115, 116syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘0)) ∈ (Base‘𝑆))
11876, 34mndcl 18768 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑋 ∈ (Base‘𝑆) ∧ (𝐶‘((ℤRHom‘𝐾)‘0)) ∈ (Base‘𝑆)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))) ∈ (Base‘𝑆))
11973, 78, 117, 118syl3anc 1370 . . . . . . . . . 10 (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))) ∈ (Base‘𝑆))
12027, 76mgpbas 20158 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑊)
121119, 120eleqtrdi 2849 . . . . . . . . 9 (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))) ∈ (Base‘𝑊))
122104, 31, 101, 108, 121mulgnn0cld 19126 . . . . . . . 8 (𝜑 → ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))) ∈ (Base‘𝑊))
123 fveq2 6907 . . . . . . . . . 10 (𝑖 = 0 → (𝐹𝑖) = (𝐹‘0))
124 2fveq3 6912 . . . . . . . . . . 11 (𝑖 = 0 → (𝐶‘((ℤRHom‘𝐾)‘𝑖)) = (𝐶‘((ℤRHom‘𝐾)‘0)))
125124oveq2d 7447 . . . . . . . . . 10 (𝑖 = 0 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))))
126123, 125oveq12d 7449 . . . . . . . . 9 (𝑖 = 0 → ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))))
127104, 126gsumsn 19987 . . . . . . . 8 ((𝑊 ∈ Mnd ∧ 0 ∈ ℤ ∧ ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))) ∈ (Base‘𝑊)) → (𝑊 Σg (𝑖 ∈ {0} ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))))
128101, 103, 122, 127syl3anc 1370 . . . . . . 7 (𝜑 → (𝑊 Σg (𝑖 ∈ {0} ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))))
12998, 128breqtrrd 5176 . . . . . 6 (𝜑𝐸 (𝑊 Σg (𝑖 ∈ {0} ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
130 fzsn 13603 . . . . . . . . . 10 (0 ∈ ℤ → (0...0) = {0})
131102, 130ax-mp 5 . . . . . . . . 9 (0...0) = {0}
132131a1i 11 . . . . . . . 8 (𝜑 → (0...0) = {0})
133132mpteq1d 5243 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ {0} ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
134133oveq2d 7447 . . . . . 6 (𝜑 → (𝑊 Σg (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑖 ∈ {0} ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
135129, 134breqtrrd 5176 . . . . 5 (𝜑𝐸 (𝑊 Σg (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
136353ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐾 ∈ Field)
137363ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑃 ∈ ℙ)
138373ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑅 ∈ ℕ)
139403ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → (𝑁 gcd 𝑅) = 1)
140393ad2ant1 1132 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑃𝑁)
141 simp3 1137 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
142 nfcv 2903 . . . . . . . . . . 11 𝑘((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))
143 nfcv 2903 . . . . . . . . . . 11 𝑖((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))
144 fveq2 6907 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝐹𝑖) = (𝐹𝑘))
145 2fveq3 6912 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → (𝐶‘((ℤRHom‘𝐾)‘𝑖)) = (𝐶‘((ℤRHom‘𝐾)‘𝑘)))
146145oveq2d 7447 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))
147144, 146oveq12d 7449 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))
148142, 143, 147cbvmpt 5259 . . . . . . . . . 10 (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑘 ∈ (0...𝑗) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))
149148oveq2i 7442 . . . . . . . . 9 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑘 ∈ (0...𝑗) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))))
150149a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑘 ∈ (0...𝑗) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))))
151141, 150breqtrd 5174 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐸 (𝑊 Σg (𝑘 ∈ (0...𝑗) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))))
15235adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐾 ∈ Field)
15336adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑃 ∈ ℙ)
15437adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑅 ∈ ℕ)
15538adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑁 ∈ ℕ)
15639adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑃𝑁)
15740adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑁 gcd 𝑅) = 1)
15841a1i 11 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐸 = ((𝑃𝑈) · ((𝑁 / 𝑃)↑𝐿)))
15937nnzd 12638 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ ℤ)
16051, 159, 603jca 1127 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝐿 ∈ ℕ0))
161159, 51, 483jca 1127 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ ∧ 𝑁 ∈ ℤ))
16248, 159jca 511 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ))
163 gcdcom 16547 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 gcd 𝑅) = (𝑅 gcd 𝑁))
164162, 163syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁 gcd 𝑅) = (𝑅 gcd 𝑁))
165 eqeq1 2739 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 gcd 𝑅) = (𝑅 gcd 𝑁) → ((𝑁 gcd 𝑅) = 1 ↔ (𝑅 gcd 𝑁) = 1))
166164, 165syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑁 gcd 𝑅) = 1 ↔ (𝑅 gcd 𝑁) = 1))
167166pm5.74i 271 . . . . . . . . . . . . . . . . . 18 ((𝜑 → (𝑁 gcd 𝑅) = 1) ↔ (𝜑 → (𝑅 gcd 𝑁) = 1))
16840, 167mpbi 230 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑅 gcd 𝑁) = 1)
16952recnd 11287 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℂ)
17053recnd 11287 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑃 ∈ ℂ)
17194, 54gtned 11394 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ≠ 0)
172169, 169, 170, 171, 47divdiv2d 12073 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 / (𝑁 / 𝑃)) = ((𝑁 · 𝑃) / 𝑁))
173169, 170mulcomd 11280 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑁 · 𝑃) = (𝑃 · 𝑁))
174173oveq1d 7446 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑁 · 𝑃) / 𝑁) = ((𝑃 · 𝑁) / 𝑁))
175170, 169, 169, 171, 171divdiv2d 12073 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑃 / (𝑁 / 𝑁)) = ((𝑃 · 𝑁) / 𝑁))
176175eqcomd 2741 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑃 · 𝑁) / 𝑁) = (𝑃 / (𝑁 / 𝑁)))
177174, 176eqtrd 2775 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 · 𝑃) / 𝑁) = (𝑃 / (𝑁 / 𝑁)))
178169, 171dividd 12039 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑁 / 𝑁) = 1)
179178oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑃 / (𝑁 / 𝑁)) = (𝑃 / 1))
180170div1d 12033 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑃 / 1) = 𝑃)
181179, 180eqtrd 2775 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑃 / (𝑁 / 𝑁)) = 𝑃)
182181, 46eqeltrd 2839 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑃 / (𝑁 / 𝑁)) ∈ ℤ)
183177, 182eqeltrd 2839 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑁 · 𝑃) / 𝑁) ∈ ℤ)
184172, 183eqeltrd 2839 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 / (𝑁 / 𝑃)) ∈ ℤ)
18594, 56gtned 11394 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 / 𝑃) ≠ 0)
186 dvdsval2 16290 . . . . . . . . . . . . . . . . . . 19 (((𝑁 / 𝑃) ∈ ℤ ∧ (𝑁 / 𝑃) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
18751, 185, 48, 186syl3anc 1370 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
188184, 187mpbird 257 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑁 / 𝑃) ∥ 𝑁)
189168, 188jca 511 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ (𝑁 / 𝑃) ∥ 𝑁))
190 rpdvds 16694 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ (𝑁 / 𝑃) ∥ 𝑁)) → (𝑅 gcd (𝑁 / 𝑃)) = 1)
191161, 189, 190syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝑅 gcd (𝑁 / 𝑃)) = 1)
192159, 51jca 511 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ))
193 gcdcom 16547 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ) → (𝑅 gcd (𝑁 / 𝑃)) = ((𝑁 / 𝑃) gcd 𝑅))
194192, 193syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑅 gcd (𝑁 / 𝑃)) = ((𝑁 / 𝑃) gcd 𝑅))
195 eqeq1 2739 . . . . . . . . . . . . . . . . 17 ((𝑅 gcd (𝑁 / 𝑃)) = ((𝑁 / 𝑃) gcd 𝑅) → ((𝑅 gcd (𝑁 / 𝑃)) = 1 ↔ ((𝑁 / 𝑃) gcd 𝑅) = 1))
196194, 195syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑅 gcd (𝑁 / 𝑃)) = 1 ↔ ((𝑁 / 𝑃) gcd 𝑅) = 1))
197196pm5.74i 271 . . . . . . . . . . . . . . 15 ((𝜑 → (𝑅 gcd (𝑁 / 𝑃)) = 1) ↔ (𝜑 → ((𝑁 / 𝑃) gcd 𝑅) = 1))
198191, 197mpbi 230 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 / 𝑃) gcd 𝑅) = 1)
199 rpexp1i 16757 . . . . . . . . . . . . . . 15 (((𝑁 / 𝑃) ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝐿 ∈ ℕ0) → (((𝑁 / 𝑃) gcd 𝑅) = 1 → (((𝑁 / 𝑃)↑𝐿) gcd 𝑅) = 1))
200199imp 406 . . . . . . . . . . . . . 14 ((((𝑁 / 𝑃) ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝐿 ∈ ℕ0) ∧ ((𝑁 / 𝑃) gcd 𝑅) = 1) → (((𝑁 / 𝑃)↑𝐿) gcd 𝑅) = 1)
201160, 198, 200syl2anc 584 . . . . . . . . . . . . 13 (𝜑 → (((𝑁 / 𝑃)↑𝐿) gcd 𝑅) = 1)
202201adantr 480 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (((𝑁 / 𝑃)↑𝐿) gcd 𝑅) = 1)
203 eqid 2735 . . . . . . . . . . . . . 14 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))
204 simpr1 1193 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑗 ∈ ℤ)
205204peano2zd 12723 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑗 + 1) ∈ ℤ)
20623, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 152, 153, 154, 157, 156, 203, 205aks6d1c1p2 42091 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑃 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
20744adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑈 ∈ ℕ0)
208159, 46, 483jca 1127 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
209168, 39jca 511 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ 𝑃𝑁))
210 rpdvds 16694 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((𝑅 gcd 𝑁) = 1 ∧ 𝑃𝑁)) → (𝑅 gcd 𝑃) = 1)
211208, 209, 210syl2anc 584 . . . . . . . . . . . . . . 15 (𝜑 → (𝑅 gcd 𝑃) = 1)
212159, 46jca 511 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ))
213 gcdcom 16547 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑅 gcd 𝑃) = (𝑃 gcd 𝑅))
214212, 213syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑅 gcd 𝑃) = (𝑃 gcd 𝑅))
215 eqeq1 2739 . . . . . . . . . . . . . . . . 17 ((𝑅 gcd 𝑃) = (𝑃 gcd 𝑅) → ((𝑅 gcd 𝑃) = 1 ↔ (𝑃 gcd 𝑅) = 1))
216214, 215syl 17 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑅 gcd 𝑃) = 1 ↔ (𝑃 gcd 𝑅) = 1))
217216pm5.74i 271 . . . . . . . . . . . . . . 15 ((𝜑 → (𝑅 gcd 𝑃) = 1) ↔ (𝜑 → (𝑃 gcd 𝑅) = 1))
218211, 217mpbi 230 . . . . . . . . . . . . . 14 (𝜑 → (𝑃 gcd 𝑅) = 1)
219218adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑃 gcd 𝑅) = 1)
22023, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 152, 153, 154, 155, 156, 157, 206, 207, 219aks6d1c1p8 42097 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑃𝑈) (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
221 2fveq3 6912 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝑗 + 1) → (𝐶‘((ℤRHom‘𝐾)‘𝑎)) = (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))
222221oveq2d 7447 . . . . . . . . . . . . . . . 16 (𝑎 = (𝑗 + 1) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
223222breq2d 5160 . . . . . . . . . . . . . . 15 (𝑎 = (𝑗 + 1) → (𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))) ↔ 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))))
224 aks6d1c1.25 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))
22523, 24, 25, 26, 28, 29, 32, 33, 35, 36, 37, 38, 39, 40, 38aks6d1c1p7 42095 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑁 𝑋)
226225, 81breqtrrd 5176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 (𝑋 + (0g𝑆)))
227226, 90breqtrrd 5176 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝜑𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))))
228227adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑎 = 0) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))))
229 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑𝑎 = 0) → 𝑎 = 0)
230229fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑎 = 0) → ((ℤRHom‘𝐾)‘𝑎) = ((ℤRHom‘𝐾)‘0))
231230fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑎 = 0) → (𝐶‘((ℤRHom‘𝐾)‘𝑎)) = (𝐶‘((ℤRHom‘𝐾)‘0)))
232231oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑎 = 0) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))))
233228, 232breqtrrd 5176 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑𝑎 = 0) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))
234233ex 412 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑎 = 0 → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))))
235234adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → (𝑎 = 0 → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))))
236 simpr 484 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))))
237 1cnd 11254 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → 1 ∈ ℂ)
238237addlidd 11460 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → (0 + 1) = 1)
239238oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → ((0 + 1)...𝐴) = (1...𝐴))
240239eleq2d 2825 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → (𝑎 ∈ ((0 + 1)...𝐴) ↔ 𝑎 ∈ (1...𝐴)))
241240imbi1d 341 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → ((𝑎 ∈ ((0 + 1)...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))) ↔ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))))
242236, 241mpbird 257 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → (𝑎 ∈ ((0 + 1)...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))))
243235, 242jaod 859 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → ((𝑎 = 0 ∨ 𝑎 ∈ ((0 + 1)...𝐴)) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))))
2442, 3jca 511 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴))
245 eluz1 12880 . . . . . . . . . . . . . . . . . . . . . . . . 25 (0 ∈ ℤ → (𝐴 ∈ (ℤ‘0) ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)))
24693, 245syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑 → (𝐴 ∈ (ℤ‘0) ↔ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴)))
247244, 246mpbird 257 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐴 ∈ (ℤ‘0))
248247adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → 𝐴 ∈ (ℤ‘0))
249 elfzp12 13640 . . . . . . . . . . . . . . . . . . . . . 22 (𝐴 ∈ (ℤ‘0) → (𝑎 ∈ (0...𝐴) ↔ (𝑎 = 0 ∨ 𝑎 ∈ ((0 + 1)...𝐴))))
250248, 249syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → (𝑎 ∈ (0...𝐴) ↔ (𝑎 = 0 ∨ 𝑎 ∈ ((0 + 1)...𝐴))))
251250imbi1d 341 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → ((𝑎 ∈ (0...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))) ↔ ((𝑎 = 0 ∨ 𝑎 ∈ ((0 + 1)...𝐴)) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))))
252243, 251mpbird 257 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → (𝑎 ∈ (0...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))))
253252ex 412 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))) → (𝑎 ∈ (0...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))))
254253ralimdv2 3161 . . . . . . . . . . . . . . . . 17 (𝜑 → (∀𝑎 ∈ (1...𝐴)𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))) → ∀𝑎 ∈ (0...𝐴)𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎)))))
255224, 254mpd 15 . . . . . . . . . . . . . . . 16 (𝜑 → ∀𝑎 ∈ (0...𝐴)𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))
256255adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ∀𝑎 ∈ (0...𝐴)𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))
257 0zd 12623 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 0 ∈ ℤ)
2582adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐴 ∈ ℤ)
259204zred 12720 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑗 ∈ ℝ)
260 1red 11260 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 1 ∈ ℝ)
261 simpr2 1194 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 0 ≤ 𝑗)
262 0le1 11784 . . . . . . . . . . . . . . . . . 18 0 ≤ 1
263262a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 0 ≤ 1)
264259, 260, 261, 263addge0d 11837 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 0 ≤ (𝑗 + 1))
265 simpr3 1195 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑗 < 𝐴)
266204, 258zltp1led 41961 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑗 < 𝐴 ↔ (𝑗 + 1) ≤ 𝐴))
267265, 266mpbid 232 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑗 + 1) ≤ 𝐴)
268257, 258, 205, 264, 267elfzd 13552 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑗 + 1) ∈ (0...𝐴))
269223, 256, 268rspcdva 3623 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
270 aks6d1c1.26 . . . . . . . . . . . . . . 15 (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingIso 𝐾))
271270adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃 𝑥)) ∈ (𝐾 RingIso 𝐾))
27223, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 152, 153, 154, 157, 156, 203, 205, 269, 271aks6d1c1p3 42092 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑁 / 𝑃) (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
27360adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐿 ∈ ℕ0)
274198adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ((𝑁 / 𝑃) gcd 𝑅) = 1)
27523, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 152, 153, 154, 155, 156, 157, 272, 273, 274aks6d1c1p8 42097 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ((𝑁 / 𝑃)↑𝐿) (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
27623, 24, 25, 26, 27, 28, 29, 30, 32, 33, 34, 152, 153, 154, 202, 156, 220, 275aks6d1c1p5 42094 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ((𝑃𝑈) · ((𝑁 / 𝑃)↑𝐿)) (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
277158, 276eqbrtrd 5170 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐸 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
27892adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐹:(0...𝐴)⟶ℕ0)
279278, 268ffvelcdmd 7105 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝐹‘(𝑗 + 1)) ∈ ℕ0)
28023, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 152, 153, 154, 155, 156, 157, 277, 279aks6d1c1p6 42096 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐸 ((𝐹‘(𝑗 + 1))𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))))
281101adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑊 ∈ Mnd)
282 ovexd 7466 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑗 + 1) ∈ V)
28373adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑆 ∈ Mnd)
28478adantr 480 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑋 ∈ (Base‘𝑆))
28575adantr 480 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐾 ∈ Ring)
286114adantr 480 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
287286, 205ffvelcdmd 7105 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ((ℤRHom‘𝐾)‘(𝑗 + 1)) ∈ (Base‘𝐾))
28824, 30, 112, 76ply1sclcl 22305 . . . . . . . . . . . . . 14 ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘(𝑗 + 1)) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))) ∈ (Base‘𝑆))
289285, 287, 288syl2anc 584 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))) ∈ (Base‘𝑆))
29076, 34mndcl 18768 . . . . . . . . . . . . 13 ((𝑆 ∈ Mnd ∧ 𝑋 ∈ (Base‘𝑆) ∧ (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))) ∈ (Base‘𝑆)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))) ∈ (Base‘𝑆))
291283, 284, 289, 290syl3anc 1370 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))) ∈ (Base‘𝑆))
292291, 120eleqtrdi 2849 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))) ∈ (Base‘𝑊))
293104, 31, 281, 279, 292mulgnn0cld 19126 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ((𝐹‘(𝑗 + 1))𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))) ∈ (Base‘𝑊))
294 fveq2 6907 . . . . . . . . . . . 12 (𝑘 = (𝑗 + 1) → (𝐹𝑘) = (𝐹‘(𝑗 + 1)))
295 2fveq3 6912 . . . . . . . . . . . . 13 (𝑘 = (𝑗 + 1) → (𝐶‘((ℤRHom‘𝐾)‘𝑘)) = (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))
296295oveq2d 7447 . . . . . . . . . . . 12 (𝑘 = (𝑗 + 1) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
297294, 296oveq12d 7449 . . . . . . . . . . 11 (𝑘 = (𝑗 + 1) → ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))) = ((𝐹‘(𝑗 + 1))𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))))
298104, 297gsumsn 19987 . . . . . . . . . 10 ((𝑊 ∈ Mnd ∧ (𝑗 + 1) ∈ V ∧ ((𝐹‘(𝑗 + 1))𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))) ∈ (Base‘𝑊)) → (𝑊 Σg (𝑘 ∈ {(𝑗 + 1)} ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))) = ((𝐹‘(𝑗 + 1))𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))))
299281, 282, 293, 298syl3anc 1370 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑊 Σg (𝑘 ∈ {(𝑗 + 1)} ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))) = ((𝐹‘(𝑗 + 1))𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))))
300280, 299breqtrrd 5176 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐸 (𝑊 Σg (𝑘 ∈ {(𝑗 + 1)} ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))))
3013003adant3 1131 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐸 (𝑊 Σg (𝑘 ∈ {(𝑗 + 1)} ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))))
30223, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 136, 137, 138, 139, 140, 151, 301aks6d1c1p4 42093 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐸 ((𝑊 Σg (𝑘 ∈ (0...𝑗) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))))(+g𝑊)(𝑊 Σg (𝑘 ∈ {(𝑗 + 1)} ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))))))
303142, 143, 147cbvmpt 5259 . . . . . . . . 9 (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑘 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))
304303a1i 11 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑘 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))))
305304oveq2d 7447 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → (𝑊 Σg (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑘 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))))
306 eqid 2735 . . . . . . . 8 (+g𝑊) = (+g𝑊)
3071003ad2ant1 1132 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑊 ∈ CMnd)
308 simp21 1205 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑗 ∈ ℤ)
309 simp22 1206 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 0 ≤ 𝑗)
310308, 309jca 511 . . . . . . . . 9 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
311 elnn0z 12624 . . . . . . . . 9 (𝑗 ∈ ℕ0 ↔ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗))
312310, 311sylibr 234 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑗 ∈ ℕ0)
3132813adant3 1131 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑊 ∈ Mnd)
314313adantr 480 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑊 ∈ Mnd)
315923ad2ant1 1132 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐹:(0...𝐴)⟶ℕ0)
316315adantr 480 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝐹:(0...𝐴)⟶ℕ0)
317 0zd 12623 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 0 ∈ ℤ)
31823ad2ant1 1132 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐴 ∈ ℤ)
319318adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝐴 ∈ ℤ)
320 elfzelz 13561 . . . . . . . . . . . 12 (𝑘 ∈ (0...(𝑗 + 1)) → 𝑘 ∈ ℤ)
321320adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑘 ∈ ℤ)
322 elfzle1 13564 . . . . . . . . . . . 12 (𝑘 ∈ (0...(𝑗 + 1)) → 0 ≤ 𝑘)
323322adantl 481 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 0 ≤ 𝑘)
324321zred 12720 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑘 ∈ ℝ)
325308adantr 480 . . . . . . . . . . . . . 14 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑗 ∈ ℤ)
326325zred 12720 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑗 ∈ ℝ)
327 1red 11260 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 1 ∈ ℝ)
328326, 327readdcld 11288 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → (𝑗 + 1) ∈ ℝ)
329319zred 12720 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝐴 ∈ ℝ)
330 elfzle2 13565 . . . . . . . . . . . . 13 (𝑘 ∈ (0...(𝑗 + 1)) → 𝑘 ≤ (𝑗 + 1))
331330adantl 481 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑘 ≤ (𝑗 + 1))
332 simpl23 1252 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑗 < 𝐴)
333325, 319zltp1led 41961 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → (𝑗 < 𝐴 ↔ (𝑗 + 1) ≤ 𝐴))
334332, 333mpbid 232 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → (𝑗 + 1) ≤ 𝐴)
335324, 328, 329, 331, 334letrd 11416 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑘𝐴)
336317, 319, 321, 323, 335elfzd 13552 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑘 ∈ (0...𝐴))
337316, 336ffvelcdmd 7105 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → (𝐹𝑘) ∈ ℕ0)
3382833adant3 1131 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑆 ∈ Mnd)
339338adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑆 ∈ Mnd)
3402843adant3 1131 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑋 ∈ (Base‘𝑆))
341340adantr 480 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝑋 ∈ (Base‘𝑆))
3422853adant3 1131 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐾 ∈ Ring)
343342adantr 480 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → 𝐾 ∈ Ring)
344343, 109, 1133syl 18 . . . . . . . . . . . . 13 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
345344, 321ffvelcdmd 7105 . . . . . . . . . . . 12 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → ((ℤRHom‘𝐾)‘𝑘) ∈ (Base‘𝐾))
34624, 30, 112, 76ply1sclcl 22305 . . . . . . . . . . . 12 ((𝐾 ∈ Ring ∧ ((ℤRHom‘𝐾)‘𝑘) ∈ (Base‘𝐾)) → (𝐶‘((ℤRHom‘𝐾)‘𝑘)) ∈ (Base‘𝑆))
347343, 345, 346syl2anc 584 . . . . . . . . . . 11 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → (𝐶‘((ℤRHom‘𝐾)‘𝑘)) ∈ (Base‘𝑆))
34876, 34mndcl 18768 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑋 ∈ (Base‘𝑆) ∧ (𝐶‘((ℤRHom‘𝐾)‘𝑘)) ∈ (Base‘𝑆)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))) ∈ (Base‘𝑆))
349339, 341, 347, 348syl3anc 1370 . . . . . . . . . 10 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))) ∈ (Base‘𝑆))
350349, 120eleqtrdi 2849 . . . . . . . . 9 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))) ∈ (Base‘𝑊))
351104, 31, 314, 337, 350mulgnn0cld 19126 . . . . . . . 8 (((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) ∧ 𝑘 ∈ (0...(𝑗 + 1))) → ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))) ∈ (Base‘𝑊))
352104, 306, 307, 312, 351gsummptfzsplit 19965 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → (𝑊 Σg (𝑘 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))) = ((𝑊 Σg (𝑘 ∈ (0...𝑗) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))))(+g𝑊)(𝑊 Σg (𝑘 ∈ {(𝑗 + 1)} ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))))))
353305, 352eqtrd 2775 . . . . . 6 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → (𝑊 Σg (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = ((𝑊 Σg (𝑘 ∈ (0...𝑗) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))))(+g𝑊)(𝑊 Σg (𝑘 ∈ {(𝑗 + 1)} ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))))))
354302, 353breqtrrd 5176 . . . . 5 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐸 (𝑊 Σg (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
35510, 14, 18, 22, 135, 354, 93, 2, 3fzindd 12718 . . . 4 ((𝜑 ∧ (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴𝐴𝐴)) → 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
356355ex 412 . . 3 (𝜑 → ((𝐴 ∈ ℤ ∧ 0 ≤ 𝐴𝐴𝐴) → 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))))
3576, 356mpd 15 . 2 (𝜑𝐸 (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
358 aks6d1c1.20 . . . 4 𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
359358a1i 11 . . 3 (𝜑𝐺 = (𝑔 ∈ (ℕ0m (0...𝐴)) ↦ (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))))
360 simplr 769 . . . . . . 7 (((𝜑𝑔 = 𝐹) ∧ 𝑖 ∈ (0...𝐴)) → 𝑔 = 𝐹)
361360fveq1d 6909 . . . . . 6 (((𝜑𝑔 = 𝐹) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔𝑖) = (𝐹𝑖))
362361oveq1d 7446 . . . . 5 (((𝜑𝑔 = 𝐹) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑔𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))
363362mpteq2dva 5248 . . . 4 ((𝜑𝑔 = 𝐹) → (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
364363oveq2d 7447 . . 3 ((𝜑𝑔 = 𝐹) → (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
365 nn0ex 12530 . . . . . 6 0 ∈ V
366365a1i 11 . . . . 5 (𝜑 → ℕ0 ∈ V)
367 ovexd 7466 . . . . 5 (𝜑 → (0...𝐴) ∈ V)
368366, 367elmapd 8879 . . . 4 (𝜑 → (𝐹 ∈ (ℕ0m (0...𝐴)) ↔ 𝐹:(0...𝐴)⟶ℕ0))
36992, 368mpbird 257 . . 3 (𝜑𝐹 ∈ (ℕ0m (0...𝐴)))
370 ovexd 7466 . . 3 (𝜑 → (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ V)
371359, 364, 369, 370fvmptd 7023 . 2 (𝜑 → (𝐺𝐹) = (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
372357, 371breqtrrd 5176 1 (𝜑𝐸 (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wral 3059  Vcvv 3478  {csn 4631   class class class wbr 5148  {copab 5210  cmpt 5231  wf 6559  cfv 6563  (class class class)co 7431  m cmap 8865  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158   < clt 11293  cle 11294   / cdiv 11918  cn 12264  0cn0 12524  cz 12611  cuz 12876  ...cfz 13544  cexp 14099  cdvds 16287   gcd cgcd 16528  cprime 16705  Basecbs 17245  +gcplusg 17298  0gc0g 17486   Σg cgsu 17487  Mndcmnd 18760  .gcmg 19098  CMndccmn 19813  mulGrpcmgp 20152  Ringcrg 20251  CRingccrg 20252   RingHom crh 20486   RingIso crs 20487  Fieldcfield 20747  ringczring 21475  ℤRHomczrh 21528  chrcchr 21530  algSccascl 21890  var1cv1 22193  Poly1cpl1 22194  eval1ce1 22334   PrimRoots cprimroots 42073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232  ax-mulf 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-tpos 8250  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-oadd 8509  df-er 8744  df-map 8867  df-pm 8868  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-sup 9480  df-inf 9481  df-oi 9548  df-dju 9939  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-dec 12732  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-fl 13829  df-mod 13907  df-seq 14040  df-exp 14100  df-fac 14310  df-bc 14339  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-dvds 16288  df-gcd 16529  df-prm 16706  df-phi 16800  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-0g 17488  df-gsum 17489  df-prds 17494  df-pws 17496  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-mhm 18809  df-submnd 18810  df-grp 18967  df-minusg 18968  df-sbg 18969  df-mulg 19099  df-subg 19154  df-ghm 19244  df-cntz 19348  df-od 19561  df-cmn 19815  df-abl 19816  df-mgp 20153  df-rng 20171  df-ur 20200  df-srg 20205  df-ring 20253  df-cring 20254  df-oppr 20351  df-dvdsr 20374  df-unit 20375  df-invr 20405  df-dvr 20418  df-rhm 20489  df-rim 20490  df-subrng 20563  df-subrg 20587  df-drng 20748  df-field 20749  df-lmod 20877  df-lss 20948  df-lsp 20988  df-cnfld 21383  df-zring 21476  df-zrh 21532  df-chr 21534  df-assa 21891  df-asp 21892  df-ascl 21893  df-psr 21947  df-mvr 21948  df-mpl 21949  df-opsr 21951  df-evls 22116  df-evl 22117  df-psr1 22197  df-vr1 22198  df-ply1 22199  df-coe1 22200  df-evl1 22336  df-primroots 42074
This theorem is referenced by:  aks6d1c1rh  42107
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