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Theorem aks6d1c1 42118
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 12641 . . . 4 (𝜑𝐴 ∈ ℤ)
31nn0ge0d 12592 . . . 4 (𝜑 → 0 ≤ 𝐴)
41nn0red 12590 . . . . 5 (𝜑𝐴 ∈ ℝ)
54leidd 11830 . . . 4 (𝜑𝐴𝐴)
62, 3, 53jca 1128 . . 3 (𝜑 → (𝐴 ∈ ℤ ∧ 0 ≤ 𝐴𝐴𝐴))
7 oveq2 7440 . . . . . . . 8 ( = 0 → (0...) = (0...0))
87mpteq1d 5236 . . . . . . 7 ( = 0 → (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
98oveq2d 7448 . . . . . 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 5154 . . . . 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 7440 . . . . . . . 8 ( = 𝑗 → (0...) = (0...𝑗))
1211mpteq1d 5236 . . . . . . 7 ( = 𝑗 → (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
1312oveq2d 7448 . . . . . 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 5154 . . . . 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 7440 . . . . . . . 8 ( = (𝑗 + 1) → (0...) = (0...(𝑗 + 1)))
1615mpteq1d 5236 . . . . . . 7 ( = (𝑗 + 1) → (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...(𝑗 + 1)) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
1716oveq2d 7448 . . . . . 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 5154 . . . . 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 7440 . . . . . . . 8 ( = 𝐴 → (0...) = (0...𝐴))
2019mpteq1d 5236 . . . . . . 7 ( = 𝐴 → (𝑖 ∈ (0...) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
2120oveq2d 7448 . . . . . 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 5154 . . . . 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 16712 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
4336, 42syl 17 . . . . . . . . . . . . . 14 (𝜑𝑃 ∈ ℕ)
44 aks6d1c1.22 . . . . . . . . . . . . . 14 (𝜑𝑈 ∈ ℕ0)
4543, 44nnexpcld 14285 . . . . . . . . . . . . 13 (𝜑 → (𝑃𝑈) ∈ ℕ)
4643nnzd 12642 . . . . . . . . . . . . . . . . . 18 (𝜑𝑃 ∈ ℤ)
4743nnne0d 12317 . . . . . . . . . . . . . . . . . 18 (𝜑𝑃 ≠ 0)
4838nnzd 12642 . . . . . . . . . . . . . . . . . 18 (𝜑𝑁 ∈ ℤ)
49 dvdsval2 16294 . . . . . . . . . . . . . . . . . 18 ((𝑃 ∈ ℤ ∧ 𝑃 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝑃𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
5046, 47, 48, 49syl3anc 1372 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃𝑁 ↔ (𝑁 / 𝑃) ∈ ℤ))
5139, 50mpbid 232 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑁 / 𝑃) ∈ ℤ)
5238nnred 12282 . . . . . . . . . . . . . . . . 17 (𝜑𝑁 ∈ ℝ)
5343nnred 12282 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 ∈ ℝ)
5438nngt0d 12316 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝑁)
5543nngt0d 12316 . . . . . . . . . . . . . . . . 17 (𝜑 → 0 < 𝑃)
5652, 53, 54, 55divgt0d 12204 . . . . . . . . . . . . . . . 16 (𝜑 → 0 < (𝑁 / 𝑃))
5751, 56jca 511 . . . . . . . . . . . . . . 15 (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 0 < (𝑁 / 𝑃)))
58 elnnz 12625 . . . . . . . . . . . . . . 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 14285 . . . . . . . . . . . . 13 (𝜑 → ((𝑁 / 𝑃)↑𝐿) ∈ ℕ)
6245, 61nnmulcld 12320 . . . . . . . . . . . 12 (𝜑 → ((𝑃𝑈) · ((𝑁 / 𝑃)↑𝐿)) ∈ ℕ)
6341, 62eqeltrid 2844 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℕ)
6423, 24, 25, 26, 28, 29, 32, 33, 35, 36, 37, 38, 39, 40, 63aks6d1c1p7 42115 . . . . . . . . . 10 (𝜑𝐸 𝑋)
6535fldcrngd 20743 . . . . . . . . . . . . . 14 (𝜑𝐾 ∈ CRing)
6624ply1crng 22201 . . . . . . . . . . . . . 14 (𝐾 ∈ CRing → 𝑆 ∈ CRing)
6765, 66syl 17 . . . . . . . . . . . . 13 (𝜑𝑆 ∈ CRing)
68 crngring 20243 . . . . . . . . . . . . . 14 (𝑆 ∈ CRing → 𝑆 ∈ Ring)
69 ringcmn 20280 . . . . . . . . . . . . . 14 (𝑆 ∈ Ring → 𝑆 ∈ CMnd)
7068, 69syl 17 . . . . . . . . . . . . 13 (𝑆 ∈ CRing → 𝑆 ∈ CMnd)
7167, 70syl 17 . . . . . . . . . . . 12 (𝜑𝑆 ∈ CMnd)
72 cmnmnd 19816 . . . . . . . . . . . 12 (𝑆 ∈ CMnd → 𝑆 ∈ Mnd)
7371, 72syl 17 . . . . . . . . . . 11 (𝜑𝑆 ∈ Mnd)
74 crngring 20243 . . . . . . . . . . . . 13 (𝐾 ∈ CRing → 𝐾 ∈ Ring)
7565, 74syl 17 . . . . . . . . . . . 12 (𝜑𝐾 ∈ Ring)
76 eqid 2736 . . . . . . . . . . . . 13 (Base‘𝑆) = (Base‘𝑆)
7726, 24, 76vr1cl 22220 . . . . . . . . . . . 12 (𝐾 ∈ Ring → 𝑋 ∈ (Base‘𝑆))
7875, 77syl 17 . . . . . . . . . . 11 (𝜑𝑋 ∈ (Base‘𝑆))
79 eqid 2736 . . . . . . . . . . . 12 (0g𝑆) = (0g𝑆)
8076, 34, 79mndrid 18769 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑋 ∈ (Base‘𝑆)) → (𝑋 + (0g𝑆)) = 𝑋)
8173, 78, 80syl2anc 584 . . . . . . . . . 10 (𝜑 → (𝑋 + (0g𝑆)) = 𝑋)
8264, 81breqtrrd 5170 . . . . . . . . 9 (𝜑𝐸 (𝑋 + (0g𝑆)))
83 eqid 2736 . . . . . . . . . . . . . 14 (ℤRHom‘𝐾) = (ℤRHom‘𝐾)
84 eqid 2736 . . . . . . . . . . . . . 14 (0g𝐾) = (0g𝐾)
8583, 84zrh0 21525 . . . . . . . . . . . . 13 (𝐾 ∈ Ring → ((ℤRHom‘𝐾)‘0) = (0g𝐾))
8675, 85syl 17 . . . . . . . . . . . 12 (𝜑 → ((ℤRHom‘𝐾)‘0) = (0g𝐾))
8786fveq2d 6909 . . . . . . . . . . 11 (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘0)) = (𝐶‘(0g𝐾)))
8824, 30, 84, 79, 75ply1ascl0 22257 . . . . . . . . . . 11 (𝜑 → (𝐶‘(0g𝐾)) = (0g𝑆))
8987, 88eqtrd 2776 . . . . . . . . . 10 (𝜑 → (𝐶‘((ℤRHom‘𝐾)‘0)) = (0g𝑆))
9089oveq2d 7448 . . . . . . . . 9 (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))) = (𝑋 + (0g𝑆)))
9182, 90breqtrrd 5170 . . . . . . . 8 (𝜑𝐸 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))))
92 aks6d1c1.19 . . . . . . . . 9 (𝜑𝐹:(0...𝐴)⟶ℕ0)
93 0zd 12627 . . . . . . . . . 10 (𝜑 → 0 ∈ ℤ)
94 0red 11265 . . . . . . . . . . 11 (𝜑 → 0 ∈ ℝ)
9594leidd 11830 . . . . . . . . . 10 (𝜑 → 0 ≤ 0)
9693, 2, 93, 95, 3elfzd 13556 . . . . . . . . 9 (𝜑 → 0 ∈ (0...𝐴))
9792, 96ffvelcdmd 7104 . . . . . . . 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 42116 . . . . . . 7 (𝜑𝐸 ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))))
9927crngmgp 20239 . . . . . . . . . 10 (𝑆 ∈ CRing → 𝑊 ∈ CMnd)
10067, 99syl 17 . . . . . . . . 9 (𝜑𝑊 ∈ CMnd)
101100cmnmndd 19823 . . . . . . . 8 (𝜑𝑊 ∈ Mnd)
102 0z 12626 . . . . . . . . 9 0 ∈ ℤ
103102a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℤ)
104 eqid 2736 . . . . . . . . 9 (Base‘𝑊) = (Base‘𝑊)
105 0le0 12368 . . . . . . . . . . . 12 0 ≤ 0
106105a1i 11 . . . . . . . . . . 11 (𝜑 → 0 ≤ 0)
107103, 2, 103, 106, 3elfzd 13556 . . . . . . . . . 10 (𝜑 → 0 ∈ (0...𝐴))
10892, 107ffvelcdmd 7104 . . . . . . . . 9 (𝜑 → (𝐹‘0) ∈ ℕ0)
10983zrhrhm 21523 . . . . . . . . . . . . . . 15 (𝐾 ∈ Ring → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
11075, 109syl 17 . . . . . . . . . . . . . 14 (𝜑 → (ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾))
111 zringbas 21465 . . . . . . . . . . . . . . 15 ℤ = (Base‘ℤring)
112 eqid 2736 . . . . . . . . . . . . . . 15 (Base‘𝐾) = (Base‘𝐾)
113111, 112rhmf 20486 . . . . . . . . . . . . . 14 ((ℤRHom‘𝐾) ∈ (ℤring RingHom 𝐾) → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
114110, 113syl 17 . . . . . . . . . . . . 13 (𝜑 → (ℤRHom‘𝐾):ℤ⟶(Base‘𝐾))
115114, 93ffvelcdmd 7104 . . . . . . . . . . . 12 (𝜑 → ((ℤRHom‘𝐾)‘0) ∈ (Base‘𝐾))
11624, 30, 112, 76ply1sclcl 22290 . . . . . . . . . . . 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 18756 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑋 ∈ (Base‘𝑆) ∧ (𝐶‘((ℤRHom‘𝐾)‘0)) ∈ (Base‘𝑆)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))) ∈ (Base‘𝑆))
11973, 78, 117, 118syl3anc 1372 . . . . . . . . . 10 (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))) ∈ (Base‘𝑆))
12027, 76mgpbas 20143 . . . . . . . . . 10 (Base‘𝑆) = (Base‘𝑊)
121119, 120eleqtrdi 2850 . . . . . . . . 9 (𝜑 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))) ∈ (Base‘𝑊))
122104, 31, 101, 108, 121mulgnn0cld 19114 . . . . . . . 8 (𝜑 → ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))) ∈ (Base‘𝑊))
123 fveq2 6905 . . . . . . . . . 10 (𝑖 = 0 → (𝐹𝑖) = (𝐹‘0))
124 2fveq3 6910 . . . . . . . . . . 11 (𝑖 = 0 → (𝐶‘((ℤRHom‘𝐾)‘𝑖)) = (𝐶‘((ℤRHom‘𝐾)‘0)))
125124oveq2d 7448 . . . . . . . . . 10 (𝑖 = 0 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))))
126123, 125oveq12d 7450 . . . . . . . . 9 (𝑖 = 0 → ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))))
127104, 126gsumsn 19973 . . . . . . . 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 1372 . . . . . . 7 (𝜑 → (𝑊 Σg (𝑖 ∈ {0} ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) = ((𝐹‘0)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0)))))
12998, 128breqtrrd 5170 . . . . . 6 (𝜑𝐸 (𝑊 Σg (𝑖 ∈ {0} ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
130 fzsn 13607 . . . . . . . . . 10 (0 ∈ ℤ → (0...0) = {0})
131102, 130ax-mp 5 . . . . . . . . 9 (0...0) = {0}
132131a1i 11 . . . . . . . 8 (𝜑 → (0...0) = {0})
133132mpteq1d 5236 . . . . . . 7 (𝜑 → (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ {0} ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
134133oveq2d 7448 . . . . . 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 5170 . . . . 5 (𝜑𝐸 (𝑊 Σg (𝑖 ∈ (0...0) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
136353ad2ant1 1133 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝐾 ∈ Field)
137363ad2ant1 1133 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑃 ∈ ℙ)
138373ad2ant1 1133 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑅 ∈ ℕ)
139403ad2ant1 1133 . . . . . . 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 1133 . . . . . . 7 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑃𝑁)
141 simp3 1138 . . . . . . . 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 2904 . . . . . . . . . . 11 𝑘((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))
143 nfcv 2904 . . . . . . . . . . 11 𝑖((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))
144 fveq2 6905 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝐹𝑖) = (𝐹𝑘))
145 2fveq3 6910 . . . . . . . . . . . . 13 (𝑖 = 𝑘 → (𝐶‘((ℤRHom‘𝐾)‘𝑖)) = (𝐶‘((ℤRHom‘𝐾)‘𝑘)))
146145oveq2d 7448 . . . . . . . . . . . 12 (𝑖 = 𝑘 → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))))
147144, 146oveq12d 7450 . . . . . . . . . . 11 (𝑖 = 𝑘 → ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))
148142, 143, 147cbvmpt 5252 . . . . . . . . . 10 (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑘 ∈ (0...𝑗) ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))
149148oveq2i 7443 . . . . . . . . 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 5168 . . . . . . 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 12642 . . . . . . . . . . . . . . 15 (𝜑𝑅 ∈ ℤ)
16051, 159, 603jca 1128 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 / 𝑃) ∈ ℤ ∧ 𝑅 ∈ ℤ ∧ 𝐿 ∈ ℕ0))
161159, 51, 483jca 1128 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑅 ∈ ℤ ∧ (𝑁 / 𝑃) ∈ ℤ ∧ 𝑁 ∈ ℤ))
16248, 159jca 511 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ))
163 gcdcom 16551 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℤ ∧ 𝑅 ∈ ℤ) → (𝑁 gcd 𝑅) = (𝑅 gcd 𝑁))
164162, 163syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑁 gcd 𝑅) = (𝑅 gcd 𝑁))
165 eqeq1 2740 . . . . . . . . . . . . . . . . . . . 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 11290 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ∈ ℂ)
17053recnd 11290 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑃 ∈ ℂ)
17194, 54gtned 11397 . . . . . . . . . . . . . . . . . . . 20 (𝜑𝑁 ≠ 0)
172169, 169, 170, 171, 47divdiv2d 12076 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 / (𝑁 / 𝑃)) = ((𝑁 · 𝑃) / 𝑁))
173169, 170mulcomd 11283 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑁 · 𝑃) = (𝑃 · 𝑁))
174173oveq1d 7447 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑁 · 𝑃) / 𝑁) = ((𝑃 · 𝑁) / 𝑁))
175170, 169, 169, 171, 171divdiv2d 12076 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑃 / (𝑁 / 𝑁)) = ((𝑃 · 𝑁) / 𝑁))
176175eqcomd 2742 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → ((𝑃 · 𝑁) / 𝑁) = (𝑃 / (𝑁 / 𝑁)))
177174, 176eqtrd 2776 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → ((𝑁 · 𝑃) / 𝑁) = (𝑃 / (𝑁 / 𝑁)))
178169, 171dividd 12042 . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑 → (𝑁 / 𝑁) = 1)
179178oveq2d 7448 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑃 / (𝑁 / 𝑁)) = (𝑃 / 1))
180170div1d 12036 . . . . . . . . . . . . . . . . . . . . . 22 (𝜑 → (𝑃 / 1) = 𝑃)
181179, 180eqtrd 2776 . . . . . . . . . . . . . . . . . . . . 21 (𝜑 → (𝑃 / (𝑁 / 𝑁)) = 𝑃)
182181, 46eqeltrd 2840 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → (𝑃 / (𝑁 / 𝑁)) ∈ ℤ)
183177, 182eqeltrd 2840 . . . . . . . . . . . . . . . . . . 19 (𝜑 → ((𝑁 · 𝑃) / 𝑁) ∈ ℤ)
184172, 183eqeltrd 2840 . . . . . . . . . . . . . . . . . 18 (𝜑 → (𝑁 / (𝑁 / 𝑃)) ∈ ℤ)
18594, 56gtned 11397 . . . . . . . . . . . . . . . . . . 19 (𝜑 → (𝑁 / 𝑃) ≠ 0)
186 dvdsval2 16294 . . . . . . . . . . . . . . . . . . 19 (((𝑁 / 𝑃) ∈ ℤ ∧ (𝑁 / 𝑃) ≠ 0 ∧ 𝑁 ∈ ℤ) → ((𝑁 / 𝑃) ∥ 𝑁 ↔ (𝑁 / (𝑁 / 𝑃)) ∈ ℤ))
18751, 185, 48, 186syl3anc 1372 . . . . . . . . . . . . . . . . . 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 16698 . . . . . . . . . . . . . . . 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 16551 . . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . . 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 16761 . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . 14 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))
204 simpr1 1194 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑗 ∈ ℤ)
205204peano2zd 12727 . . . . . . . . . . . . . 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 42111 . . . . . . . . . . . . 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 1128 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ))
209168, 39jca 511 . . . . . . . . . . . . . . . 16 (𝜑 → ((𝑅 gcd 𝑁) = 1 ∧ 𝑃𝑁))
210 rpdvds 16698 . . . . . . . . . . . . . . . 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 16551 . . . . . . . . . . . . . . . . . 18 ((𝑅 ∈ ℤ ∧ 𝑃 ∈ ℤ) → (𝑅 gcd 𝑃) = (𝑃 gcd 𝑅))
214212, 213syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑅 gcd 𝑃) = (𝑃 gcd 𝑅))
215 eqeq1 2740 . . . . . . . . . . . . . . . . 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 42117 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑃𝑈) (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
221 2fveq3 6910 . . . . . . . . . . . . . . . . 17 (𝑎 = (𝑗 + 1) → (𝐶‘((ℤRHom‘𝐾)‘𝑎)) = (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))
222221oveq2d 7448 . . . . . . . . . . . . . . . 16 (𝑎 = (𝑗 + 1) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
223222breq2d 5154 . . . . . . . . . . . . . . 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 42115 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝜑𝑁 𝑋)
226225, 81breqtrrd 5170 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝜑𝑁 (𝑋 + (0g𝑆)))
227226, 90breqtrrd 5170 . . . . . . . . . . . . . . . . . . . . . . . . 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 6909 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑𝑎 = 0) → ((ℤRHom‘𝐾)‘𝑎) = ((ℤRHom‘𝐾)‘0))
231230fveq2d 6909 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑𝑎 = 0) → (𝐶‘((ℤRHom‘𝐾)‘𝑎)) = (𝐶‘((ℤRHom‘𝐾)‘0)))
232231oveq2d 7448 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑𝑎 = 0) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘0))))
233228, 232breqtrrd 5170 . . . . . . . . . . . . . . . . . . . . . . 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 11257 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → 1 ∈ ℂ)
238237addlidd 11463 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → (0 + 1) = 1)
239238oveq1d 7447 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝜑 ∧ (𝑎 ∈ (1...𝐴) → 𝑁 (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑎))))) → ((0 + 1)...𝐴) = (1...𝐴))
240239eleq2d 2826 . . . . . . . . . . . . . . . . . . . . . . 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 12883 . . . . . . . . . . . . . . . . . . . . . . . . 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 13644 . . . . . . . . . . . . . . . . . . . . . 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 3162 . . . . . . . . . . . . . . . . 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 12627 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 0 ∈ ℤ)
2582adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐴 ∈ ℤ)
259204zred 12724 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑗 ∈ ℝ)
260 1red 11263 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 1 ∈ ℝ)
261 simpr2 1195 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 0 ≤ 𝑗)
262 0le1 11787 . . . . . . . . . . . . . . . . . 18 0 ≤ 1
263262a1i 11 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 0 ≤ 1)
264259, 260, 261, 263addge0d 11840 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 0 ≤ (𝑗 + 1))
265 simpr3 1196 . . . . . . . . . . . . . . . . 17 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝑗 < 𝐴)
266204, 258zltp1led 41981 . . . . . . . . . . . . . . . . 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 13556 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑗 + 1) ∈ (0...𝐴))
269223, 256, 268rspcdva 3622 . . . . . . . . . . . . . 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 42112 . . . . . . . . . . . . 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 42117 . . . . . . . . . . . 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 42114 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ((𝑃𝑈) · ((𝑁 / 𝑃)↑𝐿)) (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
277158, 276eqbrtrd 5164 . . . . . . . . . 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 7104 . . . . . . . . . 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 42116 . . . . . . . . 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 7467 . . . . . . . . . 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 7104 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ((ℤRHom‘𝐾)‘(𝑗 + 1)) ∈ (Base‘𝐾))
28824, 30, 112, 76ply1sclcl 22290 . . . . . . . . . . . . . 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 18756 . . . . . . . . . . . . 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 1372 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))) ∈ (Base‘𝑆))
292291, 120eleqtrdi 2850 . . . . . . . . . . 11 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))) ∈ (Base‘𝑊))
293104, 31, 281, 279, 292mulgnn0cld 19114 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → ((𝐹‘(𝑗 + 1))𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))) ∈ (Base‘𝑊))
294 fveq2 6905 . . . . . . . . . . . 12 (𝑘 = (𝑗 + 1) → (𝐹𝑘) = (𝐹‘(𝑗 + 1)))
295 2fveq3 6910 . . . . . . . . . . . . 13 (𝑘 = (𝑗 + 1) → (𝐶‘((ℤRHom‘𝐾)‘𝑘)) = (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))
296295oveq2d 7448 . . . . . . . . . . . 12 (𝑘 = (𝑗 + 1) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))) = (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1)))))
297294, 296oveq12d 7450 . . . . . . . . . . 11 (𝑘 = (𝑗 + 1) → ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))) = ((𝐹‘(𝑗 + 1))𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘(𝑗 + 1))))))
298104, 297gsumsn 19973 . . . . . . . . . 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 1372 . . . . . . . . 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 5170 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴)) → 𝐸 (𝑊 Σg (𝑘 ∈ {(𝑗 + 1)} ↦ ((𝐹𝑘)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘)))))))
3013003adant3 1132 . . . . . . 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 42113 . . . . . 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 5252 . . . . . . . . 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 7448 . . . . . . 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 2736 . . . . . . . 8 (+g𝑊) = (+g𝑊)
3071003ad2ant1 1133 . . . . . . . 8 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑊 ∈ CMnd)
308 simp21 1206 . . . . . . . . . 10 ((𝜑 ∧ (𝑗 ∈ ℤ ∧ 0 ≤ 𝑗𝑗 < 𝐴) ∧ 𝐸 (𝑊 Σg (𝑖 ∈ (0...𝑗) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))) → 𝑗 ∈ ℤ)
309 simp22 1207 . . . . . . . . . 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 12628 . . . . . . . . 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 1132 . . . . . . . . . 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 1133 . . . . . . . . . . 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 12627 . . . . . . . . . . 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 1133 . . . . . . . . . . . 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 13565 . . . . . . . . . . . 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 13568 . . . . . . . . . . . 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 12724 . . . . . . . . . . . 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 12724 . . . . . . . . . . . . 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 11263 . . . . . . . . . . . . 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 11291 . . . . . . . . . . . 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 12724 . . . . . . . . . . . 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 13569 . . . . . . . . . . . . 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 1253 . . . . . . . . . . . . 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 41981 . . . . . . . . . . . . 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 11419 . . . . . . . . . . 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 13556 . . . . . . . . . 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 7104 . . . . . . . . 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 1132 . . . . . . . . . . . 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 1132 . . . . . . . . . . . 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 1132 . . . . . . . . . . . . 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 7104 . . . . . . . . . . . 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 22290 . . . . . . . . . . . 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 18756 . . . . . . . . . . 11 ((𝑆 ∈ Mnd ∧ 𝑋 ∈ (Base‘𝑆) ∧ (𝐶‘((ℤRHom‘𝐾)‘𝑘)) ∈ (Base‘𝑆)) → (𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑘))) ∈ (Base‘𝑆))
349339, 341, 347, 348syl3anc 1372 . . . . . . . . . 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 2850 . . . . . . . . 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 19114 . . . . . . . 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 19951 . . . . . . 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 2776 . . . . . 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 5170 . . . . 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 12722 . . . 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 768 . . . . . . 7 (((𝜑𝑔 = 𝐹) ∧ 𝑖 ∈ (0...𝐴)) → 𝑔 = 𝐹)
361360fveq1d 6907 . . . . . 6 (((𝜑𝑔 = 𝐹) ∧ 𝑖 ∈ (0...𝐴)) → (𝑔𝑖) = (𝐹𝑖))
362361oveq1d 7447 . . . . 5 (((𝜑𝑔 = 𝐹) ∧ 𝑖 ∈ (0...𝐴)) → ((𝑔𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))) = ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))
363362mpteq2dva 5241 . . . 4 ((𝜑𝑔 = 𝐹) → (𝑖 ∈ (0...𝐴) ↦ ((𝑔𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))) = (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖))))))
364363oveq2d 7448 . . 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 12534 . . . . . 6 0 ∈ V
366365a1i 11 . . . . 5 (𝜑 → ℕ0 ∈ V)
367 ovexd 7467 . . . . 5 (𝜑 → (0...𝐴) ∈ V)
368366, 367elmapd 8881 . . . 4 (𝜑 → (𝐹 ∈ (ℕ0m (0...𝐴)) ↔ 𝐹:(0...𝐴)⟶ℕ0))
36992, 368mpbird 257 . . 3 (𝜑𝐹 ∈ (ℕ0m (0...𝐴)))
370 ovexd 7467 . . 3 (𝜑 → (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))) ∈ V)
371359, 364, 369, 370fvmptd 7022 . 2 (𝜑 → (𝐺𝐹) = (𝑊 Σg (𝑖 ∈ (0...𝐴) ↦ ((𝐹𝑖)𝐷(𝑋 + (𝐶‘((ℤRHom‘𝐾)‘𝑖)))))))
372357, 371breqtrrd 5170 1 (𝜑𝐸 (𝐺𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  Vcvv 3479  {csn 4625   class class class wbr 5142  {copab 5204  cmpt 5224  wf 6556  cfv 6560  (class class class)co 7432  m cmap 8867  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161   < clt 11296  cle 11297   / cdiv 11921  cn 12267  0cn0 12528  cz 12615  cuz 12879  ...cfz 13548  cexp 14103  cdvds 16291   gcd cgcd 16532  cprime 16709  Basecbs 17248  +gcplusg 17298  0gc0g 17485   Σg cgsu 17486  Mndcmnd 18748  .gcmg 19086  CMndccmn 19799  mulGrpcmgp 20138  Ringcrg 20231  CRingccrg 20232   RingHom crh 20470   RingIso crs 20471  Fieldcfield 20731  ringczring 21458  ℤRHomczrh 21511  chrcchr 21513  algSccascl 21873  var1cv1 22178  Poly1cpl1 22179  eval1ce1 22319   PrimRoots cprimroots 42093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234  ax-addf 11235  ax-mulf 11236
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-tp 4630  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-iin 4993  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-of 7698  df-ofr 7699  df-om 7889  df-1st 8015  df-2nd 8016  df-supp 8187  df-tpos 8252  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-2o 8508  df-oadd 8511  df-er 8746  df-map 8869  df-pm 8870  df-ixp 8939  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-fsupp 9403  df-sup 9483  df-inf 9484  df-oi 9551  df-dju 9942  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-4 12332  df-5 12333  df-6 12334  df-7 12335  df-8 12336  df-9 12337  df-n0 12529  df-xnn0 12602  df-z 12616  df-dec 12736  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-fl 13833  df-mod 13911  df-seq 14044  df-exp 14104  df-fac 14314  df-bc 14343  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-dvds 16292  df-gcd 16533  df-prm 16710  df-phi 16804  df-struct 17185  df-sets 17202  df-slot 17220  df-ndx 17232  df-base 17249  df-ress 17276  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 17487  df-gsum 17488  df-prds 17493  df-pws 17495  df-mre 17630  df-mrc 17631  df-acs 17633  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-mhm 18797  df-submnd 18798  df-grp 18955  df-minusg 18956  df-sbg 18957  df-mulg 19087  df-subg 19142  df-ghm 19232  df-cntz 19336  df-od 19547  df-cmn 19801  df-abl 19802  df-mgp 20139  df-rng 20151  df-ur 20180  df-srg 20185  df-ring 20233  df-cring 20234  df-oppr 20335  df-dvdsr 20358  df-unit 20359  df-invr 20389  df-dvr 20402  df-rhm 20473  df-rim 20474  df-subrng 20547  df-subrg 20571  df-drng 20732  df-field 20733  df-lmod 20861  df-lss 20931  df-lsp 20971  df-cnfld 21366  df-zring 21459  df-zrh 21515  df-chr 21517  df-assa 21874  df-asp 21875  df-ascl 21876  df-psr 21930  df-mvr 21931  df-mpl 21932  df-opsr 21934  df-evls 22099  df-evl 22100  df-psr1 22182  df-vr1 22183  df-ply1 22184  df-coe1 22185  df-evl1 22321  df-primroots 42094
This theorem is referenced by:  aks6d1c1rh  42127
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