Theorem aks6d1c2 43160

Description: Claim 2 of Theorem 6.1 of https://www3.nd.edu/%7eandyp/notes/AKS.pdf (Contributed by metakunt, 2-May-2025.)

Hypotheses

Ref	Expression
aks6d1c2a.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)))}
aks6d1c2a.2	⊢ 𝑃 = (chr‘𝐾)
aks6d1c2a.3	⊢ (𝜑 → 𝐾 ∈ Field)
aks6d1c2a.4	⊢ (𝜑 → 𝑃 ∈ ℙ)
aks6d1c2a.5	⊢ (𝜑 → 𝑅 ∈ ℕ)
aks6d1c2a.6	⊢ (𝜑 → 𝑁 ∈ ℕ)
aks6d1c2a.7	⊢ (𝜑 → 𝑃 ∥ 𝑁)
aks6d1c2a.8	⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
aks6d1c2a.10	⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
aks6d1c2a.11	⊢ (𝜑 → 𝐴 ∈ ℕ0)
aks6d1c2a.12	⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
aks6d1c2a.13	⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
aks6d1c2a.14	⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
aks6d1c2a.15	⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
aks6d1c2a.16	⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
aks6d1c2a.17	⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
aks6d1c2a.18	⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
aks6d1c2a.19	⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))
aks6d1c2a.20	⊢ (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁 ∧ 𝑃 ≠ 𝑄))

Assertion

Ref	Expression
aks6d1c2	⊢ (𝜑 → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))

Distinct variable groups:   ∼ ,𝑎   𝐴,𝑎   𝐴,𝑔,𝑖   𝐴,ℎ   𝐴,𝑘,𝑙   𝑥,𝐴   𝐵,𝑎   𝐵,𝑔,𝑖   𝐵,𝑘,𝑙   𝑥,𝐵   𝐶,𝑎   𝐶,𝑔,𝑖   𝐶,ℎ   𝐶,𝑘,𝑙   𝑥,𝐶   𝐸,𝑎   𝑔,𝐸,𝑖   𝑘,𝐸,𝑙   𝑥,𝐸   𝑒,𝐺,𝑓,𝑦   ℎ,𝐺   𝐾,𝑎   𝑒,𝐾,𝑓,𝑦   𝑔,𝐾,𝑖   ℎ,𝐾   𝑥,𝐾   ℎ,𝑀   𝑦,𝑀   𝑁,𝑎   𝑒,𝑁,𝑓,𝑦   𝑘,𝑁,𝑙   𝑥,𝑁   𝑃,𝑒,𝑓,𝑦   𝑃,𝑘,𝑙   𝑥,𝑃   𝑅,𝑎   𝑅,𝑒,𝑓,𝑦   𝑅,𝑔,𝑖   𝑅,ℎ   𝑅,𝑘,𝑙   𝑥,𝑅   𝜑,𝑎   𝜑,𝑔,𝑖   𝜑,ℎ   𝜑,𝑘,𝑙   𝜑,𝑥

Allowed substitution hints:   𝜑(𝑦, 𝑒, 𝑓)   𝐴(𝑦, 𝑒, 𝑓)   𝐵(𝑦, 𝑒, 𝑓, ℎ)   𝐶(𝑦, 𝑒, 𝑓)   𝑃(𝑔, ℎ, 𝑖, 𝑎)   𝑄(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙)   ∼ (𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑙)   𝐸(𝑦, 𝑒, 𝑓, ℎ)   𝐺(𝑥, 𝑔, 𝑖, 𝑘, 𝑎, 𝑙)   𝐻(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙)   𝐾(𝑘, 𝑙)   𝐿(𝑥, 𝑦, 𝑒, 𝑓, 𝑔, ℎ, 𝑖, 𝑘, 𝑎, 𝑙)   𝑀(𝑥, 𝑒, 𝑓, 𝑔, 𝑖, 𝑘, 𝑎, 𝑙)   𝑁(𝑔, ℎ, 𝑖)

Proof of Theorem aks6d1c2

Dummy variables 𝑏 𝑐 𝑑 𝑗 𝑢 𝑤 𝑠 𝑡 𝑜 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl 488	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → ((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶))
2		simprl 783	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → 𝑏 < 𝑐)
3	1, 2	jca 521	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐))
4		simprr 785	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))
5	3, 4	jca 521	. . 3 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅))))
6		aks6d1c2a.1	. . . . . . 7 ⊢ ∼ = {⟨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)))}
7		aks6d1c2a.2	. . . . . . 7 ⊢ 𝑃 = (chr‘𝐾)
8		aks6d1c2a.3	. . . . . . . 8 ⊢ (𝜑 → 𝐾 ∈ Field)
9	8	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝐾 ∈ Field)
10		aks6d1c2a.4	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℙ)
11	10	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑃 ∈ ℙ)
12		aks6d1c2a.5	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ ℕ)
13	12	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑅 ∈ ℕ)
14		aks6d1c2a.6	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ)
15	14	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑁 ∈ ℕ)
16		aks6d1c2a.7	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∥ 𝑁)
17	16	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑃 ∥ 𝑁)
18		aks6d1c2a.8	. . . . . . . 8 ⊢ (𝜑 → (𝑁 gcd 𝑅) = 1)
19	18	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → (𝑁 gcd 𝑅) = 1)
20		0nn0 12614	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
21	20	a1i 11	. . . . . . . 8 ⊢ (((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) ∧ 𝑗 ∈ (0...𝐴)) → 0 ∈ ℕ0)
22		eqid 2761	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝐴) ↦ 0) = (𝑗 ∈ (0...𝐴) ↦ 0)
23	21, 22	fmptd 7112	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → (𝑗 ∈ (0...𝐴) ↦ 0):(0...𝐴)⟶ℕ0)
24		aks6d1c2a.10	. . . . . . 7 ⊢ 𝐺 = (𝑔 ∈ (ℕ0 ↑m (0...𝐴)) ↦ ((mulGrp‘(Poly1‘𝐾)) Σg (𝑖 ∈ (0...𝐴) ↦ ((𝑔‘𝑖)(.g‘(mulGrp‘(Poly1‘𝐾)))((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑖)))))))
25		aks6d1c2a.11	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℕ0)
26	25	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝐴 ∈ ℕ0)
27		aks6d1c2a.12	. . . . . . 7 ⊢ 𝐸 = (𝑘 ∈ ℕ0, 𝑙 ∈ ℕ0 ↦ ((𝑃↑𝑘) · ((𝑁 / 𝑃)↑𝑙)))
28		aks6d1c2a.13	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘(ℤ/nℤ‘𝑅))
29		aks6d1c2a.14	. . . . . . . 8 ⊢ (𝜑 → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
30	29	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → ∀𝑎 ∈ (1...𝐴)𝑁 ∼ ((var1‘𝐾)(+g‘(Poly1‘𝐾))((algSc‘(Poly1‘𝐾))‘((ℤRHom‘𝐾)‘𝑎))))
31		aks6d1c2a.15	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
32	31	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → (𝑥 ∈ (Base‘𝐾) ↦ (𝑃(.g‘(mulGrp‘𝐾))𝑥)) ∈ (𝐾 RingIso 𝐾))
33		aks6d1c2a.16	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
34	33	ad5antr 747	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑀 ∈ ((mulGrp‘𝐾) PrimRoots 𝑅))
35		aks6d1c2a.17	. . . . . . 7 ⊢ 𝐻 = (ℎ ∈ (ℕ0 ↑m (0...𝐴)) ↦ (((eval1‘𝐾)‘(𝐺‘ℎ))‘𝑀))
36		aks6d1c2a.18	. . . . . . 7 ⊢ 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
37		aks6d1c2a.19	. . . . . . 7 ⊢ 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵)))
38		simp-5r 798	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑏 ∈ 𝐶)
39		simp-4r 796	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑐 ∈ 𝐶)
40		simpllr 788	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑏 < 𝑐)
41		eqid 2761	. . . . . . 7 ⊢ (.g‘(mulGrp‘(Poly1‘𝐾))) = (.g‘(mulGrp‘(Poly1‘𝐾)))
42		eqid 2761	. . . . . . 7 ⊢ (var1‘𝐾) = (var1‘𝐾)
43		eqid 2761	. . . . . . 7 ⊢ ((𝑐(.g‘(mulGrp‘(Poly1‘𝐾)))(var1‘𝐾))(-g‘(Poly1‘𝐾))(𝑏(.g‘(mulGrp‘(Poly1‘𝐾)))(var1‘𝐾))) = ((𝑐(.g‘(mulGrp‘(Poly1‘𝐾)))(var1‘𝐾))(-g‘(Poly1‘𝐾))(𝑏(.g‘(mulGrp‘(Poly1‘𝐾)))(var1‘𝐾)))
44		simplr 781	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑑 ∈ ℕ)
45		simpr 490	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑐 = (𝑏 + (𝑑 · 𝑅)))
46	6, 7, 9, 11, 13, 15, 17, 19, 23, 24, 26, 27, 28, 30, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45	aks6d1c2lem4 43157	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))
47	46	ex 418	. . . . 5 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) → (𝑐 = (𝑏 + (𝑑 · 𝑅)) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵)))
48	47	rexlimdva 3164	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) → (∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵)))
49	48	imp 412	. . 3 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))
50	5, 49	syl 18	. 2 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))
51		simprr 785	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑏 < 𝑐)
52		nfcv 2923	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑠(𝐿‘𝑡)
53		nfcv 2923	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝐿‘𝑠)
54		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (𝐿‘𝑡) = (𝐿‘𝑠))
55	52, 53, 54	cbvmpt 5207	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡)) = (𝑠 ∈ 𝐶 ↦ (𝐿‘𝑠))
56	55	a1i 11	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡)) = (𝑠 ∈ 𝐶 ↦ (𝐿‘𝑠)))
57		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ 𝑠 = 𝑏) → 𝑠 = 𝑏)
58	57	fveq2d 6887	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ 𝑠 = 𝑏) → (𝐿‘𝑠) = (𝐿‘𝑏))
59		simpllr 788	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑏 ∈ 𝐶)
60		fvexd 6898	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝐿‘𝑏) ∈ V)
61	56, 58, 59, 60	fvmptd 6999	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = (𝐿‘𝑏))
62	61	eqcomd 2767	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝐿‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏))
63		simprl 783	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐))
64		simpr 490	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ 𝑠 = 𝑐) → 𝑠 = 𝑐)
65	64	fveq2d 6887	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ 𝑠 = 𝑐) → (𝐿‘𝑠) = (𝐿‘𝑐))
66		simplr 781	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑐 ∈ 𝐶)
67		fvexd 6898	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝐿‘𝑐) ∈ V)
68	56, 65, 66, 67	fvmptd 6999	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) = (𝐿‘𝑐))
69	63, 68	eqtrd 2796	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = (𝐿‘𝑐))
70	62, 69	eqtrd 2796	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝐿‘𝑏) = (𝐿‘𝑐))
71	70	eqcomd 2767	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝐿‘𝑐) = (𝐿‘𝑏))
72	12	nnnn0d 12660	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
73	72	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑅 ∈ ℕ0)
74	73	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑅 ∈ ℕ0)
75	74	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑅 ∈ ℕ0)
76		fz0ssnn0 13749	. . . . . . . . . . . . . . . . . 18 ⊢ (0...𝐵) ⊆ ℕ0
77	76	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...𝐵) ⊆ ℕ0)
78	77, 77	jca 521	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...𝐵) ⊆ ℕ0 ∧ (0...𝐵) ⊆ ℕ0))
79		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
80	14, 10, 16, 12, 18, 27, 28, 79	hashscontpowcl 43150	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0)
81	80	nn0red 12661	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ)
82	80	nn0ge0d 12663	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
83	81, 82	resqrtcld 15578	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∈ ℝ)
84	83	flcld 13931	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ)
85	81, 82	sqrtge0d 15581	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
86		0zd 12698	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ∈ ℤ)
87		flge 13938	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
88	83, 86, 87	syl2anc 596	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
89	85, 88	mpbid 235	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
90	84, 89	jca 521	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
91		elnn0z 12699	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℕ0 ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
92	90, 91	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℕ0)
93	36	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
94	93	eleq1d 2846	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐵 ∈ ℕ0 ↔ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℕ0))
95	92, 94	mpbird 260	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐵 ∈ ℕ0)
96	95	nn0ge0d 12663	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ≤ 𝐵)
97	95	nn0zd 12711	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐵 ∈ ℤ)
98		eluz 12972	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ (ℤ≥‘0) ↔ 0 ≤ 𝐵))
99	86, 97, 98	syl2anc 596	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 ∈ (ℤ≥‘0) ↔ 0 ≤ 𝐵))
100	96, 99	mpbird 260	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘0))
101		fzn0 13664	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0...𝐵) ≠ ∅ ↔ 𝐵 ∈ (ℤ≥‘0))
102	100, 101	sylibr 237	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0...𝐵) ≠ ∅)
103	102, 102	jca 521	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅))
104		xpnz 6150	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅) ↔ ((0...𝐵) × (0...𝐵)) ≠ ∅)
105	104	biimpi 219	. . . . . . . . . . . . . . . . . 18 ⊢ (((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅) → ((0...𝐵) × (0...𝐵)) ≠ ∅)
106	103, 105	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ≠ ∅)
107		ssxpb 6166	. . . . . . . . . . . . . . . . 17 ⊢ (((0...𝐵) × (0...𝐵)) ≠ ∅ → (((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0) ↔ ((0...𝐵) ⊆ ℕ0 ∧ (0...𝐵) ⊆ ℕ0)))
108	106, 107	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0) ↔ ((0...𝐵) ⊆ ℕ0 ∧ (0...𝐵) ⊆ ℕ0)))
109	78, 108	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0))
110		imass2 6055	. . . . . . . . . . . . . . 15 ⊢ (((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0) → (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ (𝐸 “ (ℕ0 × ℕ0)))
111	109, 110	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ (𝐸 “ (ℕ0 × ℕ0)))
112		nfv 1947	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑜𝜑
113		aks6d1c2a.20	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁 ∧ 𝑃 ≠ 𝑄))
114	113	simp1d 1160	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑄 ∈ ℙ)
115	113	simp2d 1161	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑄 ∥ 𝑁)
116	113	simp3d 1162	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
117	14, 10, 16, 27, 114, 115, 116	aks6d1c2p2 43149	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)–1-1→ℕ)
118		f1f 6776	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸:(ℕ0 × ℕ0)–1-1→ℕ → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
119	117, 118	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
120	119	ffnd 6708	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 Fn (ℕ0 × ℕ0))
121	120	fnfund 6638	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐸)
122	119	ffvelcdmda 7082	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑜 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑜) ∈ ℕ)
123	112, 121, 122	funimassd 6949	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℕ)
124	111, 123	sstrd 3941	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ ℕ)
125	37	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵))))
126	125	sseq1d 3962	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ⊆ ℕ ↔ (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ ℕ))
127	124, 126	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ⊆ ℕ)
128	127	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝐶 ⊆ ℕ)
129		simpr 490	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ 𝐶)
130	128, 129	sseldd 3932	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ ℕ)
131	130	nnzd 12712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ ℤ)
132	131	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑐 ∈ ℤ)
133		simplr 781	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑏 ∈ 𝐶)
134	128, 133	sseldd 3932	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑏 ∈ ℕ)
135	134	nnzd 12712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑏 ∈ ℤ)
136	135	adantr 486	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑏 ∈ ℤ)
137	79, 28	zndvds 21848	. . . . . . . 8 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑐 ∈ ℤ ∧ 𝑏 ∈ ℤ) → ((𝐿‘𝑐) = (𝐿‘𝑏) ↔ 𝑅 ∥ (𝑐 − 𝑏)))
138	75, 132, 136, 137	syl3anc 1398	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → ((𝐿‘𝑐) = (𝐿‘𝑏) ↔ 𝑅 ∥ (𝑐 − 𝑏)))
139	71, 138	mpbid 235	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑅 ∥ (𝑐 − 𝑏))
140	75	nn0zd 12711	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑅 ∈ ℤ)
141	132, 136	zsubcld 12801	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝑐 − 𝑏) ∈ ℤ)
142		divides 16417	. . . . . . . 8 ⊢ ((𝑅 ∈ ℤ ∧ (𝑐 − 𝑏) ∈ ℤ) → (𝑅 ∥ (𝑐 − 𝑏) ↔ ∃𝑑 ∈ ℤ (𝑑 · 𝑅) = (𝑐 − 𝑏)))
143	140, 141, 142	syl2anc 596	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝑅 ∥ (𝑐 − 𝑏) ↔ ∃𝑑 ∈ ℤ (𝑑 · 𝑅) = (𝑐 − 𝑏)))
144	143	biimpd 232	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝑅 ∥ (𝑐 − 𝑏) → ∃𝑑 ∈ ℤ (𝑑 · 𝑅) = (𝑐 − 𝑏)))
145	139, 144	mpd 16	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → ∃𝑑 ∈ ℤ (𝑑 · 𝑅) = (𝑐 − 𝑏))
146		simprl 783	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑑 ∈ ℤ)
147	130	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑐 ∈ ℕ)
148	147	nnred 12343	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑐 ∈ ℝ)
149	134	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑏 ∈ ℕ)
150	149	nnred 12343	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑏 ∈ ℝ)
151	148, 150	resubcld 11737	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑐 − 𝑏) ∈ ℝ)
152	12	nnrpd 13155	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
153	152	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑅 ∈ ℝ+)
154	153	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑅 ∈ ℝ+)
155	154	adantr 486	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑅 ∈ ℝ+)
156	155	adantr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑅 ∈ ℝ+)
157	156	rpred 13157	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑅 ∈ ℝ)
158	51	adantr 486	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑏 < 𝑐)
159	150, 148	posdifd 11896	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑏 < 𝑐 ↔ 0 < (𝑐 − 𝑏)))
160	158, 159	mpbid 235	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 0 < (𝑐 − 𝑏))
161	156	rpgt0d 13160	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 0 < 𝑅)
162	151, 157, 160, 161	divgt0d 12245	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 0 < ((𝑐 − 𝑏) / 𝑅))
163	157	recnd 11330	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑅 ∈ ℂ)
164	146	zred 12796	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑑 ∈ ℝ)
165	164	recnd 11330	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑑 ∈ ℂ)
166	163, 165	mulcomd 11323	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑅 · 𝑑) = (𝑑 · 𝑅))
167		simprr 785	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑑 · 𝑅) = (𝑐 − 𝑏))
168	166, 167	eqtrd 2796	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑅 · 𝑑) = (𝑐 − 𝑏))
169	151	recnd 11330	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑐 − 𝑏) ∈ ℂ)
170	161	gt0ne0d 11873	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑅 ≠ 0)
171	169, 163, 165, 170	divmuld 12108	. . . . . . . . 9 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (((𝑐 − 𝑏) / 𝑅) = 𝑑 ↔ (𝑅 · 𝑑) = (𝑐 − 𝑏)))
172	168, 171	mpbird 260	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → ((𝑐 − 𝑏) / 𝑅) = 𝑑)
173	162, 172	breqtrd 5131	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 0 < 𝑑)
174	146, 173	jca 521	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑑 ∈ ℤ ∧ 0 < 𝑑))
175		elnnz 12696	. . . . . 6 ⊢ (𝑑 ∈ ℕ ↔ (𝑑 ∈ ℤ ∧ 0 < 𝑑))
176	174, 175	sylibr 237	. . . . 5 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑑 ∈ ℕ)
177	167	eqcomd 2767	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑐 − 𝑏) = (𝑑 · 𝑅))
178	148	recnd 11330	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑐 ∈ ℂ)
179	150	recnd 11330	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑏 ∈ ℂ)
180	167, 169	eqeltrd 2861	. . . . . . . 8 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑑 · 𝑅) ∈ ℂ)
181	178, 179, 180	subaddd 11680	. . . . . . 7 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → ((𝑐 − 𝑏) = (𝑑 · 𝑅) ↔ (𝑏 + (𝑑 · 𝑅)) = 𝑐))
182	177, 181	mpbid 235	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → (𝑏 + (𝑑 · 𝑅)) = 𝑐)
183	182	eqcomd 2767	. . . . 5 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) ∧ (𝑑 ∈ ℤ ∧ (𝑑 · 𝑅) = (𝑐 − 𝑏))) → 𝑐 = (𝑏 + (𝑑 · 𝑅)))
184	145, 176, 183	reximssdv 3181	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))
185	51, 184	jca 521	. . 3 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅))))
186		fzfid 14109	. . . . . . 7 ⊢ (𝜑 → (0...𝐵) ∈ Fin)
187		xpfi 9304	. . . . . . 7 ⊢ (((0...𝐵) ∈ Fin ∧ (0...𝐵) ∈ Fin) → ((0...𝐵) × (0...𝐵)) ∈ Fin)
188	186, 186, 187	syl2anc 596	. . . . . 6 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ∈ Fin)
189		imafi 9300	. . . . . 6 ⊢ ((Fun 𝐸 ∧ ((0...𝐵) × (0...𝐵)) ∈ Fin) → (𝐸 “ ((0...𝐵) × (0...𝐵))) ∈ Fin)
190	121, 188, 189	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) ∈ Fin)
191	125	eleq1d 2846	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Fin ↔ (𝐸 “ ((0...𝐵) × (0...𝐵))) ∈ Fin))
192	190, 191	mpbird 260	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Fin)
193	79	zncrng 21843	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℕ0 → (ℤ/nℤ‘𝑅) ∈ CRing)
194	72, 193	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
195		crngring 20465	. . . . . . . . 9 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
196	194, 195	syl 18	. . . . . . . 8 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ Ring)
197	28	zrhrhm 21810	. . . . . . . 8 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
198	196, 197	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
199	198	imaexd 7926	. . . . . 6 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V)
200		hashclb 14495	. . . . . 6 ⊢ ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V → ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin ↔ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0))
201	199, 200	syl 18	. . . . 5 ⊢ (𝜑 → ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin ↔ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0))
202	80, 201	mpbird 260	. . . 4 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin)
203		hashcl 14493	. . . . . . . . . . . . 13 ⊢ ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0)
204	202, 203	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0)
205	204	nn0red 12661	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ)
206	204	nn0ge0d 12663	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
207		sqrtmsq 15430	. . . . . . . . . . 11 ⊢ (((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ ∧ 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
208	205, 206, 207	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
209	208	eqcomd 2767	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) = (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
210	205, 206	jca 521	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ ∧ 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
211		sqrtmul 15419	. . . . . . . . . . 11 ⊢ ((((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ ∧ 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∧ ((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ ∧ 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
212	210, 210, 211	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
213	205, 206	resqrtcld 15578	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∈ ℝ)
214	213	flcld 13931	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ)
215	205, 206	sqrtge0d 15581	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
216	213, 86, 87	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
217	215, 216	mpbid 235	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
218	214, 217	jca 521	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
219	218, 91	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℕ0)
220	219, 94	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℕ0)
221	220	nn0red 12661	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ)
222		1red 11302	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
223	221, 222	readdcld 11331	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 1) ∈ ℝ)
224		flltp1 13933	. . . . . . . . . . . . 13 ⊢ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∈ ℝ → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) < ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) + 1))
225	213, 224	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) < ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) + 1))
226	93	oveq1d 7433	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 1) = ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) + 1))
227	225, 226	breqtrrd 5133	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) < (𝐵 + 1))
228	213, 223, 213, 223, 215, 227, 215, 227	ltmul12ad 12251	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) < ((𝐵 + 1) · (𝐵 + 1)))
229	212, 228	eqbrtrd 5127	. . . . . . . . 9 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) < ((𝐵 + 1) · (𝐵 + 1)))
230	209, 229	eqbrtrd 5127	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < ((𝐵 + 1) · (𝐵 + 1)))
231		hashfz0 14570	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ0 → (♯‘(0...𝐵)) = (𝐵 + 1))
232	220, 231	syl 18	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(0...𝐵)) = (𝐵 + 1))
233	232, 232	oveq12d 7436	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(0...𝐵)) · (♯‘(0...𝐵))) = ((𝐵 + 1) · (𝐵 + 1)))
234	230, 233	breqtrrd 5133	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < ((♯‘(0...𝐵)) · (♯‘(0...𝐵))))
235	186, 186	jca 521	. . . . . . . 8 ⊢ (𝜑 → ((0...𝐵) ∈ Fin ∧ (0...𝐵) ∈ Fin))
236		hashxp 14572	. . . . . . . 8 ⊢ (((0...𝐵) ∈ Fin ∧ (0...𝐵) ∈ Fin) → (♯‘((0...𝐵) × (0...𝐵))) = ((♯‘(0...𝐵)) · (♯‘(0...𝐵))))
237	235, 236	syl 18	. . . . . . 7 ⊢ (𝜑 → (♯‘((0...𝐵) × (0...𝐵))) = ((♯‘(0...𝐵)) · (♯‘(0...𝐵))))
238	234, 237	breqtrrd 5133	. . . . . 6 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < (♯‘((0...𝐵) × (0...𝐵))))
239		ovexd 7453	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝐵) ∈ V)
240	239, 239	jca 521	. . . . . . . . . 10 ⊢ (𝜑 → ((0...𝐵) ∈ V ∧ (0...𝐵) ∈ V))
241		xpexg 7762	. . . . . . . . . 10 ⊢ (((0...𝐵) ∈ V ∧ (0...𝐵) ∈ V) → ((0...𝐵) × (0...𝐵)) ∈ V)
242	240, 241	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ∈ V)
243	242	mptexd 7228	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) ∈ V)
244	120	adantr 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ((0...𝐵) × (0...𝐵))) → 𝐸 Fn (ℕ0 × ℕ0))
245	109	sselda 3931	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ((0...𝐵) × (0...𝐵))) → 𝑤 ∈ (ℕ0 × ℕ0))
246		simpr 490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ((0...𝐵) × (0...𝐵))) → 𝑤 ∈ ((0...𝐵) × (0...𝐵)))
247	244, 245, 246	fnfvimad 7238	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ((0...𝐵) × (0...𝐵))) → (𝐸‘𝑤) ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
248		eqid 2761	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) = (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤))
249	247, 248	fmptd 7112	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))⟶(𝐸 “ ((0...𝐵) × (0...𝐵))))
250	119, 109	feqresmpt 6952	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ↾ ((0...𝐵) × (0...𝐵))) = (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)))
251	250	feq1d 6689	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐸 ↾ ((0...𝐵) × (0...𝐵))):((0...𝐵) × (0...𝐵))⟶(𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))⟶(𝐸 “ ((0...𝐵) × (0...𝐵)))))
252	249, 251	mpbird 260	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ↾ ((0...𝐵) × (0...𝐵))):((0...𝐵) × (0...𝐵))⟶(𝐸 “ ((0...𝐵) × (0...𝐵))))
253		f1resf1 6786	. . . . . . . . . . . 12 ⊢ ((𝐸:(ℕ0 × ℕ0)–1-1→ℕ ∧ ((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0) ∧ (𝐸 ↾ ((0...𝐵) × (0...𝐵))):((0...𝐵) × (0...𝐵))⟶(𝐸 “ ((0...𝐵) × (0...𝐵)))) → (𝐸 ↾ ((0...𝐵) × (0...𝐵))):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵))))
254	117, 109, 252, 253	syl3anc 1398	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ↾ ((0...𝐵) × (0...𝐵))):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵))))
255		eqidd 2762	. . . . . . . . . . . 12 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) = ((0...𝐵) × (0...𝐵)))
256		eqidd 2762	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) = (𝐸 “ ((0...𝐵) × (0...𝐵))))
257	250, 255, 256	f1eq123d 6814	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝐸 ↾ ((0...𝐵) × (0...𝐵))):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵)))))
258	254, 257	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵))))
259		df-ima 5664	. . . . . . . . . . . 12 ⊢ (𝐸 “ ((0...𝐵) × (0...𝐵))) = ran (𝐸 ↾ ((0...𝐵) × (0...𝐵)))
260	259	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) = ran (𝐸 ↾ ((0...𝐵) × (0...𝐵))))
261	250	rneqd 5920	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐸 ↾ ((0...𝐵) × (0...𝐵))) = ran (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)))
262	260, 261	eqtr2d 2797	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) = (𝐸 “ ((0...𝐵) × (0...𝐵))))
263	258, 262	jca 521	. . . . . . . . 9 ⊢ (𝜑 → ((𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵))) ∧ ran (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) = (𝐸 “ ((0...𝐵) × (0...𝐵)))))
264		dff1o5 6832	. . . . . . . . 9 ⊢ ((𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ ((𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵))) ∧ ran (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) = (𝐸 “ ((0...𝐵) × (0...𝐵)))))
265	263, 264	sylibr 237	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵))))
266		f1oeq1 6810	. . . . . . . 8 ⊢ (𝑢 = (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) → (𝑢:((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵))) ↔ (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵)))))
267	243, 265, 266	spcedv 3553	. . . . . . 7 ⊢ (𝜑 → ∃𝑢 𝑢:((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵))))
268		hasheqf1oi 14488	. . . . . . . 8 ⊢ (((0...𝐵) × (0...𝐵)) ∈ V → (∃𝑢 𝑢:((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵))) → (♯‘((0...𝐵) × (0...𝐵))) = (♯‘(𝐸 “ ((0...𝐵) × (0...𝐵))))))
269	242, 268	syl 18	. . . . . . 7 ⊢ (𝜑 → (∃𝑢 𝑢:((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵))) → (♯‘((0...𝐵) × (0...𝐵))) = (♯‘(𝐸 “ ((0...𝐵) × (0...𝐵))))))
270	267, 269	mpd 16	. . . . . 6 ⊢ (𝜑 → (♯‘((0...𝐵) × (0...𝐵))) = (♯‘(𝐸 “ ((0...𝐵) × (0...𝐵)))))
271	238, 270	breqtrd 5131	. . . . 5 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < (♯‘(𝐸 “ ((0...𝐵) × (0...𝐵)))))
272	125	fveq2d 6887	. . . . 5 ⊢ (𝜑 → (♯‘𝐶) = (♯‘(𝐸 “ ((0...𝐵) × (0...𝐵)))))
273	271, 272	breqtrrd 5133	. . . 4 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < (♯‘𝐶))
274		zringbas 21752	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring)
275		eqid 2761	. . . . . . . . . 10 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
276	274, 275	rhmf 20708	. . . . . . . . 9 ⊢ (𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)) → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
277	198, 276	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
278	277	ffnd 6708	. . . . . . 7 ⊢ (𝜑 → 𝐿 Fn ℤ)
279	278	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐶) → 𝐿 Fn ℤ)
280		resss 5992	. . . . . . . . . . . 12 ⊢ (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ 𝐸
281	280	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ 𝐸)
282		rnss 5921	. . . . . . . . . . 11 ⊢ ((𝐸 ↾ (ℕ0 × ℕ0)) ⊆ 𝐸 → ran (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ ran 𝐸)
283	281, 282	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ ran 𝐸)
284		df-ima 5664	. . . . . . . . . . . 12 ⊢ (𝐸 “ (ℕ0 × ℕ0)) = ran (𝐸 ↾ (ℕ0 × ℕ0))
285	284	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) = ran (𝐸 ↾ (ℕ0 × ℕ0)))
286	285	sseq1d 3962	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 “ (ℕ0 × ℕ0)) ⊆ ran 𝐸 ↔ ran (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ ran 𝐸))
287	283, 286	mpbird 260	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ran 𝐸)
288		frn 6715	. . . . . . . . . 10 ⊢ (𝐸:(ℕ0 × ℕ0)⟶ℕ → ran 𝐸 ⊆ ℕ)
289	119, 288	syl 18	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐸 ⊆ ℕ)
290	287, 289	sstrd 3941	. . . . . . . 8 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℕ)
291		nnssz 12708	. . . . . . . . 9 ⊢ ℕ ⊆ ℤ
292	291	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ⊆ ℤ)
293	290, 292	sstrd 3941	. . . . . . 7 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ)
294	293	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐶) → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ)
295	125	sseq1d 3962	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ⊆ (𝐸 “ (ℕ0 × ℕ0)) ↔ (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ (𝐸 “ (ℕ0 × ℕ0))))
296	111, 295	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ⊆ (𝐸 “ (ℕ0 × ℕ0)))
297	296	sseld 3930	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ 𝐶 → 𝑡 ∈ (𝐸 “ (ℕ0 × ℕ0))))
298	297	imp 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐶) → 𝑡 ∈ (𝐸 “ (ℕ0 × ℕ0)))
299		fnfvima 7237	. . . . . 6 ⊢ ((𝐿 Fn ℤ ∧ (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ ∧ 𝑡 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝐿‘𝑡) ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
300	279, 294, 298, 299	syl3anc 1398	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐶) → (𝐿‘𝑡) ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
301		eqid 2761	. . . . 5 ⊢ (𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡)) = (𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))
302	300, 301	fmptd 7112	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡)):𝐶⟶(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
303		nnssre 12332	. . . . . 6 ⊢ ℕ ⊆ ℝ
304	303	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ⊆ ℝ)
305	127, 304	sstrd 3941	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ ℝ)
306	192, 202, 273, 302, 305	hashnexinjle 43159	. . 3 ⊢ (𝜑 → ∃𝑏 ∈ 𝐶 ∃𝑐 ∈ 𝐶 (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐))
307	185, 306	reximddv2 3222	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝐶 ∃𝑐 ∈ 𝐶 (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅))))
308	50, 307	r19.29vva 3223	1 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899  ∅c0 4279   class class class wbr 5103  {copab 5167   ↦ cmpt 5186   × cxp 5649  ran crn 5652   ↾ cres 5653   “ cima 5654  Fun wfun 6531   Fn wfn 6532  ⟶wf 6533  –1-1→wf1 6534  –1-1-onto→wf1o 6536  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420   ↑_m cmap 8840  Fincfn 8966  ℂcc 11191  ℝcr 11192  0cc0 11193  1c1 11194   + caddc 11196   · cmul 11198   < clt 11336   ≤ cle 11337   − cmin 11534   / cdiv 11966  ℕcn 12328  ℕ₀cn0 12599  ℤcz 12686  ℤ_≥cuz 12958  ℝ⁺crp 13113  ...cfz 13632  ⌊cfl 13923  ↑cexp 14197  ♯chash 14467  √csqrt 15393   ∥ cdvds 16415   gcd cgcd 16657  ℙcprime 16839  Basecbs 17380  +_gcplusg 17421   Σ_g cgsu 17604  -_gcsg 19139  ._gcmg 19270  mulGrpcmgp 20353  Ringcrg 20452  CRingccrg 20453   RingHom crh 20692   RingIso crs 20693  Fieldcfield 20974  ℤ_ringczring 21745  ℤRHomczrh 21798  chrcchr 21800  ℤ/nℤczn 21801  algSccascl 22153  var₁cv1 22487  Poly₁cpl1 22488  eval₁ce1 22625   PrimRoots cprimroots 43121

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271  ax-addf 11272  ax-mulf 11273

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-of 7691  df-ofr 7692  df-om 7876  df-1st 7999  df-2nd 8000  df-supp 8171  df-tpos 8236  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8710  df-ec 8712  df-qs 8716  df-map 8842  df-pm 8843  df-ixp 8919  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-fsupp 9347  df-sup 9427  df-inf 9428  df-oi 9497  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-4 12400  df-5 12401  df-6 12402  df-7 12403  df-8 12404  df-9 12405  df-n0 12600  df-xnn0 12673  df-z 12687  df-dec 12808  df-uz 12959  df-q 13069  df-rp 13114  df-fz 13633  df-fzo 13782  df-fl 13925  df-mod 14003  df-seq 14138  df-exp 14198  df-fac 14411  df-bc 14440  df-hash 14468  df-cj 15259  df-re 15260  df-im 15261  df-sqrt 15395  df-abs 15396  df-dvds 16416  df-gcd 16658  df-prm 16840  df-phi 16936  df-pc 17008  df-struct 17318  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-mulr 17435  df-starv 17436  df-sca 17437  df-vsca 17438  df-ip 17439  df-tset 17440  df-ple 17441  df-ds 17443  df-unif 17444  df-hom 17445  df-cco 17446  df-0g 17605  df-gsum 17606  df-prds 17611  df-pws 17613  df-imas 17673  df-qus 17674  df-mre 17749  df-mrc 17750  df-acs 17752  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-mhm 18971  df-submnd 18972  df-grp 19140  df-minusg 19141  df-sbg 19142  df-mulg 19271  df-subg 19326  df-nsg 19327  df-eqg 19328  df-ghm 19421  df-cntz 19524  df-od 19735  df-cmn 19989  df-abl 19990  df-mgp 20354  df-rng 20368  df-ur 20401  df-srg 20406  df-ring 20454  df-cring 20455  df-oppr 20560  df-dvdsr 20580  df-unit 20581  df-invr 20611  df-dvr 20624  df-rhm 20695  df-rim 20696  df-nzr 20756  df-subrng 20791  df-subrg 20815  df-rlreg 20939  df-domn 20940  df-idom 20941  df-drng 20975  df-field 20976  df-lmod 21130  df-lss 21200  df-lsp 21240  df-sra 21441  df-rgmod 21442  df-lidl 21479  df-rsp 21480  df-2idl 21536  df-cnfld 21672  df-zring 21746  df-zrh 21802  df-chr 21804  df-zn 21805  df-assa 22154  df-asp 22155  df-ascl 22156  df-psr 22210  df-mvr 22211  df-mpl 22212  df-opsr 22214  df-evls 22376  df-evl 22377  df-psr1 22491  df-vr1 22492  df-ply1 22493  df-coe1 22494  df-evl1 22627  df-mdeg 26366  df-deg1 26367  df-mon1 26442  df-uc1p 26443  df-q1p 26444  df-r1p 26445  df-primroots 43122

This theorem is used by:  aks6d1c7lem2  43211