Theorem aks6d1c2 42131

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 482	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → ((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶))
2		simprl 771	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → 𝑏 < 𝑐)
3	1, 2	jca 511	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐))
4		simprr 773	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))
5	3, 4	jca 511	. . 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 734	. . . . . . 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 734	. . . . . . 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 734	. . . . . . 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 734	. . . . . . 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 734	. . . . . . 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 734	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → (𝑁 gcd 𝑅) = 1)
20		0nn0 12541	. . . . . . . . 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 2737	. . . . . . . 8 ⊢ (𝑗 ∈ (0...𝐴) ↦ 0) = (𝑗 ∈ (0...𝐴) ↦ 0)
23	21, 22	fmptd 7134	. . . . . . 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 734	. . . . . . 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 734	. . . . . . 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 734	. . . . . . 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 734	. . . . . . 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 786	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑏 ∈ 𝐶)
39		simp-4r 784	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑐 ∈ 𝐶)
40		simpllr 776	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑏 < 𝑐)
41		eqid 2737	. . . . . . 7 ⊢ (.g‘(mulGrp‘(Poly1‘𝐾))) = (.g‘(mulGrp‘(Poly1‘𝐾)))
42		eqid 2737	. . . . . . 7 ⊢ (var1‘𝐾) = (var1‘𝐾)
43		eqid 2737	. . . . . . 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 769	. . . . . . 7 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → 𝑑 ∈ ℕ)
45		simpr 484	. . . . . . 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 42128	. . . . . 6 ⊢ ((((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) ∧ 𝑐 = (𝑏 + (𝑑 · 𝑅))) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))
47	46	ex 412	. . . . 5 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) ∧ 𝑑 ∈ ℕ) → (𝑐 = (𝑏 + (𝑑 · 𝑅)) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵)))
48	47	rexlimdva 3155	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ 𝑏 < 𝑐) → (∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵)))
49	48	imp 406	. . 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 17	. 2 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅)))) → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))
51		simprr 773	. . . 4 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑏 < 𝑐)
52		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑠(𝐿‘𝑡)
53		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡(𝐿‘𝑠)
54		fveq2 6906	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑠 → (𝐿‘𝑡) = (𝐿‘𝑠))
55	52, 53, 54	cbvmpt 5253	. . . . . . . . . . . 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 484	. . . . . . . . . . . 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 6910	. . . . . . . . . . 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 776	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑏 ∈ 𝐶)
60		fvexd 6921	. . . . . . . . . . 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 7023	. . . . . . . . . 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 2743	. . . . . . . . 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 771	. . . . . . . . . 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 484	. . . . . . . . . . . 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 6910	. . . . . . . . . . 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 769	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑐 ∈ 𝐶)
67		fvexd 6921	. . . . . . . . . . 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 7023	. . . . . . . . . 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 2777	. . . . . . . . 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 2777	. . . . . . . 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 2743	. . . . . . 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 12587	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑅 ∈ ℕ0)
73	72	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑅 ∈ ℕ0)
74	73	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑅 ∈ ℕ0)
75	74	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑅 ∈ ℕ0)
76		fz0ssnn0 13662	. . . . . . . . . . . . . . . . . 18 ⊢ (0...𝐵) ⊆ ℕ0
77	76	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...𝐵) ⊆ ℕ0)
78	77, 77	jca 511	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((0...𝐵) ⊆ ℕ0 ∧ (0...𝐵) ⊆ ℕ0))
79		eqid 2737	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (ℤ/nℤ‘𝑅) = (ℤ/nℤ‘𝑅)
80	14, 10, 16, 12, 18, 27, 28, 79	hashscontpowcl 42121	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0)
81	80	nn0red 12588	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ)
82	80	nn0ge0d 12590	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
83	81, 82	resqrtcld 15456	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∈ ℝ)
84	83	flcld 13838	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ)
85	81, 82	sqrtge0d 15459	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
86		0zd 12625	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝜑 → 0 ∈ ℤ)
87		flge 13845	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∈ ℝ ∧ 0 ∈ ℤ) → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
88	83, 86, 87	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
89	85, 88	mpbid 232	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
90	84, 89	jca 511	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
91		elnn0z 12626	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℕ0 ↔ ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
92	90, 91	sylibr 234	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℕ0)
93	36	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → 𝐵 = (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
94	93	eleq1d 2826	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → (𝐵 ∈ ℕ0 ↔ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℕ0))
95	92, 94	mpbird 257	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐵 ∈ ℕ0)
96	95	nn0ge0d 12590	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 0 ≤ 𝐵)
97	95	nn0zd 12639	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → 𝐵 ∈ ℤ)
98		eluz 12892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((0 ∈ ℤ ∧ 𝐵 ∈ ℤ) → (𝐵 ∈ (ℤ≥‘0) ↔ 0 ≤ 𝐵))
99	86, 97, 98	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (𝐵 ∈ (ℤ≥‘0) ↔ 0 ≤ 𝐵))
100	96, 99	mpbird 257	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘0))
101		fzn0 13578	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((0...𝐵) ≠ ∅ ↔ 𝐵 ∈ (ℤ≥‘0))
102	100, 101	sylibr 234	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → (0...𝐵) ≠ ∅)
103	102, 102	jca 511	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅))
104		xpnz 6179	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅) ↔ ((0...𝐵) × (0...𝐵)) ≠ ∅)
105	104	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (((0...𝐵) ≠ ∅ ∧ (0...𝐵) ≠ ∅) → ((0...𝐵) × (0...𝐵)) ≠ ∅)
106	103, 105	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ≠ ∅)
107		ssxpb 6194	. . . . . . . . . . . . . . . . 17 ⊢ (((0...𝐵) × (0...𝐵)) ≠ ∅ → (((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0) ↔ ((0...𝐵) ⊆ ℕ0 ∧ (0...𝐵) ⊆ ℕ0)))
108	106, 107	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0) ↔ ((0...𝐵) ⊆ ℕ0 ∧ (0...𝐵) ⊆ ℕ0)))
109	78, 108	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0))
110		imass2 6120	. . . . . . . . . . . . . . 15 ⊢ (((0...𝐵) × (0...𝐵)) ⊆ (ℕ0 × ℕ0) → (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ (𝐸 “ (ℕ0 × ℕ0)))
111	109, 110	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ (𝐸 “ (ℕ0 × ℕ0)))
112		nfv 1914	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑜𝜑
113		aks6d1c2a.20	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (𝑄 ∈ ℙ ∧ 𝑄 ∥ 𝑁 ∧ 𝑃 ≠ 𝑄))
114	113	simp1d 1143	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑄 ∈ ℙ)
115	113	simp2d 1144	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑄 ∥ 𝑁)
116	113	simp3d 1145	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
117	14, 10, 16, 27, 114, 115, 116	aks6d1c2p2 42120	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)–1-1→ℕ)
118		f1f 6804	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐸:(ℕ0 × ℕ0)–1-1→ℕ → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
119	117, 118	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐸:(ℕ0 × ℕ0)⟶ℕ)
120	119	ffnd 6737	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐸 Fn (ℕ0 × ℕ0))
121	120	fnfund 6669	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → Fun 𝐸)
122	119	ffvelcdmda 7104	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑜 ∈ (ℕ0 × ℕ0)) → (𝐸‘𝑜) ∈ ℕ)
123	112, 121, 122	funimassd 6975	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℕ)
124	111, 123	sstrd 3994	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ ℕ)
125	37	a1i 11	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐶 = (𝐸 “ ((0...𝐵) × (0...𝐵))))
126	125	sseq1d 4015	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ⊆ ℕ ↔ (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ ℕ))
127	124, 126	mpbird 257	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐶 ⊆ ℕ)
128	127	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝐶 ⊆ ℕ)
129		simpr 484	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ 𝐶)
130	128, 129	sseldd 3984	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ ℕ)
131	130	nnzd 12640	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑐 ∈ ℤ)
132	131	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑐 ∈ ℤ)
133		simplr 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑏 ∈ 𝐶)
134	128, 133	sseldd 3984	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑏 ∈ ℕ)
135	134	nnzd 12640	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑏 ∈ ℤ)
136	135	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑏 ∈ ℤ)
137	79, 28	zndvds 21568	. . . . . . . 8 ⊢ ((𝑅 ∈ ℕ0 ∧ 𝑐 ∈ ℤ ∧ 𝑏 ∈ ℤ) → ((𝐿‘𝑐) = (𝐿‘𝑏) ↔ 𝑅 ∥ (𝑐 − 𝑏)))
138	75, 132, 136, 137	syl3anc 1373	. . . . . . 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 232	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑅 ∥ (𝑐 − 𝑏))
140	75	nn0zd 12639	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑅 ∈ ℤ)
141	132, 136	zsubcld 12727	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → (𝑐 − 𝑏) ∈ ℤ)
142		divides 16292	. . . . . . . 8 ⊢ ((𝑅 ∈ ℤ ∧ (𝑐 − 𝑏) ∈ ℤ) → (𝑅 ∥ (𝑐 − 𝑏) ↔ ∃𝑑 ∈ ℤ (𝑑 · 𝑅) = (𝑐 − 𝑏)))
143	140, 141, 142	syl2anc 584	. . . . . . 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 229	. . . . . 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 15	. . . . 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 771	. . . . . . 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 726	. . . . . . . . . . 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 12281	. . . . . . . . . 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 726	. . . . . . . . . . 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 12281	. . . . . . . . . 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 11691	. . . . . . . . 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 13075	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ∈ ℝ+)
153	152	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐶) → 𝑅 ∈ ℝ+)
154	153	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) → 𝑅 ∈ ℝ+)
155	154	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐶) ∧ 𝑐 ∈ 𝐶) ∧ (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐)) → 𝑅 ∈ ℝ+)
156	155	adantr 480	. . . . . . . . . 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 13077	. . . . . . . . 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 480	. . . . . . . . . 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 11850	. . . . . . . . . 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 232	. . . . . . . . 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 13080	. . . . . . . . 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 12203	. . . . . . . 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 11289	. . . . . . . . . . 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 12722	. . . . . . . . . . . 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 11289	. . . . . . . . . . 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 11282	. . . . . . . . . 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 773	. . . . . . . . . 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 2777	. . . . . . . . 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 11289	. . . . . . . . . 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 11827	. . . . . . . . . 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 12065	. . . . . . . . 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 257	. . . . . . . 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 5169	. . . . . . 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 511	. . . . . 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 12623	. . . . . 6 ⊢ (𝑑 ∈ ℕ ↔ (𝑑 ∈ ℤ ∧ 0 < 𝑑))
176	174, 175	sylibr 234	. . . . 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 2743	. . . . . . 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 11289	. . . . . . . 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 11289	. . . . . . . 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 2841	. . . . . . . 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 11638	. . . . . . 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 232	. . . . . 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 2743	. . . . 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 3173	. . . 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 511	. . 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 14014	. . . . . . 7 ⊢ (𝜑 → (0...𝐵) ∈ Fin)
187		xpfi 9358	. . . . . . 7 ⊢ (((0...𝐵) ∈ Fin ∧ (0...𝐵) ∈ Fin) → ((0...𝐵) × (0...𝐵)) ∈ Fin)
188	186, 186, 187	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ∈ Fin)
189		imafi 9353	. . . . . 6 ⊢ ((Fun 𝐸 ∧ ((0...𝐵) × (0...𝐵)) ∈ Fin) → (𝐸 “ ((0...𝐵) × (0...𝐵))) ∈ Fin)
190	121, 188, 189	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) ∈ Fin)
191	125	eleq1d 2826	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ Fin ↔ (𝐸 “ ((0...𝐵) × (0...𝐵))) ∈ Fin))
192	190, 191	mpbird 257	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Fin)
193	79	zncrng 21563	. . . . . . . . . 10 ⊢ (𝑅 ∈ ℕ0 → (ℤ/nℤ‘𝑅) ∈ CRing)
194	72, 193	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ CRing)
195		crngring 20242	. . . . . . . . 9 ⊢ ((ℤ/nℤ‘𝑅) ∈ CRing → (ℤ/nℤ‘𝑅) ∈ Ring)
196	194, 195	syl 17	. . . . . . . 8 ⊢ (𝜑 → (ℤ/nℤ‘𝑅) ∈ Ring)
197	28	zrhrhm 21522	. . . . . . . 8 ⊢ ((ℤ/nℤ‘𝑅) ∈ Ring → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
198	196, 197	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)))
199	198	imaexd 7938	. . . . . 6 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V)
200		hashclb 14397	. . . . . 6 ⊢ ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ V → ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin ↔ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0))
201	199, 200	syl 17	. . . . 5 ⊢ (𝜑 → ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin ↔ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0))
202	80, 201	mpbird 257	. . . 4 ⊢ (𝜑 → (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin)
203		hashcl 14395	. . . . . . . . . . . . 13 ⊢ ((𝐿 “ (𝐸 “ (ℕ0 × ℕ0))) ∈ Fin → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0)
204	202, 203	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℕ0)
205	204	nn0red 12588	. . . . . . . . . . 11 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ)
206	204	nn0ge0d 12590	. . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
207		sqrtmsq 15309	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) = (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))
209	208	eqcomd 2743	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) = (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
210	205, 206	jca 511	. . . . . . . . . . 11 ⊢ (𝜑 → ((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) ∈ ℝ ∧ 0 ≤ (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
211		sqrtmul 15298	. . . . . . . . . . 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 584	. . . . . . . . . 10 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) = ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
213	205, 206	resqrtcld 15456	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∈ ℝ)
214	213	flcld 13838	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ)
215	205, 206	sqrtge0d 15459	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))
216	213, 86, 87	syl2anc 584	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0 ≤ (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ↔ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
217	215, 216	mpbid 232	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))))
218	214, 217	jca 511	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℤ ∧ 0 ≤ (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))))))
219	218, 91	sylibr 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) ∈ ℕ0)
220	219, 94	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℕ0)
221	220	nn0red 12588	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ)
222		1red 11262	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
223	221, 222	readdcld 11290	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐵 + 1) ∈ ℝ)
224		flltp1 13840	. . . . . . . . . . . . 13 ⊢ ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) ∈ ℝ → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) < ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) + 1))
225	213, 224	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) < ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) + 1))
226	93	oveq1d 7446	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐵 + 1) = ((⌊‘(√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) + 1))
227	225, 226	breqtrrd 5171	. . . . . . . . . . 11 ⊢ (𝜑 → (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) < (𝐵 + 1))
228	213, 223, 213, 223, 215, 227, 215, 227	ltmul12ad 12209	. . . . . . . . . 10 ⊢ (𝜑 → ((√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))) · (√‘(♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) < ((𝐵 + 1) · (𝐵 + 1)))
229	212, 228	eqbrtrd 5165	. . . . . . . . 9 ⊢ (𝜑 → (√‘((♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) · (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))))) < ((𝐵 + 1) · (𝐵 + 1)))
230	209, 229	eqbrtrd 5165	. . . . . . . 8 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < ((𝐵 + 1) · (𝐵 + 1)))
231		hashfz0 14471	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℕ0 → (♯‘(0...𝐵)) = (𝐵 + 1))
232	220, 231	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (♯‘(0...𝐵)) = (𝐵 + 1))
233	232, 232	oveq12d 7449	. . . . . . . 8 ⊢ (𝜑 → ((♯‘(0...𝐵)) · (♯‘(0...𝐵))) = ((𝐵 + 1) · (𝐵 + 1)))
234	230, 233	breqtrrd 5171	. . . . . . 7 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < ((♯‘(0...𝐵)) · (♯‘(0...𝐵))))
235	186, 186	jca 511	. . . . . . . 8 ⊢ (𝜑 → ((0...𝐵) ∈ Fin ∧ (0...𝐵) ∈ Fin))
236		hashxp 14473	. . . . . . . 8 ⊢ (((0...𝐵) ∈ Fin ∧ (0...𝐵) ∈ Fin) → (♯‘((0...𝐵) × (0...𝐵))) = ((♯‘(0...𝐵)) · (♯‘(0...𝐵))))
237	235, 236	syl 17	. . . . . . 7 ⊢ (𝜑 → (♯‘((0...𝐵) × (0...𝐵))) = ((♯‘(0...𝐵)) · (♯‘(0...𝐵))))
238	234, 237	breqtrrd 5171	. . . . . 6 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < (♯‘((0...𝐵) × (0...𝐵))))
239		ovexd 7466	. . . . . . . . . . 11 ⊢ (𝜑 → (0...𝐵) ∈ V)
240	239, 239	jca 511	. . . . . . . . . 10 ⊢ (𝜑 → ((0...𝐵) ∈ V ∧ (0...𝐵) ∈ V))
241		xpexg 7770	. . . . . . . . . 10 ⊢ (((0...𝐵) ∈ V ∧ (0...𝐵) ∈ V) → ((0...𝐵) × (0...𝐵)) ∈ V)
242	240, 241	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) ∈ V)
243	242	mptexd 7244	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) ∈ V)
244	120	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ((0...𝐵) × (0...𝐵))) → 𝐸 Fn (ℕ0 × ℕ0))
245	109	sselda 3983	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ((0...𝐵) × (0...𝐵))) → 𝑤 ∈ (ℕ0 × ℕ0))
246		simpr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑤 ∈ ((0...𝐵) × (0...𝐵))) → 𝑤 ∈ ((0...𝐵) × (0...𝐵)))
247	244, 245, 246	fnfvimad 7254	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑤 ∈ ((0...𝐵) × (0...𝐵))) → (𝐸‘𝑤) ∈ (𝐸 “ ((0...𝐵) × (0...𝐵))))
248		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) = (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤))
249	247, 248	fmptd 7134	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))⟶(𝐸 “ ((0...𝐵) × (0...𝐵))))
250	119, 109	feqresmpt 6978	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝐸 ↾ ((0...𝐵) × (0...𝐵))) = (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)))
251	250	feq1d 6720	. . . . . . . . . . . . 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 257	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ↾ ((0...𝐵) × (0...𝐵))):((0...𝐵) × (0...𝐵))⟶(𝐸 “ ((0...𝐵) × (0...𝐵))))
253		f1resf1 6812	. . . . . . . . . . . 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 1373	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ↾ ((0...𝐵) × (0...𝐵))):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵))))
255		eqidd 2738	. . . . . . . . . . . 12 ⊢ (𝜑 → ((0...𝐵) × (0...𝐵)) = ((0...𝐵) × (0...𝐵)))
256		eqidd 2738	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 “ ((0...𝐵) × (0...𝐵))) = (𝐸 “ ((0...𝐵) × (0...𝐵))))
257	250, 255, 256	f1eq123d 6840	. . . . . . . . . . 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 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1→(𝐸 “ ((0...𝐵) × (0...𝐵))))
259		df-ima 5698	. . . . . . . . . . . 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 5949	. . . . . . . . . . 11 ⊢ (𝜑 → ran (𝐸 ↾ ((0...𝐵) × (0...𝐵))) = ran (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)))
262	260, 261	eqtr2d 2778	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)) = (𝐸 “ ((0...𝐵) × (0...𝐵))))
263	258, 262	jca 511	. . . . . . . . 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 6857	. . . . . . . . 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 234	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ∈ ((0...𝐵) × (0...𝐵)) ↦ (𝐸‘𝑤)):((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵))))
266		f1oeq1 6836	. . . . . . . 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 3598	. . . . . . 7 ⊢ (𝜑 → ∃𝑢 𝑢:((0...𝐵) × (0...𝐵))–1-1-onto→(𝐸 “ ((0...𝐵) × (0...𝐵))))
268		hasheqf1oi 14390	. . . . . . . 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 17	. . . . . . 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 15	. . . . . 6 ⊢ (𝜑 → (♯‘((0...𝐵) × (0...𝐵))) = (♯‘(𝐸 “ ((0...𝐵) × (0...𝐵)))))
271	238, 270	breqtrd 5169	. . . . 5 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < (♯‘(𝐸 “ ((0...𝐵) × (0...𝐵)))))
272	125	fveq2d 6910	. . . . 5 ⊢ (𝜑 → (♯‘𝐶) = (♯‘(𝐸 “ ((0...𝐵) × (0...𝐵)))))
273	271, 272	breqtrrd 5171	. . . 4 ⊢ (𝜑 → (♯‘(𝐿 “ (𝐸 “ (ℕ0 × ℕ0)))) < (♯‘𝐶))
274		zringbas 21464	. . . . . . . . . 10 ⊢ ℤ = (Base‘ℤring)
275		eqid 2737	. . . . . . . . . 10 ⊢ (Base‘(ℤ/nℤ‘𝑅)) = (Base‘(ℤ/nℤ‘𝑅))
276	274, 275	rhmf 20485	. . . . . . . . 9 ⊢ (𝐿 ∈ (ℤring RingHom (ℤ/nℤ‘𝑅)) → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
277	198, 276	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘(ℤ/nℤ‘𝑅)))
278	277	ffnd 6737	. . . . . . 7 ⊢ (𝜑 → 𝐿 Fn ℤ)
279	278	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐶) → 𝐿 Fn ℤ)
280		resss 6019	. . . . . . . . . . . 12 ⊢ (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ 𝐸
281	280	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ 𝐸)
282		rnss 5950	. . . . . . . . . . 11 ⊢ ((𝐸 ↾ (ℕ0 × ℕ0)) ⊆ 𝐸 → ran (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ ran 𝐸)
283	281, 282	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ran (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ ran 𝐸)
284		df-ima 5698	. . . . . . . . . . . 12 ⊢ (𝐸 “ (ℕ0 × ℕ0)) = ran (𝐸 ↾ (ℕ0 × ℕ0))
285	284	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) = ran (𝐸 ↾ (ℕ0 × ℕ0)))
286	285	sseq1d 4015	. . . . . . . . . 10 ⊢ (𝜑 → ((𝐸 “ (ℕ0 × ℕ0)) ⊆ ran 𝐸 ↔ ran (𝐸 ↾ (ℕ0 × ℕ0)) ⊆ ran 𝐸))
287	283, 286	mpbird 257	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ran 𝐸)
288		frn 6743	. . . . . . . . . 10 ⊢ (𝐸:(ℕ0 × ℕ0)⟶ℕ → ran 𝐸 ⊆ ℕ)
289	119, 288	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ran 𝐸 ⊆ ℕ)
290	287, 289	sstrd 3994	. . . . . . . 8 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℕ)
291		nnssz 12635	. . . . . . . . 9 ⊢ ℕ ⊆ ℤ
292	291	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℕ ⊆ ℤ)
293	290, 292	sstrd 3994	. . . . . . 7 ⊢ (𝜑 → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ)
294	293	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐶) → (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ)
295	125	sseq1d 4015	. . . . . . . . 9 ⊢ (𝜑 → (𝐶 ⊆ (𝐸 “ (ℕ0 × ℕ0)) ↔ (𝐸 “ ((0...𝐵) × (0...𝐵))) ⊆ (𝐸 “ (ℕ0 × ℕ0))))
296	111, 295	mpbird 257	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ⊆ (𝐸 “ (ℕ0 × ℕ0)))
297	296	sseld 3982	. . . . . . 7 ⊢ (𝜑 → (𝑡 ∈ 𝐶 → 𝑡 ∈ (𝐸 “ (ℕ0 × ℕ0))))
298	297	imp 406	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐶) → 𝑡 ∈ (𝐸 “ (ℕ0 × ℕ0)))
299		fnfvima 7253	. . . . . 6 ⊢ ((𝐿 Fn ℤ ∧ (𝐸 “ (ℕ0 × ℕ0)) ⊆ ℤ ∧ 𝑡 ∈ (𝐸 “ (ℕ0 × ℕ0))) → (𝐿‘𝑡) ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
300	279, 294, 298, 299	syl3anc 1373	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝐶) → (𝐿‘𝑡) ∈ (𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
301		eqid 2737	. . . . 5 ⊢ (𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡)) = (𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))
302	300, 301	fmptd 7134	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡)):𝐶⟶(𝐿 “ (𝐸 “ (ℕ0 × ℕ0))))
303		nnssre 12270	. . . . . 6 ⊢ ℕ ⊆ ℝ
304	303	a1i 11	. . . . 5 ⊢ (𝜑 → ℕ ⊆ ℝ)
305	127, 304	sstrd 3994	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ ℝ)
306	192, 202, 273, 302, 305	hashnexinjle 42130	. . 3 ⊢ (𝜑 → ∃𝑏 ∈ 𝐶 ∃𝑐 ∈ 𝐶 (((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑏) = ((𝑡 ∈ 𝐶 ↦ (𝐿‘𝑡))‘𝑐) ∧ 𝑏 < 𝑐))
307	185, 306	reximddv2 3215	. 2 ⊢ (𝜑 → ∃𝑏 ∈ 𝐶 ∃𝑐 ∈ 𝐶 (𝑏 < 𝑐 ∧ ∃𝑑 ∈ ℕ 𝑐 = (𝑏 + (𝑑 · 𝑅))))
308	50, 307	r19.29vva 3216	1 ⊢ (𝜑 → (♯‘(𝐻 “ (ℕ0 ↑m (0...𝐴)))) ≤ (𝑁↑𝐵))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143  {copab 5205   ↦ cmpt 5225   × cxp 5683  ran crn 5686   ↾ cres 5687   “ cima 5688  Fun wfun 6555   Fn wfn 6556  ⟶wf 6557  –1-1→wf1 6558  –1-1-onto→wf1o 6560  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433   ↑_m cmap 8866  Fincfn 8985  ℂcc 11153  ℝcr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295   ≤ cle 11296   − cmin 11492   / cdiv 11920  ℕcn 12266  ℕ₀cn0 12526  ℤcz 12613  ℤ_≥cuz 12878  ℝ⁺crp 13034  ...cfz 13547  ⌊cfl 13830  ↑cexp 14102  ♯chash 14369  √csqrt 15272   ∥ cdvds 16290   gcd cgcd 16531  ℙcprime 16708  Basecbs 17247  +_gcplusg 17297   Σ_g cgsu 17485  -_gcsg 18953  ._gcmg 19085  mulGrpcmgp 20137  Ringcrg 20230  CRingccrg 20231   RingHom crh 20469   RingIso crs 20470  Fieldcfield 20730  ℤ_ringczring 21457  ℤRHomczrh 21510  chrcchr 21512  ℤ/nℤczn 21513  algSccascl 21872  var₁cv1 22177  Poly₁cpl1 22178  eval₁ce1 22318   PrimRoots cprimroots 42092

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234  ax-mulf 11235

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-ofr 7698  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-tpos 8251  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-ec 8747  df-qs 8751  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-fz 13548  df-fzo 13695  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-fac 14313  df-bc 14342  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-gcd 16532  df-prm 16709  df-phi 16803  df-pc 16875  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-0g 17486  df-gsum 17487  df-prds 17492  df-pws 17494  df-imas 17553  df-qus 17554  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-mhm 18796  df-submnd 18797  df-grp 18954  df-minusg 18955  df-sbg 18956  df-mulg 19086  df-subg 19141  df-nsg 19142  df-eqg 19143  df-ghm 19231  df-cntz 19335  df-od 19546  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-srg 20184  df-ring 20232  df-cring 20233  df-oppr 20334  df-dvdsr 20357  df-unit 20358  df-invr 20388  df-dvr 20401  df-rhm 20472  df-rim 20473  df-nzr 20513  df-subrng 20546  df-subrg 20570  df-rlreg 20694  df-domn 20695  df-idom 20696  df-drng 20731  df-field 20732  df-lmod 20860  df-lss 20930  df-lsp 20970  df-sra 21172  df-rgmod 21173  df-lidl 21218  df-rsp 21219  df-2idl 21260  df-cnfld 21365  df-zring 21458  df-zrh 21514  df-chr 21516  df-zn 21517  df-assa 21873  df-asp 21874  df-ascl 21875  df-psr 21929  df-mvr 21930  df-mpl 21931  df-opsr 21933  df-evls 22098  df-evl 22099  df-psr1 22181  df-vr1 22182  df-ply1 22183  df-coe1 22184  df-evl1 22320  df-mdeg 26094  df-deg1 26095  df-mon1 26170  df-uc1p 26171  df-q1p 26172  df-r1p 26173  df-primroots 42093

This theorem is referenced by:  aks6d1c7lem2  42182