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Theorem lgsquadlem3 27345

Description: Lemma for lgsquad 27346. (Contributed by Mario Carneiro, 18-Jun-2015.)

**Hypotheses**
Ref	Expression
lgseisen.1	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
lgseisen.2	⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
lgseisen.3	⊢ (𝜑 → 𝑃 ≠ 𝑄)
lgsquad.4	⊢ 𝑀 = ((𝑃 − 1) / 2)
lgsquad.5	⊢ 𝑁 = ((𝑄 − 1) / 2)
lgsquad.6	⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}

**Assertion**
Ref	Expression
lgsquadlem3	⊢ (𝜑 → ((𝑃 /_L 𝑄) · (𝑄 /_L 𝑃)) = (-1↑(𝑀 · 𝑁)))

Distinct variable groups: 𝑥,𝑦,𝑃 𝜑,𝑥,𝑦 𝑦,𝑀 𝑥,𝑁,𝑦 𝑥,𝑄,𝑦 𝑥,𝑆 𝑥,𝑀 𝑦,𝑆

Proof of Theorem lgsquadlem3 Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lgseisen.2	. . . . 5 ⊢ (𝜑 → 𝑄 ∈ (ℙ ∖ {2}))
2		lgseisen.1	. . . . 5 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
3		lgseisen.3	. . . . . 6 ⊢ (𝜑 → 𝑃 ≠ 𝑄)
4	3	necomd 2987	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃)
5		lgsquad.5	. . . . 5 ⊢ 𝑁 = ((𝑄 − 1) / 2)
6		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
7		eleq1w 2819	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (1...𝑀) ↔ 𝑧 ∈ (1...𝑀)))
8		eleq1w 2819	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (1...𝑁) ↔ 𝑤 ∈ (1...𝑁)))
9	7, 8	bi2anan9 639	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (𝑧 ∈ (1...𝑀) ∧ 𝑤 ∈ (1...𝑁))))
10	9	biancomd 463	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (𝑤 ∈ (1...𝑁) ∧ 𝑧 ∈ (1...𝑀))))
11		oveq1 7374	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 · 𝑄) = (𝑧 · 𝑄))
12		oveq1 7374	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 · 𝑃) = (𝑤 · 𝑃))
13	11, 12	breqan12d 5101	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ↔ (𝑧 · 𝑄) < (𝑤 · 𝑃)))
14	10, 13	anbi12d 633	. . . . . . 7 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃)) ↔ ((𝑤 ∈ (1...𝑁) ∧ 𝑧 ∈ (1...𝑀)) ∧ (𝑧 · 𝑄) < (𝑤 · 𝑃))))
15	14	ancoms 458	. . . . . 6 ⊢ ((𝑦 = 𝑤 ∧ 𝑥 = 𝑧) → (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃)) ↔ ((𝑤 ∈ (1...𝑁) ∧ 𝑧 ∈ (1...𝑀)) ∧ (𝑧 · 𝑄) < (𝑤 · 𝑃))))
16	15	cbvopabv 5158	. . . . 5 ⊢ {⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} = {⟨𝑤, 𝑧⟩ ∣ ((𝑤 ∈ (1...𝑁) ∧ 𝑧 ∈ (1...𝑀)) ∧ (𝑧 · 𝑄) < (𝑤 · 𝑃))}
17	1, 2, 4, 5, 6, 16	lgsquadlem2 27344	. . . 4 ⊢ (𝜑 → (𝑃 /_L 𝑄) = (-1↑(♯‘{⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})))
18		relopabv 5777	. . . . . . . 8 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}
19		fzfid 13935	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
20		fzfid 13935	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
21		xpfi 9230	. . . . . . . . . 10 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑀) × (1...𝑁)) ∈ Fin)
22	19, 20, 21	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) × (1...𝑁)) ∈ Fin)
23		opabssxp 5723	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ⊆ ((1...𝑀) × (1...𝑁))
24		ssfi 9107	. . . . . . . . 9 ⊢ ((((1...𝑀) × (1...𝑁)) ∈ Fin ∧ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ⊆ ((1...𝑀) × (1...𝑁))) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin)
25	22, 23, 24	sylancl 587	. . . . . . . 8 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin)
26		cnven 8980	. . . . . . . 8 ⊢ ((Rel {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∧ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin) → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ≈ ^◡{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})
27	18, 25, 26	sylancr 588	. . . . . . 7 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ≈ ^◡{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})
28		cnvopab 6100	. . . . . . 7 ⊢ ^◡{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} = {⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}
29	27, 28	breqtrdi 5126	. . . . . 6 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ≈ {⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})
30		hasheni 14310	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ≈ {⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} → (♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) = (♯‘{⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}))
31	29, 30	syl 17	. . . . 5 ⊢ (𝜑 → (♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) = (♯‘{⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}))
32	31	oveq2d 7383	. . . 4 ⊢ (𝜑 → (-1↑(♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})) = (-1↑(♯‘{⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})))
33	17, 32	eqtr4d 2774	. . 3 ⊢ (𝜑 → (𝑃 /_L 𝑄) = (-1↑(♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})))
34		lgsquad.6	. . . 4 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}
35	2, 1, 3, 6, 5, 34	lgsquadlem2 27344	. . 3 ⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑(♯‘𝑆)))
36	33, 35	oveq12d 7385	. 2 ⊢ (𝜑 → ((𝑃 /_L 𝑄) · (𝑄 /_L 𝑃)) = ((-1↑(♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})) · (-1↑(♯‘𝑆))))
37		neg1cn 12144	. . . 4 ⊢ -1 ∈ ℂ
38	37	a1i 11	. . 3 ⊢ (𝜑 → -1 ∈ ℂ)
39		opabssxp 5723	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⊆ ((1...𝑀) × (1...𝑁))
40	34, 39	eqsstri 3968	. . . . 5 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
41		ssfi 9107	. . . . 5 ⊢ ((((1...𝑀) × (1...𝑁)) ∈ Fin ∧ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))) → 𝑆 ∈ Fin)
42	22, 40, 41	sylancl 587	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin)
43		hashcl 14318	. . . 4 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ₀)
44	42, 43	syl 17	. . 3 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
45		hashcl 14318	. . . 4 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin → (♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) ∈ ℕ₀)
46	25, 45	syl 17	. . 3 ⊢ (𝜑 → (♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) ∈ ℕ₀)
47	38, 44, 46	expaddd 14110	. 2 ⊢ (𝜑 → (-1↑((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆))) = ((-1↑(♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})) · (-1↑(♯‘𝑆))))
48	1	eldifad 3901	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ ℙ)
49	48	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℙ)
50		prmnn 16643	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
51	49, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
52	1, 5	gausslemma2dlem0b 27320	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
53	52	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℕ)
54	53	nnzd 12550	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℤ)
55		prmz 16644	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
56	49, 55	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℤ)
57		peano2zm 12570	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ∈ ℤ → (𝑄 − 1) ∈ ℤ)
58	56, 57	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℤ)
59	53	nnred 12189	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℝ)
60	58	zred 12633	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ)
61		prmuz2 16665	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ (ℤ_≥‘2))
62	49, 61	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ (ℤ_≥‘2))
63		uz2m1nn 12873	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑄 ∈ (ℤ_≥‘2) → (𝑄 − 1) ∈ ℕ)
64	62, 63	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℕ)
65	64	nnrpd 12984	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ⁺)
66		rphalflt 12973	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 − 1) ∈ ℝ⁺ → ((𝑄 − 1) / 2) < (𝑄 − 1))
67	65, 66	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 − 1) / 2) < (𝑄 − 1))
68	5, 67	eqbrtrid 5120	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 < (𝑄 − 1))
69	59, 60, 68	ltled 11294	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ≤ (𝑄 − 1))
70		eluz2 12794	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 − 1) ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑄 − 1) ∈ ℤ ∧ 𝑁 ≤ (𝑄 − 1)))
71	54, 58, 69, 70	syl3anbrc 1345	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ (ℤ_≥‘𝑁))
72		fzss2 13518	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 − 1) ∈ (ℤ_≥‘𝑁) → (1...𝑁) ⊆ (1...(𝑄 − 1)))
73	71, 72	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (1...𝑁) ⊆ (1...(𝑄 − 1)))
74		simprr 773	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...𝑁))
75	73, 74	sseldd 3922	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...(𝑄 − 1)))
76		fzm1ndvds 16291	. . . . . . . . . . . . . . 15 ⊢ ((𝑄 ∈ ℕ ∧ 𝑦 ∈ (1...(𝑄 − 1))) → ¬ 𝑄 ∥ 𝑦)
77	51, 75, 76	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ 𝑦)
78	4	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ≠ 𝑃)
79	2	eldifad 3901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℙ)
80	79	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℙ)
81		prmrp 16682	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ) → ((𝑄 gcd 𝑃) = 1 ↔ 𝑄 ≠ 𝑃))
82	49, 80, 81	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 gcd 𝑃) = 1 ↔ 𝑄 ≠ 𝑃))
83	78, 82	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 gcd 𝑃) = 1)
84		prmz 16644	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
85	80, 84	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℤ)
86		elfzelz 13478	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℤ)
87	86	ad2antll 730	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℤ)
88		coprmdvds 16622	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
89	56, 85, 87, 88	syl3anc 1374	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
90	83, 89	mpan2d 695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 ∥ (𝑃 · 𝑦) → 𝑄 ∥ 𝑦))
91	77, 90	mtod 198	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ (𝑃 · 𝑦))
92		prmnn 16643	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
93	80, 92	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
94	93	nncnd 12190	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℂ)
95		elfznn 13507	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
96	95	ad2antll 730	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
97	96	nncnd 12190	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℂ)
98	94, 97	mulcomd 11166	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑃 · 𝑦) = (𝑦 · 𝑃))
99	98	breq2d 5097	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 ∥ (𝑃 · 𝑦) ↔ 𝑄 ∥ (𝑦 · 𝑃)))
100	91, 99	mtbid 324	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ (𝑦 · 𝑃))
101		elfzelz 13478	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℤ)
102	101	ad2antrl 729	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℤ)
103		dvdsmul2 16247	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑥 · 𝑄))
104	102, 56, 103	syl2anc 585	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∥ (𝑥 · 𝑄))
105		breq2 5089	. . . . . . . . . . . . . 14 ⊢ ((𝑥 · 𝑄) = (𝑦 · 𝑃) → (𝑄 ∥ (𝑥 · 𝑄) ↔ 𝑄 ∥ (𝑦 · 𝑃)))
106	104, 105	syl5ibcom 245	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) = (𝑦 · 𝑃) → 𝑄 ∥ (𝑦 · 𝑃)))
107	106	necon3bd 2946	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (¬ 𝑄 ∥ (𝑦 · 𝑃) → (𝑥 · 𝑄) ≠ (𝑦 · 𝑃)))
108	100, 107	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ≠ (𝑦 · 𝑃))
109		elfznn 13507	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
110	109	ad2antrl 729	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
111	110, 51	nnmulcld 12230	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
112	111	nnred 12189	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℝ)
113	96, 93	nnmulcld 12230	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
114	113	nnred 12189	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℝ)
115	112, 114	lttri2d 11285	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) ≠ (𝑦 · 𝑃) ↔ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
116	108, 115	mpbid 232	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
117	116	ex 412	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
118	117	pm4.71rd 562	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)) ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)))))
119		ancom 460	. . . . . . . 8 ⊢ ((((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)) ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) ↔ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
120	118, 119	bitr2di 288	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))) ↔ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))))
121	120	opabbidv 5151	. . . . . 6 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)))} = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))})
122		unopab 5165	. . . . . . 7 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}) = {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃)) ∨ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))}
123	34	uneq2i 4105	. . . . . . 7 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))})
124		andi 1010	. . . . . . . 8 ⊢ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))) ↔ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃)) ∨ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
125	124	opabbii 5152	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)))} = {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃)) ∨ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))}
126	122, 123, 125	3eqtr4i 2769	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)))}
127		df-xp 5637	. . . . . 6 ⊢ ((1...𝑀) × (1...𝑁)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))}
128	121, 126, 127	3eqtr4g 2796	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆) = ((1...𝑀) × (1...𝑁)))
129	128	fveq2d 6844	. . . 4 ⊢ (𝜑 → (♯‘({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆)) = (♯‘((1...𝑀) × (1...𝑁))))
130		inopab 5785	. . . . . . 7 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}) = {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃)) ∧ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))}
131	34	ineq2i 4157	. . . . . . 7 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ 𝑆) = ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))})
132		anandi 677	. . . . . . . 8 ⊢ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))) ↔ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃)) ∧ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
133	132	opabbii 5152	. . . . . . 7 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))} = {⟨𝑥, 𝑦⟩ ∣ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃)) ∧ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))}
134	130, 131, 133	3eqtr4i 2769	. . . . . 6 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ 𝑆) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))}
135		ltnsym2 11245	. . . . . . . . . . . 12 ⊢ (((𝑥 · 𝑄) ∈ ℝ ∧ (𝑦 · 𝑃) ∈ ℝ) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
136	112, 114, 135	syl2anc 585	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
137	136	ex 412	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
138		imnan 399	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))) ↔ ¬ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
139	137, 138	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
140	139	nexdv 1938	. . . . . . . 8 ⊢ (𝜑 → ¬ ∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
141	140	nexdv 1938	. . . . . . 7 ⊢ (𝜑 → ¬ ∃𝑥∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
142		opabn0 5508	. . . . . . . 8 ⊢ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))} ≠ ∅ ↔ ∃𝑥∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
143	142	necon1bbii 2981	. . . . . . 7 ⊢ (¬ ∃𝑥∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))) ↔ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))} = ∅)
144	141, 143	sylib 218	. . . . . 6 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))} = ∅)
145	134, 144	eqtrid 2783	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ 𝑆) = ∅)
146		hashun 14344	. . . . 5 ⊢ (({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin ∧ 𝑆 ∈ Fin ∧ ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ 𝑆) = ∅) → (♯‘({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆)) = ((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆)))
147	25, 42, 145, 146	syl3anc 1374	. . . 4 ⊢ (𝜑 → (♯‘({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆)) = ((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆)))
148		hashxp 14396	. . . . . 6 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
149	19, 20, 148	syl2anc 585	. . . . 5 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
150	2, 6	gausslemma2dlem0b 27320	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
151	150	nnnn0d 12498	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
152		hashfz1 14308	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
154	52	nnnn0d 12498	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
155		hashfz1 14308	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
156	154, 155	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
157	153, 156	oveq12d 7385	. . . . 5 ⊢ (𝜑 → ((♯‘(1...𝑀)) · (♯‘(1...𝑁))) = (𝑀 · 𝑁))
158	149, 157	eqtrd 2771	. . . 4 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = (𝑀 · 𝑁))
159	129, 147, 158	3eqtr3d 2779	. . 3 ⊢ (𝜑 → ((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆)) = (𝑀 · 𝑁))
160	159	oveq2d 7383	. 2 ⊢ (𝜑 → (-1↑((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆))) = (-1↑(𝑀 · 𝑁)))
161	36, 47, 160	3eqtr2d 2777	1 ⊢ (𝜑 → ((𝑃 /_L 𝑄) · (𝑄 /_L 𝑃)) = (-1↑(𝑀 · 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2932 ∖ cdif 3886 ∪ cun 3887 ∩ cin 3888 ⊆ wss 3889 ∅c0 4273 {csn 4567 class class class wbr 5085 {copab 5147 × cxp 5629 ^◡ccnv 5630 Rel wrel 5636 ‘cfv 6498 (class class class)co 7367 ≈ cen 8890 Fincfn 8893 ℂcc 11036 ℝcr 11037 1c1 11039 + caddc 11041 · cmul 11043 < clt 11179 ≤ cle 11180 − cmin 11377 -cneg 11378 / cdiv 11807 ℕcn 12174 2c2 12236 ℕ₀cn0 12437 ℤcz 12524 ℤ_≥cuz 12788 ℝ⁺crp 12942 ...cfz 13461 ↑cexp 14023 ♯chash 14292 ∥ cdvds 16221 gcd cgcd 16463 ℙcprime 16640 /_L clgs 27257

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-disj 5053 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-isom 6507 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-of 7631 df-om 7818 df-1st 7942 df-2nd 7943 df-supp 8111 df-tpos 8176 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-2o 8406 df-oadd 8409 df-er 8643 df-ec 8645 df-qs 8649 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-fsupp 9275 df-sup 9355 df-inf 9356 df-oi 9425 df-dju 9825 df-card 9863 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-div 11808 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-7 12249 df-8 12250 df-9 12251 df-n0 12438 df-xnn0 12511 df-z 12525 df-dec 12645 df-uz 12789 df-q 12899 df-rp 12943 df-fz 13462 df-fzo 13609 df-fl 13751 df-mod 13829 df-seq 13964 df-exp 14024 df-hash 14293 df-cj 15061 df-re 15062 df-im 15063 df-sqrt 15197 df-abs 15198 df-clim 15450 df-sum 15649 df-dvds 16222 df-gcd 16464 df-prm 16641 df-phi 16736 df-pc 16808 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-starv 17235 df-sca 17236 df-vsca 17237 df-ip 17238 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-0g 17404 df-gsum 17405 df-imas 17472 df-qus 17473 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-mhm 18751 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-mulg 19044 df-subg 19099 df-nsg 19100 df-eqg 19101 df-ghm 19188 df-cntz 19292 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-cring 20217 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-rhm 20452 df-nzr 20490 df-subrng 20523 df-subrg 20547 df-rlreg 20671 df-domn 20672 df-idom 20673 df-drng 20708 df-field 20709 df-lmod 20857 df-lss 20927 df-lsp 20967 df-sra 21168 df-rgmod 21169 df-lidl 21206 df-rsp 21207 df-2idl 21248 df-cnfld 21353 df-zring 21427 df-zrh 21483 df-zn 21486 df-lgs 27258

This theorem is referenced by: lgsquad 27346

W3C validator