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Theorem lgsquadlem3 27511

Description: Lemma for lgsquad 27512. (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 3019	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃)
5		lgsquad.5	. . . . 5 ⊢ 𝑁 = ((𝑄 − 1) / 2)
6		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
7		eleq1w 2852	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (1...𝑀) ↔ 𝑧 ∈ (1...𝑀)))
8		eleq1w 2852	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (1...𝑁) ↔ 𝑤 ∈ (1...𝑁)))
9	7, 8	bi2anan9 649	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (𝑧 ∈ (1...𝑀) ∧ 𝑤 ∈ (1...𝑁))))
10	9	biancomd 468	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (𝑤 ∈ (1...𝑁) ∧ 𝑧 ∈ (1...𝑀))))
11		oveq1 7418	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 · 𝑄) = (𝑧 · 𝑄))
12		oveq1 7418	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 · 𝑃) = (𝑤 · 𝑃))
13	11, 12	breqan12d 5129	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ↔ (𝑧 · 𝑄) < (𝑤 · 𝑃)))
14	10, 13	anbi12d 643	. . . . . . 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 463	. . . . . 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 5188	. . . . 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 27510	. . . 4 ⊢ (𝜑 → (𝑃 /_L 𝑄) = (-1↑(♯‘{⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})))
18		relopabv 5809	. . . . . . . 8 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}
19		fzfid 14008	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
20		fzfid 14008	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
21		xpfi 9278	. . . . . . . . . 10 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑀) × (1...𝑁)) ∈ Fin)
22	19, 20, 21	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) × (1...𝑁)) ∈ Fin)
23		opabssxp 5754	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ⊆ ((1...𝑀) × (1...𝑁))
24		ssfi 9156	. . . . . . . . 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 597	. . . . . . . 8 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin)
26		cnven 9029	. . . . . . . 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 598	. . . . . . 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 6138	. . . . . . 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 5156	. . . . . 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 14383	. . . . . 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 18	. . . . 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 7427	. . . 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 2807	. . 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 27510	. . 3 ⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑(♯‘𝑆)))
36	33, 35	oveq12d 7429	. 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 12202	. . . 4 ⊢ -1 ∈ ℂ
38	37	a1i 11	. . 3 ⊢ (𝜑 → -1 ∈ ℂ)
39		opabssxp 5754	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⊆ ((1...𝑀) × (1...𝑁))
40	34, 39	eqsstri 3991	. . . . 5 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
41		ssfi 9156	. . . . 5 ⊢ ((((1...𝑀) × (1...𝑁)) ∈ Fin ∧ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))) → 𝑆 ∈ Fin)
42	22, 40, 41	sylancl 597	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin)
43		hashcl 14391	. . . 4 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ₀)
44	42, 43	syl 18	. . 3 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
45		hashcl 14391	. . . 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 18	. . 3 ⊢ (𝜑 → (♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) ∈ ℕ₀)
47	38, 44, 46	expaddd 14183	. 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 3925	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ ℙ)
49	48	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℙ)
50		prmnn 16731	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
51	49, 50	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
52	1, 5	gausslemma2dlem0b 27486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
53	52	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℕ)
54	53	nnzd 12616	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℤ)
55		prmz 16732	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
56	49, 55	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℤ)
57		peano2zm 12636	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ∈ ℤ → (𝑄 − 1) ∈ ℤ)
58	56, 57	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℤ)
59	53	nnred 12247	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℝ)
60	58	zred 12699	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ)
61		prmuz2 16753	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ (ℤ_≥‘2))
62	49, 61	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ (ℤ_≥‘2))
63		uz2m1nn 12946	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑄 ∈ (ℤ_≥‘2) → (𝑄 − 1) ∈ ℕ)
64	62, 63	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℕ)
65	64	nnrpd 13057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ⁺)
66		rphalflt 13046	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 − 1) ∈ ℝ⁺ → ((𝑄 − 1) / 2) < (𝑄 − 1))
67	65, 66	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 − 1) / 2) < (𝑄 − 1))
68	5, 67	eqbrtrid 5150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 < (𝑄 − 1))
69	59, 60, 68	ltled 11357	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ≤ (𝑄 − 1))
70		eluz2 12867	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 − 1) ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑄 − 1) ∈ ℤ ∧ 𝑁 ≤ (𝑄 − 1)))
71	54, 58, 69, 70	syl3anbrc 1360	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ (ℤ_≥‘𝑁))
72		fzss2 13591	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 − 1) ∈ (ℤ_≥‘𝑁) → (1...𝑁) ⊆ (1...(𝑄 − 1)))
73	71, 72	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (1...𝑁) ⊆ (1...(𝑄 − 1)))
74		simprr 784	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...𝑁))
75	73, 74	sseldd 3946	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...(𝑄 − 1)))
76		fzm1ndvds 16379	. . . . . . . . . . . . . . 15 ⊢ ((𝑄 ∈ ℕ ∧ 𝑦 ∈ (1...(𝑄 − 1))) → ¬ 𝑄 ∥ 𝑦)
77	51, 75, 76	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ 𝑦)
78	4	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ≠ 𝑃)
79	2	eldifad 3925	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℙ)
80	79	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℙ)
81		prmrp 16770	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ) → ((𝑄 gcd 𝑃) = 1 ↔ 𝑄 ≠ 𝑃))
82	49, 80, 81	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 gcd 𝑃) = 1 ↔ 𝑄 ≠ 𝑃))
83	78, 82	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 gcd 𝑃) = 1)
84		prmz 16732	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
85	80, 84	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℤ)
86		elfzelz 13551	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℤ)
87	86	ad2antll 741	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℤ)
88		coprmdvds 16710	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
89	56, 85, 87, 88	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
90	83, 89	mpan2d 706	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 ∥ (𝑃 · 𝑦) → 𝑄 ∥ 𝑦))
91	77, 90	mtod 201	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ (𝑃 · 𝑦))
92		prmnn 16731	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
93	80, 92	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
94	93	nncnd 12248	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℂ)
95		elfznn 13580	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
96	95	ad2antll 741	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
97	96	nncnd 12248	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℂ)
98	94, 97	mulcomd 11229	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑃 · 𝑦) = (𝑦 · 𝑃))
99	98	breq2d 5125	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 ∥ (𝑃 · 𝑦) ↔ 𝑄 ∥ (𝑦 · 𝑃)))
100	91, 99	mtbid 327	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ (𝑦 · 𝑃))
101		elfzelz 13551	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℤ)
102	101	ad2antrl 740	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℤ)
103		dvdsmul2 16335	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑥 · 𝑄))
104	102, 56, 103	syl2anc 595	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∥ (𝑥 · 𝑄))
105		breq2 5117	. . . . . . . . . . . . . 14 ⊢ ((𝑥 · 𝑄) = (𝑦 · 𝑃) → (𝑄 ∥ (𝑥 · 𝑄) ↔ 𝑄 ∥ (𝑦 · 𝑃)))
106	104, 105	syl5ibcom 248	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) = (𝑦 · 𝑃) → 𝑄 ∥ (𝑦 · 𝑃)))
107	106	necon3bd 2978	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (¬ 𝑄 ∥ (𝑦 · 𝑃) → (𝑥 · 𝑄) ≠ (𝑦 · 𝑃)))
108	100, 107	mpd 16	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ≠ (𝑦 · 𝑃))
109		elfznn 13580	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
110	109	ad2antrl 740	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
111	110, 51	nnmulcld 12288	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
112	111	nnred 12247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℝ)
113	96, 93	nnmulcld 12288	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
114	113	nnred 12247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℝ)
115	112, 114	lttri2d 11348	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) ≠ (𝑦 · 𝑃) ↔ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
116	108, 115	mpbid 235	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
117	116	ex 417	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
118	117	pm4.71rd 571	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)) ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)))))
119		ancom 465	. . . . . . . 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 291	. . . . . . 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 5181	. . . . . 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 5195	. . . . . . 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 4127	. . . . . . 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 1023	. . . . . . . 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 5182	. . . . . . 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 2802	. . . . . 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 5668	. . . . . 6 ⊢ ((1...𝑀) × (1...𝑁)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))}
128	121, 126, 127	3eqtr4g 2829	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆) = ((1...𝑀) × (1...𝑁)))
129	128	fveq2d 6886	. . . 4 ⊢ (𝜑 → (♯‘({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆)) = (♯‘((1...𝑀) × (1...𝑁))))
130		inopab 5817	. . . . . . 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 4178	. . . . . . 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 688	. . . . . . . 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 5182	. . . . . . 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 2802	. . . . . 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 11308	. . . . . . . . . . . 12 ⊢ (((𝑥 · 𝑄) ∈ ℝ ∧ (𝑦 · 𝑃) ∈ ℝ) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
136	112, 114, 135	syl2anc 595	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
137	136	ex 417	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
138		imnan 404	. . . . . . . . . 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 221	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
140	139	nexdv 1963	. . . . . . . 8 ⊢ (𝜑 → ¬ ∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
141	140	nexdv 1963	. . . . . . 7 ⊢ (𝜑 → ¬ ∃𝑥∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
142		opabn0 5539	. . . . . . . 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 3013	. . . . . . 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 221	. . . . . 6 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))} = ∅)
145	134, 144	eqtrid 2816	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ 𝑆) = ∅)
146		hashun 14417	. . . . 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 1396	. . . 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 14470	. . . . . 6 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
149	19, 20, 148	syl2anc 595	. . . . 5 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
150	2, 6	gausslemma2dlem0b 27486	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
151	150	nnnn0d 12564	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
152		hashfz1 14381	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
153	151, 152	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
154	52	nnnn0d 12564	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
155		hashfz1 14381	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
156	154, 155	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
157	153, 156	oveq12d 7429	. . . . 5 ⊢ (𝜑 → ((♯‘(1...𝑀)) · (♯‘(1...𝑁))) = (𝑀 · 𝑁))
158	149, 157	eqtrd 2804	. . . 4 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = (𝑀 · 𝑁))
159	129, 147, 158	3eqtr3d 2812	. . 3 ⊢ (𝜑 → ((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆)) = (𝑀 · 𝑁))
160	159	oveq2d 7427	. 2 ⊢ (𝜑 → (-1↑((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆))) = (-1↑(𝑀 · 𝑁)))
161	36, 47, 160	3eqtr2d 2810	1 ⊢ (𝜑 → ((𝑃 /_L 𝑄) · (𝑄 /_L 𝑃)) = (-1↑(𝑀 · 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 = wceq 1567 ∃wex 1806 ∈ wcel 2149 ≠ wne 2964 ∖ cdif 3910 ∪ cun 3911 ∩ cin 3912 ⊆ wss 3913 ∅c0 4294 {csn 4594 class class class wbr 5113 {copab 5177 × cxp 5660 ^◡ccnv 5661 Rel wrel 5667 ‘cfv 6537 (class class class)co 7411 ≈ cen 8939 Fincfn 8942 ℂcc 11097 ℝcr 11098 1c1 11100 + caddc 11102 · cmul 11104 < clt 11242 ≤ cle 11243 − cmin 11440 -cneg 11441 / cdiv 11870 ℕcn 12232 2c2 12294 ℕ₀cn0 12503 ℤcz 12590 ℤ_≥cuz 12861 ℝ⁺crp 13015 ...cfz 13534 ↑cexp 14096 ♯chash 14365 ∥ cdvds 16309 gcd cgcd 16551 ℙcprime 16728 /_L clgs 27423

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178 ax-mulf 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-disj 5081 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-tpos 8221 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-oadd 8456 df-er 8693 df-ec 8695 df-qs 8699 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-sup 9401 df-inf 9402 df-oi 9471 df-dju 9886 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-xnn0 12577 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-fz 13535 df-fzo 13682 df-fl 13824 df-mod 13902 df-seq 14037 df-exp 14097 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-sum 15737 df-dvds 16310 df-gcd 16552 df-prm 16729 df-phi 16824 df-pc 16896 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-0g 17493 df-gsum 17494 df-imas 17561 df-qus 17562 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-mhm 18840 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-mulg 19133 df-subg 19188 df-nsg 19189 df-eqg 19190 df-ghm 19283 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-cring 20317 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-rhm 20553 df-nzr 20595 df-subrng 20630 df-subrg 20654 df-rlreg 20778 df-domn 20779 df-idom 20780 df-drng 20814 df-field 20815 df-lmod 20960 df-lss 21030 df-lsp 21070 df-sra 21271 df-rgmod 21272 df-lidl 21309 df-rsp 21310 df-2idl 21359 df-cnfld 21491 df-zring 21565 df-zrh 21621 df-zn 21624 df-lgs 27424

This theorem is referenced by: lgsquad 27512

W3C validator