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Theorem lgsquadlem3 27326

Description: Lemma for lgsquad 27327. (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 2980	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃)
5		lgsquad.5	. . . . 5 ⊢ 𝑁 = ((𝑄 − 1) / 2)
6		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
7		eleq1w 2811	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (1...𝑀) ↔ 𝑧 ∈ (1...𝑀)))
8		eleq1w 2811	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (1...𝑁) ↔ 𝑤 ∈ (1...𝑁)))
9	7, 8	bi2anan9 638	. . . . . . . . 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 7376	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 · 𝑄) = (𝑧 · 𝑄))
12		oveq1 7376	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 · 𝑃) = (𝑤 · 𝑃))
13	11, 12	breqan12d 5118	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ↔ (𝑧 · 𝑄) < (𝑤 · 𝑃)))
14	10, 13	anbi12d 632	. . . . . . 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 5175	. . . . 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 27325	. . . 4 ⊢ (𝜑 → (𝑃 /_L 𝑄) = (-1↑(♯‘{⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})))
18		relopabv 5775	. . . . . . . 8 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}
19		fzfid 13914	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
20		fzfid 13914	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
21		xpfi 9245	. . . . . . . . . 10 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑀) × (1...𝑁)) ∈ Fin)
22	19, 20, 21	syl2anc 584	. . . . . . . . 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 9114	. . . . . . . . 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 586	. . . . . . . 8 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin)
26		cnven 8981	. . . . . . . 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 587	. . . . . . 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 6098	. . . . . . 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 5143	. . . . . 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 14289	. . . . . 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 7385	. . . 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 2767	. . 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 27325	. . 3 ⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑(♯‘𝑆)))
36	33, 35	oveq12d 7387	. 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 12147	. . . 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 3990	. . . . 5 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
41		ssfi 9114	. . . . 5 ⊢ ((((1...𝑀) × (1...𝑁)) ∈ Fin ∧ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))) → 𝑆 ∈ Fin)
42	22, 40, 41	sylancl 586	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin)
43		hashcl 14297	. . . 4 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ₀)
44	42, 43	syl 17	. . 3 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
45		hashcl 14297	. . . 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 14089	. 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 3923	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ ℙ)
49	48	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℙ)
50		prmnn 16620	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
51	49, 50	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
52	1, 5	gausslemma2dlem0b 27301	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
53	52	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℕ)
54	53	nnzd 12532	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℤ)
55		prmz 16621	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
56	49, 55	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℤ)
57		peano2zm 12552	. . . . . . . . . . . . . . . . . . 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 12177	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℝ)
60	58	zred 12614	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ)
61		prmuz2 16642	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ (ℤ_≥‘2))
62	49, 61	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ (ℤ_≥‘2))
63		uz2m1nn 12858	. . . . . . . . . . . . . . . . . . . . . . 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 12969	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ⁺)
66		rphalflt 12958	. . . . . . . . . . . . . . . . . . . . 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 5137	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 < (𝑄 − 1))
69	59, 60, 68	ltled 11298	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ≤ (𝑄 − 1))
70		eluz2 12775	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 − 1) ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑄 − 1) ∈ ℤ ∧ 𝑁 ≤ (𝑄 − 1)))
71	54, 58, 69, 70	syl3anbrc 1344	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ (ℤ_≥‘𝑁))
72		fzss2 13501	. . . . . . . . . . . . . . . . 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 772	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...𝑁))
75	73, 74	sseldd 3944	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...(𝑄 − 1)))
76		fzm1ndvds 16268	. . . . . . . . . . . . . . 15 ⊢ ((𝑄 ∈ ℕ ∧ 𝑦 ∈ (1...(𝑄 − 1))) → ¬ 𝑄 ∥ 𝑦)
77	51, 75, 76	syl2anc 584	. . . . . . . . . . . . . 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 3923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℙ)
80	79	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℙ)
81		prmrp 16658	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ) → ((𝑄 gcd 𝑃) = 1 ↔ 𝑄 ≠ 𝑃))
82	49, 80, 81	syl2anc 584	. . . . . . . . . . . . . . . 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 16621	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
85	80, 84	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℤ)
86		elfzelz 13461	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℤ)
87	86	ad2antll 729	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℤ)
88		coprmdvds 16599	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
89	56, 85, 87, 88	syl3anc 1373	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
90	83, 89	mpan2d 694	. . . . . . . . . . . . . 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 16620	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
93	80, 92	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
94	93	nncnd 12178	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℂ)
95		elfznn 13490	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
96	95	ad2antll 729	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
97	96	nncnd 12178	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℂ)
98	94, 97	mulcomd 11171	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑃 · 𝑦) = (𝑦 · 𝑃))
99	98	breq2d 5114	. . . . . . . . . . . . 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 13461	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℤ)
102	101	ad2antrl 728	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℤ)
103		dvdsmul2 16224	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑥 · 𝑄))
104	102, 56, 103	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∥ (𝑥 · 𝑄))
105		breq2 5106	. . . . . . . . . . . . . 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 2939	. . . . . . . . . . . 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 13490	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
110	109	ad2antrl 728	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
111	110, 51	nnmulcld 12215	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
112	111	nnred 12177	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℝ)
113	96, 93	nnmulcld 12215	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
114	113	nnred 12177	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℝ)
115	112, 114	lttri2d 11289	. . . . . . . . . . 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 5168	. . . . . 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 5182	. . . . . . 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 4124	. . . . . . 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 1009	. . . . . . . 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 5169	. . . . . . 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 2762	. . . . . 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 2789	. . . . 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 5783	. . . . . . 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 4176	. . . . . . 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 676	. . . . . . . 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 5169	. . . . . . 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 2762	. . . . . 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 11249	. . . . . . . . . . . 12 ⊢ (((𝑥 · 𝑄) ∈ ℝ ∧ (𝑦 · 𝑃) ∈ ℝ) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
136	112, 114, 135	syl2anc 584	. . . . . . . . . . 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 1936	. . . . . . . 8 ⊢ (𝜑 → ¬ ∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
141	140	nexdv 1936	. . . . . . 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 2974	. . . . . . 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 2776	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ 𝑆) = ∅)
146		hashun 14323	. . . . 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 1373	. . . 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 14375	. . . . . 6 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
149	19, 20, 148	syl2anc 584	. . . . 5 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
150	2, 6	gausslemma2dlem0b 27301	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
151	150	nnnn0d 12479	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
152		hashfz1 14287	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
153	151, 152	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
154	52	nnnn0d 12479	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
155		hashfz1 14287	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
156	154, 155	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
157	153, 156	oveq12d 7387	. . . . 5 ⊢ (𝜑 → ((♯‘(1...𝑀)) · (♯‘(1...𝑁))) = (𝑀 · 𝑁))
158	149, 157	eqtrd 2764	. . . 4 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = (𝑀 · 𝑁))
159	129, 147, 158	3eqtr3d 2772	. . 3 ⊢ (𝜑 → ((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆)) = (𝑀 · 𝑁))
160	159	oveq2d 7385	. 2 ⊢ (𝜑 → (-1↑((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆))) = (-1↑(𝑀 · 𝑁)))
161	36, 47, 160	3eqtr2d 2770	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 847 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3908 ∪ cun 3909 ∩ cin 3910 ⊆ wss 3911 ∅c0 4292 {csn 4585 class class class wbr 5102 {copab 5164 × cxp 5629 ^◡ccnv 5630 Rel wrel 5636 ‘cfv 6499 (class class class)co 7369 ≈ cen 8892 Fincfn 8895 ℂcc 11042 ℝcr 11043 1c1 11045 + caddc 11047 · cmul 11049 < clt 11184 ≤ cle 11185 − cmin 11381 -cneg 11382 / cdiv 11811 ℕcn 12162 2c2 12217 ℕ₀cn0 12418 ℤcz 12505 ℤ_≥cuz 12769 ℝ⁺crp 12927 ...cfz 13444 ↑cexp 14002 ♯chash 14271 ∥ cdvds 16198 gcd cgcd 16440 ℙcprime 16617 /_L clgs 27238

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123 ax-mulf 11124

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-disj 5070 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 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 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-tpos 8182 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-oadd 8415 df-er 8648 df-ec 8650 df-qs 8654 df-map 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-sup 9369 df-inf 9370 df-oi 9439 df-dju 9830 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-fz 13445 df-fzo 13592 df-fl 13730 df-mod 13808 df-seq 13943 df-exp 14003 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-sum 15629 df-dvds 16199 df-gcd 16441 df-prm 16618 df-phi 16712 df-pc 16784 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-0g 17380 df-gsum 17381 df-imas 17447 df-qus 17448 df-mgm 18549 df-sgrp 18628 df-mnd 18644 df-mhm 18692 df-submnd 18693 df-grp 18850 df-minusg 18851 df-sbg 18852 df-mulg 18982 df-subg 19037 df-nsg 19038 df-eqg 19039 df-ghm 19127 df-cntz 19231 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-cring 20156 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-rhm 20392 df-nzr 20433 df-subrng 20466 df-subrg 20490 df-rlreg 20614 df-domn 20615 df-idom 20616 df-drng 20651 df-field 20652 df-lmod 20800 df-lss 20870 df-lsp 20910 df-sra 21112 df-rgmod 21113 df-lidl 21150 df-rsp 21151 df-2idl 21192 df-cnfld 21297 df-zring 21389 df-zrh 21445 df-zn 21448 df-lgs 27239

This theorem is referenced by: lgsquad 27327

W3C validator