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Theorem lgsquadlem3 27597

Description: Lemma for lgsquad 27598. (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 3015	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃)
5		lgsquad.5	. . . . 5 ⊢ 𝑁 = ((𝑄 − 1) / 2)
6		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
7		eleq1w 2848	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (1...𝑀) ↔ 𝑧 ∈ (1...𝑀)))
8		eleq1w 2848	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (1...𝑁) ↔ 𝑤 ∈ (1...𝑁)))
9	7, 8	bi2anan9 650	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (𝑧 ∈ (1...𝑀) ∧ 𝑤 ∈ (1...𝑁))))
10	9	biancomd 469	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (𝑤 ∈ (1...𝑁) ∧ 𝑧 ∈ (1...𝑀))))
11		oveq1 7426	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 · 𝑄) = (𝑧 · 𝑄))
12		oveq1 7426	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 · 𝑃) = (𝑤 · 𝑃))
13	11, 12	breqan12d 5127	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ↔ (𝑧 · 𝑄) < (𝑤 · 𝑃)))
14	10, 13	anbi12d 644	. . . . . . 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 464	. . . . . 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 5186	. . . . 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 27596	. . . 4 ⊢ (𝜑 → (𝑃 /_L 𝑄) = (-1↑(♯‘{⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})))
18		relopabv 5810	. . . . . . . 8 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}
19		fzfid 14027	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
20		fzfid 14027	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
21		xpfi 9286	. . . . . . . . . 10 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑀) × (1...𝑁)) ∈ Fin)
22	19, 20, 21	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) × (1...𝑁)) ∈ Fin)
23		opabssxp 5755	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ⊆ ((1...𝑀) × (1...𝑁))
24		ssfi 9164	. . . . . . . . 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 598	. . . . . . . 8 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin)
26		cnven 9037	. . . . . . . 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 599	. . . . . . 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 6139	. . . . . . 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 5154	. . . . . 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 14402	. . . . . 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 7435	. . . 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 2803	. . 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 27596	. . 3 ⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑(♯‘𝑆)))
36	33, 35	oveq12d 7437	. 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 12218	. . . 4 ⊢ -1 ∈ ℂ
38	37	a1i 11	. . 3 ⊢ (𝜑 → -1 ∈ ℂ)
39		opabssxp 5755	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⊆ ((1...𝑀) × (1...𝑁))
40	34, 39	eqsstri 3984	. . . . 5 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
41		ssfi 9164	. . . . 5 ⊢ ((((1...𝑀) × (1...𝑁)) ∈ Fin ∧ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))) → 𝑆 ∈ Fin)
42	22, 40, 41	sylancl 598	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin)
43		hashcl 14410	. . . 4 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ₀)
44	42, 43	syl 18	. . 3 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
45		hashcl 14410	. . . 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 14202	. 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 3918	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ ℙ)
49	48	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℙ)
50		prmnn 16754	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
51	49, 50	syl 18	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
52	1, 5	gausslemma2dlem0b 27572	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
53	52	adantr 486	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℕ)
54	53	nnzd 12632	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℤ)
55		prmz 16755	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
56	49, 55	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℤ)
57		peano2zm 12652	. . . . . . . . . . . . . . . . . . 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 12263	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℝ)
60	58	zred 12716	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ)
61		prmuz2 16776	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ (ℤ_≥‘2))
62	49, 61	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ (ℤ_≥‘2))
63		uz2m1nn 12963	. . . . . . . . . . . . . . . . . . . . . . 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 13074	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ⁺)
66		rphalflt 13063	. . . . . . . . . . . . . . . . . . . . 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 5148	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 < (𝑄 − 1))
69	59, 60, 68	ltled 11373	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ≤ (𝑄 − 1))
70		eluz2 12884	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 − 1) ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑄 − 1) ∈ ℤ ∧ 𝑁 ≤ (𝑄 − 1)))
71	54, 58, 69, 70	syl3anbrc 1362	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ (ℤ_≥‘𝑁))
72		fzss2 13609	. . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...𝑁))
75	73, 74	sseldd 3939	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...(𝑄 − 1)))
76		fzm1ndvds 16402	. . . . . . . . . . . . . . 15 ⊢ ((𝑄 ∈ ℕ ∧ 𝑦 ∈ (1...(𝑄 − 1))) → ¬ 𝑄 ∥ 𝑦)
77	51, 75, 76	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ 𝑦)
78	4	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ≠ 𝑃)
79	2	eldifad 3918	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℙ)
80	79	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℙ)
81		prmrp 16793	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ) → ((𝑄 gcd 𝑃) = 1 ↔ 𝑄 ≠ 𝑃))
82	49, 80, 81	syl2anc 596	. . . . . . . . . . . . . . . 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 16755	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
85	80, 84	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℤ)
86		elfzelz 13568	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℤ)
87	86	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℤ)
88		coprmdvds 16733	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
89	56, 85, 87, 88	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
90	83, 89	mpan2d 707	. . . . . . . . . . . . . 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 16754	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
93	80, 92	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
94	93	nncnd 12264	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℂ)
95		elfznn 13598	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
96	95	ad2antll 742	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
97	96	nncnd 12264	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℂ)
98	94, 97	mulcomd 11245	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑃 · 𝑦) = (𝑦 · 𝑃))
99	98	breq2d 5123	. . . . . . . . . . . . 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 13568	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℤ)
102	101	ad2antrl 741	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℤ)
103		dvdsmul2 16358	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑥 · 𝑄))
104	102, 56, 103	syl2anc 596	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∥ (𝑥 · 𝑄))
105		breq2 5115	. . . . . . . . . . . . . 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 2974	. . . . . . . . . . . 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 13598	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
110	109	ad2antrl 741	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
111	110, 51	nnmulcld 12304	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
112	111	nnred 12263	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℝ)
113	96, 93	nnmulcld 12304	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
114	113	nnred 12263	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℝ)
115	112, 114	lttri2d 11364	. . . . . . . . . . 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 418	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
118	117	pm4.71rd 572	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)) ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)))))
119		ancom 466	. . . . . . . 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 5179	. . . . . 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 5193	. . . . . . 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 4119	. . . . . . 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 1025	. . . . . . . 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 5180	. . . . . . 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 2798	. . . . . 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 5669	. . . . . 6 ⊢ ((1...𝑀) × (1...𝑁)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))}
128	121, 126, 127	3eqtr4g 2825	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆) = ((1...𝑀) × (1...𝑁)))
129	128	fveq2d 6889	. . . 4 ⊢ (𝜑 → (♯‘({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆)) = (♯‘((1...𝑀) × (1...𝑁))))
130		inopab 5818	. . . . . . 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 4170	. . . . . . 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 689	. . . . . . . 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 5180	. . . . . . 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 2798	. . . . . 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 11324	. . . . . . . . . . . 12 ⊢ (((𝑥 · 𝑄) ∈ ℝ ∧ (𝑦 · 𝑃) ∈ ℝ) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
136	112, 114, 135	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
137	136	ex 418	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
138		imnan 405	. . . . . . . . . 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 1969	. . . . . . . 8 ⊢ (𝜑 → ¬ ∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
141	140	nexdv 1969	. . . . . . 7 ⊢ (𝜑 → ¬ ∃𝑥∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
142		opabn0 5540	. . . . . . . 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 3009	. . . . . . 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 2812	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ 𝑆) = ∅)
146		hashun 14436	. . . . 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 1398	. . . 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 14489	. . . . . 6 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
149	19, 20, 148	syl2anc 596	. . . . 5 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
150	2, 6	gausslemma2dlem0b 27572	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
151	150	nnnn0d 12580	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
152		hashfz1 14400	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
153	151, 152	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
154	52	nnnn0d 12580	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
155		hashfz1 14400	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
156	154, 155	syl 18	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
157	153, 156	oveq12d 7437	. . . . 5 ⊢ (𝜑 → ((♯‘(1...𝑀)) · (♯‘(1...𝑁))) = (𝑀 · 𝑁))
158	149, 157	eqtrd 2800	. . . 4 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = (𝑀 · 𝑁))
159	129, 147, 158	3eqtr3d 2808	. . 3 ⊢ (𝜑 → ((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆)) = (𝑀 · 𝑁))
160	159	oveq2d 7435	. 2 ⊢ (𝜑 → (-1↑((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆))) = (-1↑(𝑀 · 𝑁)))
161	36, 47, 160	3eqtr2d 2806	1 ⊢ (𝜑 → ((𝑃 /_L 𝑄) · (𝑄 /_L 𝑃)) = (-1↑(𝑀 · 𝑁)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∃wex 1812 ∈ wcel 2146 ≠ wne 2960 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 {csn 4591 class class class wbr 5111 {copab 5175 × cxp 5661 ^◡ccnv 5662 Rel wrel 5668 ‘cfv 6540 (class class class)co 7419 ≈ cen 8946 Fincfn 8949 ℂcc 11113 ℝcr 11114 1c1 11116 + caddc 11118 · cmul 11120 < clt 11258 ≤ cle 11259 − cmin 11456 -cneg 11457 / cdiv 11886 ℕcn 12248 2c2 12310 ℕ₀cn0 12519 ℤcz 12606 ℤ_≥cuz 12878 ℝ⁺crp 13032 ...cfz 13551 ↑cexp 14115 ♯chash 14384 ∥ cdvds 16332 gcd cgcd 16574 ℙcprime 16751 /_L clgs 27509

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-inf2 9617 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-pre-sup 11193 ax-addf 11194 ax-mulf 11195

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-disj 5079 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8163 df-tpos 8228 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-oadd 8463 df-er 8700 df-ec 8702 df-qs 8706 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-fsupp 9329 df-sup 9409 df-inf 9410 df-oi 9479 df-dju 9903 df-card 9941 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-div 11887 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-7 12323 df-8 12324 df-9 12325 df-n0 12520 df-xnn0 12593 df-z 12607 df-dec 12728 df-uz 12879 df-q 12989 df-rp 13033 df-fz 13552 df-fzo 13700 df-fl 13843 df-mod 13921 df-seq 14056 df-exp 14116 df-hash 14385 df-cj 15174 df-re 15175 df-im 15176 df-sqrt 15310 df-abs 15311 df-clim 15563 df-sum 15762 df-dvds 16333 df-gcd 16575 df-prm 16752 df-phi 16847 df-pc 16919 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-starv 17347 df-sca 17348 df-vsca 17349 df-ip 17350 df-tset 17351 df-ple 17352 df-ds 17354 df-unif 17355 df-0g 17516 df-gsum 17517 df-imas 17584 df-qus 17585 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-mhm 18878 df-submnd 18879 df-grp 19047 df-minusg 19048 df-sbg 19049 df-mulg 19178 df-subg 19233 df-nsg 19234 df-eqg 19235 df-ghm 19328 df-cntz 19431 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-ring 20361 df-cring 20362 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-invr 20516 df-dvr 20529 df-rhm 20600 df-nzr 20660 df-subrng 20695 df-subrg 20719 df-rlreg 20843 df-domn 20844 df-idom 20845 df-drng 20879 df-field 20880 df-lmod 21033 df-lss 21103 df-lsp 21143 df-sra 21344 df-rgmod 21345 df-lidl 21382 df-rsp 21383 df-2idl 21439 df-cnfld 21573 df-zring 21647 df-zrh 21703 df-zn 21706 df-lgs 27510

This theorem is used by: lgsquad 27598

W3C validator