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Theorem lgsquadlem3 25398

Description: Lemma for lgsquad 25399. (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 2992	. . . . 5 ⊢ (𝜑 → 𝑄 ≠ 𝑃)
5		lgsquad.5	. . . . 5 ⊢ 𝑁 = ((𝑄 − 1) / 2)
6		lgsquad.4	. . . . 5 ⊢ 𝑀 = ((𝑃 − 1) / 2)
7		eleq1w 2827	. . . . . . . . . 10 ⊢ (𝑥 = 𝑧 → (𝑥 ∈ (1...𝑀) ↔ 𝑧 ∈ (1...𝑀)))
8		eleq1w 2827	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦 ∈ (1...𝑁) ↔ 𝑤 ∈ (1...𝑁)))
9	7, 8	bi2anan9 629	. . . . . . . . 9 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (𝑧 ∈ (1...𝑀) ∧ 𝑤 ∈ (1...𝑁))))
10		ancom 452	. . . . . . . . 9 ⊢ ((𝑧 ∈ (1...𝑀) ∧ 𝑤 ∈ (1...𝑁)) ↔ (𝑤 ∈ (1...𝑁) ∧ 𝑧 ∈ (1...𝑀)))
11	9, 10	syl6bb 278	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (𝑤 ∈ (1...𝑁) ∧ 𝑧 ∈ (1...𝑀))))
12		oveq1 6849	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 · 𝑄) = (𝑧 · 𝑄))
13		oveq1 6849	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦 · 𝑃) = (𝑤 · 𝑃))
14	12, 13	breqan12d 4825	. . . . . . . 8 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ↔ (𝑧 · 𝑄) < (𝑤 · 𝑃)))
15	11, 14	anbi12d 624	. . . . . . 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))))
16	15	ancoms 450	. . . . . 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))))
17	16	cbvopabv 4881	. . . . 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))}
18	1, 2, 4, 5, 6, 17	lgsquadlem2 25397	. . . 4 ⊢ (𝜑 → (𝑃 /_L 𝑄) = (-1↑(♯‘{⟨𝑦, 𝑥⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})))
19		relopab 5416	. . . . . . . 8 ⊢ Rel {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}
20		fzfid 12980	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑀) ∈ Fin)
21		fzfid 12980	. . . . . . . . . 10 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
22		xpfi 8438	. . . . . . . . . 10 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → ((1...𝑀) × (1...𝑁)) ∈ Fin)
23	20, 21, 22	syl2anc 579	. . . . . . . . 9 ⊢ (𝜑 → ((1...𝑀) × (1...𝑁)) ∈ Fin)
24		opabssxp 5363	. . . . . . . . 9 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ⊆ ((1...𝑀) × (1...𝑁))
25		ssfi 8387	. . . . . . . . 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)
26	23, 24, 25	sylancl 580	. . . . . . . 8 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∈ Fin)
27		cnven 8236	. . . . . . . 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))})
28	19, 26, 27	sylancr 581	. . . . . . 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))})
29		cnvopab 5716	. . . . . . 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))}
30	28, 29	syl6breq 4850	. . . . . 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))})
31		hasheni 13340	. . . . . 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))}))
32	30, 31	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))}))
33	32	oveq2d 6858	. . . 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))})))
34	18, 33	eqtr4d 2802	. . 3 ⊢ (𝜑 → (𝑃 /_L 𝑄) = (-1↑(♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})))
35		lgsquad.6	. . . 4 ⊢ 𝑆 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))}
36	2, 1, 3, 6, 5, 35	lgsquadlem2 25397	. . 3 ⊢ (𝜑 → (𝑄 /_L 𝑃) = (-1↑(♯‘𝑆)))
37	34, 36	oveq12d 6860	. 2 ⊢ (𝜑 → ((𝑃 /_L 𝑄) · (𝑄 /_L 𝑃)) = ((-1↑(♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))})) · (-1↑(♯‘𝑆))))
38		neg1cn 11393	. . . 4 ⊢ -1 ∈ ℂ
39	38	a1i 11	. . 3 ⊢ (𝜑 → -1 ∈ ℂ)
40		opabssxp 5363	. . . . . 6 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))} ⊆ ((1...𝑀) × (1...𝑁))
41	35, 40	eqsstri 3795	. . . . 5 ⊢ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))
42		ssfi 8387	. . . . 5 ⊢ ((((1...𝑀) × (1...𝑁)) ∈ Fin ∧ 𝑆 ⊆ ((1...𝑀) × (1...𝑁))) → 𝑆 ∈ Fin)
43	23, 41, 42	sylancl 580	. . . 4 ⊢ (𝜑 → 𝑆 ∈ Fin)
44		hashcl 13349	. . . 4 ⊢ (𝑆 ∈ Fin → (♯‘𝑆) ∈ ℕ₀)
45	43, 44	syl 17	. . 3 ⊢ (𝜑 → (♯‘𝑆) ∈ ℕ₀)
46		hashcl 13349	. . . 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)
47	26, 46	syl 17	. . 3 ⊢ (𝜑 → (♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) ∈ ℕ₀)
48	39, 45, 47	expaddd 13217	. 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))))
49	1	eldifad 3744	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑄 ∈ ℙ)
50	49	adantr 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℙ)
51		prmnn 15670	. . . . . . . . . . . . . . . 16 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℕ)
52	50, 51	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℕ)
53		oddprm 15796	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑄 ∈ (ℙ ∖ {2}) → ((𝑄 − 1) / 2) ∈ ℕ)
54	1, 53	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → ((𝑄 − 1) / 2) ∈ ℕ)
55	5, 54	syl5eqel 2848	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝑁 ∈ ℕ)
56	55	adantr 472	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℕ)
57	56	nnzd 11728	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℤ)
58		prmz 15671	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ ℤ)
59	50, 58	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ ℤ)
60		peano2zm 11667	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑄 ∈ ℤ → (𝑄 − 1) ∈ ℤ)
61	59, 60	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℤ)
62	56	nnred 11291	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ∈ ℝ)
63	61	zred 11729	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ)
64		prmuz2 15690	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑄 ∈ ℙ → 𝑄 ∈ (ℤ_≥‘2))
65	50, 64	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∈ (ℤ_≥‘2))
66		uz2m1nn 11964	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑄 ∈ (ℤ_≥‘2) → (𝑄 − 1) ∈ ℕ)
67	65, 66	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℕ)
68	67	nnrpd 12068	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ ℝ⁺)
69		rphalflt 12058	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑄 − 1) ∈ ℝ⁺ → ((𝑄 − 1) / 2) < (𝑄 − 1))
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 − 1) / 2) < (𝑄 − 1))
71	5, 70	syl5eqbr 4844	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 < (𝑄 − 1))
72	62, 63, 71	ltled 10439	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑁 ≤ (𝑄 − 1))
73		eluz2 11892	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑄 − 1) ∈ (ℤ_≥‘𝑁) ↔ (𝑁 ∈ ℤ ∧ (𝑄 − 1) ∈ ℤ ∧ 𝑁 ≤ (𝑄 − 1)))
74	57, 61, 72, 73	syl3anbrc 1443	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 − 1) ∈ (ℤ_≥‘𝑁))
75		fzss2 12588	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 − 1) ∈ (ℤ_≥‘𝑁) → (1...𝑁) ⊆ (1...(𝑄 − 1)))
76	74, 75	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (1...𝑁) ⊆ (1...(𝑄 − 1)))
77		simprr 789	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...𝑁))
78	76, 77	sseldd 3762	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ (1...(𝑄 − 1)))
79		fzm1ndvds 15331	. . . . . . . . . . . . . . 15 ⊢ ((𝑄 ∈ ℕ ∧ 𝑦 ∈ (1...(𝑄 − 1))) → ¬ 𝑄 ∥ 𝑦)
80	52, 78, 79	syl2anc 579	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ 𝑦)
81	4	adantr 472	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ≠ 𝑃)
82	2	eldifad 3744	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑃 ∈ ℙ)
83	82	adantr 472	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℙ)
84		prmrp 15705	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑄 ∈ ℙ ∧ 𝑃 ∈ ℙ) → ((𝑄 gcd 𝑃) = 1 ↔ 𝑄 ≠ 𝑃))
85	50, 83, 84	syl2anc 579	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 gcd 𝑃) = 1 ↔ 𝑄 ≠ 𝑃))
86	81, 85	mpbird 248	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 gcd 𝑃) = 1)
87		prmz 15671	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
88	83, 87	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℤ)
89		elfzelz 12549	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℤ)
90	89	ad2antll 720	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℤ)
91		coprmdvds 15649	. . . . . . . . . . . . . . . 16 ⊢ ((𝑄 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑦 ∈ ℤ) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
92	59, 88, 90, 91	syl3anc 1490	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑄 ∥ (𝑃 · 𝑦) ∧ (𝑄 gcd 𝑃) = 1) → 𝑄 ∥ 𝑦))
93	86, 92	mpan2d 685	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 ∥ (𝑃 · 𝑦) → 𝑄 ∥ 𝑦))
94	80, 93	mtod 189	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ (𝑃 · 𝑦))
95		prmnn 15670	. . . . . . . . . . . . . . . . 17 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
96	83, 95	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℕ)
97	96	nncnd 11292	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑃 ∈ ℂ)
98		elfznn 12577	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ (1...𝑁) → 𝑦 ∈ ℕ)
99	98	ad2antll 720	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℕ)
100	99	nncnd 11292	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑦 ∈ ℂ)
101	97, 100	mulcomd 10315	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑃 · 𝑦) = (𝑦 · 𝑃))
102	101	breq2d 4821	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑄 ∥ (𝑃 · 𝑦) ↔ 𝑄 ∥ (𝑦 · 𝑃)))
103	94, 102	mtbid 315	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ 𝑄 ∥ (𝑦 · 𝑃))
104		elfzelz 12549	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℤ)
105	104	ad2antrl 719	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℤ)
106		dvdsmul2 15291	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ ℤ ∧ 𝑄 ∈ ℤ) → 𝑄 ∥ (𝑥 · 𝑄))
107	105, 59, 106	syl2anc 579	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑄 ∥ (𝑥 · 𝑄))
108		breq2 4813	. . . . . . . . . . . . . 14 ⊢ ((𝑥 · 𝑄) = (𝑦 · 𝑃) → (𝑄 ∥ (𝑥 · 𝑄) ↔ 𝑄 ∥ (𝑦 · 𝑃)))
109	107, 108	syl5ibcom 236	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) = (𝑦 · 𝑃) → 𝑄 ∥ (𝑦 · 𝑃)))
110	109	necon3bd 2951	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (¬ 𝑄 ∥ (𝑦 · 𝑃) → (𝑥 · 𝑄) ≠ (𝑦 · 𝑃)))
111	103, 110	mpd 15	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ≠ (𝑦 · 𝑃))
112		elfznn 12577	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ (1...𝑀) → 𝑥 ∈ ℕ)
113	112	ad2antrl 719	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → 𝑥 ∈ ℕ)
114	113, 52	nnmulcld 11325	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℕ)
115	114	nnred 11291	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑥 · 𝑄) ∈ ℝ)
116	99, 96	nnmulcld 11325	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℕ)
117	116	nnred 11291	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → (𝑦 · 𝑃) ∈ ℝ)
118	115, 117	lttri2d 10430	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) ≠ (𝑦 · 𝑃) ↔ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
119	111, 118	mpbid 223	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
120	119	ex 401	. . . . . . . . 9 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
121	120	pm4.71rd 558	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ↔ (((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄)) ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)))))
122		ancom 452	. . . . . . . 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))))
123	121, 122	syl6rbb 279	. . . . . . 7 ⊢ (𝜑 → (((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∨ (𝑦 · 𝑃) < (𝑥 · 𝑄))) ↔ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))))
124	123	opabbidv 4875	. . . . . 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))})
125		unopab 4887	. . . . . . 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)))}
126	35	uneq2i 3926	. . . . . . 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))})
127		andi 1030	. . . . . . . 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))))
128	127	opabbii 4876	. . . . . . 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)))}
129	125, 126, 128	3eqtr4i 2797	. . . . . 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)))}
130		df-xp 5283	. . . . . 6 ⊢ ((1...𝑀) × (1...𝑁)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))}
131	124, 129, 130	3eqtr4g 2824	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆) = ((1...𝑀) × (1...𝑁)))
132	131	fveq2d 6379	. . . 4 ⊢ (𝜑 → (♯‘({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∪ 𝑆)) = (♯‘((1...𝑀) × (1...𝑁))))
133		inopab 5421	. . . . . . 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)))}
134	35	ineq2i 3973	. . . . . . 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))})
135		anandi 666	. . . . . . . 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))))
136	135	opabbii 4876	. . . . . . 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)))}
137	133, 134, 136	3eqtr4i 2797	. . . . . 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)))}
138		ltnsym2 10390	. . . . . . . . . . . 12 ⊢ (((𝑥 · 𝑄) ∈ ℝ ∧ (𝑦 · 𝑃) ∈ ℝ) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
139	115, 117, 138	syl2anc 579	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁))) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))
140	139	ex 401	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) → ¬ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
141		imnan 388	. . . . . . . . . 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))))
142	140, 141	sylib 209	. . . . . . . . 9 ⊢ (𝜑 → ¬ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
143	142	nexdv 2031	. . . . . . . 8 ⊢ (𝜑 → ¬ ∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
144	143	nexdv 2031	. . . . . . 7 ⊢ (𝜑 → ¬ ∃𝑥∃𝑦((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄))))
145		opabn0 5167	. . . . . . . 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))))
146	145	necon1bbii 2986	. . . . . . 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)))} = ∅)
147	144, 146	sylib 209	. . . . . 6 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ ((𝑥 · 𝑄) < (𝑦 · 𝑃) ∧ (𝑦 · 𝑃) < (𝑥 · 𝑄)))} = ∅)
148	137, 147	syl5eq 2811	. . . . 5 ⊢ (𝜑 → ({⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))} ∩ 𝑆) = ∅)
149		hashun 13373	. . . . 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)))
150	26, 43, 148, 149	syl3anc 1490	. . . 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)))
151		hashxp 13422	. . . . . 6 ⊢ (((1...𝑀) ∈ Fin ∧ (1...𝑁) ∈ Fin) → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
152	20, 21, 151	syl2anc 579	. . . . 5 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = ((♯‘(1...𝑀)) · (♯‘(1...𝑁))))
153		oddprm 15796	. . . . . . . . . 10 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ((𝑃 − 1) / 2) ∈ ℕ)
154	2, 153	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ((𝑃 − 1) / 2) ∈ ℕ)
155	6, 154	syl5eqel 2848	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
156	155	nnnn0d 11598	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
157		hashfz1 13338	. . . . . . 7 ⊢ (𝑀 ∈ ℕ₀ → (♯‘(1...𝑀)) = 𝑀)
158	156, 157	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑀)) = 𝑀)
159	55	nnnn0d 11598	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
160		hashfz1 13338	. . . . . . 7 ⊢ (𝑁 ∈ ℕ₀ → (♯‘(1...𝑁)) = 𝑁)
161	159, 160	syl 17	. . . . . 6 ⊢ (𝜑 → (♯‘(1...𝑁)) = 𝑁)
162	158, 161	oveq12d 6860	. . . . 5 ⊢ (𝜑 → ((♯‘(1...𝑀)) · (♯‘(1...𝑁))) = (𝑀 · 𝑁))
163	152, 162	eqtrd 2799	. . . 4 ⊢ (𝜑 → (♯‘((1...𝑀) × (1...𝑁))) = (𝑀 · 𝑁))
164	132, 150, 163	3eqtr3d 2807	. . 3 ⊢ (𝜑 → ((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆)) = (𝑀 · 𝑁))
165	164	oveq2d 6858	. 2 ⊢ (𝜑 → (-1↑((♯‘{⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ (1...𝑀) ∧ 𝑦 ∈ (1...𝑁)) ∧ (𝑥 · 𝑄) < (𝑦 · 𝑃))}) + (♯‘𝑆))) = (-1↑(𝑀 · 𝑁)))
166	37, 48, 165	3eqtr2d 2805	1 ⊢ (𝜑 → ((𝑃 /_L 𝑄) · (𝑄 /_L 𝑃)) = (-1↑(𝑀 · 𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 197 ∧ wa 384 ∨ wo 873 = wceq 1652 ∃wex 1874 ∈ wcel 2155 ≠ wne 2937 ∖ cdif 3729 ∪ cun 3730 ∩ cin 3731 ⊆ wss 3732 ∅c0 4079 {csn 4334 class class class wbr 4809 {copab 4871 × cxp 5275 ^◡ccnv 5276 Rel wrel 5282 ‘cfv 6068 (class class class)co 6842 ≈ cen 8157 Fincfn 8160 ℂcc 10187 ℝcr 10188 1c1 10190 + caddc 10192 · cmul 10194 < clt 10328 ≤ cle 10329 − cmin 10520 -cneg 10521 / cdiv 10938 ℕcn 11274 2c2 11327 ℕ₀cn0 11538 ℤcz 11624 ℤ_≥cuz 11886 ℝ⁺crp 12028 ...cfz 12533 ↑cexp 13067 ♯chash 13321 ∥ cdvds 15267 gcd cgcd 15499 ℙcprime 15667 /_L clgs 25310

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1890 ax-4 1904 ax-5 2005 ax-6 2069 ax-7 2105 ax-8 2157 ax-9 2164 ax-10 2183 ax-11 2198 ax-12 2211 ax-13 2352 ax-ext 2743 ax-rep 4930 ax-sep 4941 ax-nul 4949 ax-pow 5001 ax-pr 5062 ax-un 7147 ax-inf2 8753 ax-cnex 10245 ax-resscn 10246 ax-1cn 10247 ax-icn 10248 ax-addcl 10249 ax-addrcl 10250 ax-mulcl 10251 ax-mulrcl 10252 ax-mulcom 10253 ax-addass 10254 ax-mulass 10255 ax-distr 10256 ax-i2m1 10257 ax-1ne0 10258 ax-1rid 10259 ax-rnegex 10260 ax-rrecex 10261 ax-cnre 10262 ax-pre-lttri 10263 ax-pre-lttrn 10264 ax-pre-ltadd 10265 ax-pre-mulgt0 10266 ax-pre-sup 10267 ax-addf 10268 ax-mulf 10269

This theorem depends on definitions: df-bi 198 df-an 385 df-or 874 df-3or 1108 df-3an 1109 df-tru 1656 df-fal 1666 df-ex 1875 df-nf 1879 df-sb 2062 df-mo 2565 df-eu 2582 df-clab 2752 df-cleq 2758 df-clel 2761 df-nfc 2896 df-ne 2938 df-nel 3041 df-ral 3060 df-rex 3061 df-reu 3062 df-rmo 3063 df-rab 3064 df-v 3352 df-sbc 3597 df-csb 3692 df-dif 3735 df-un 3737 df-in 3739 df-ss 3746 df-pss 3748 df-nul 4080 df-if 4244 df-pw 4317 df-sn 4335 df-pr 4337 df-tp 4339 df-op 4341 df-uni 4595 df-int 4634 df-iun 4678 df-disj 4778 df-br 4810 df-opab 4872 df-mpt 4889 df-tr 4912 df-id 5185 df-eprel 5190 df-po 5198 df-so 5199 df-fr 5236 df-se 5237 df-we 5238 df-xp 5283 df-rel 5284 df-cnv 5285 df-co 5286 df-dm 5287 df-rn 5288 df-res 5289 df-ima 5290 df-pred 5865 df-ord 5911 df-on 5912 df-lim 5913 df-suc 5914 df-iota 6031 df-fun 6070 df-fn 6071 df-f 6072 df-f1 6073 df-fo 6074 df-f1o 6075 df-fv 6076 df-isom 6077 df-riota 6803 df-ov 6845 df-oprab 6846 df-mpt2 6847 df-of 7095 df-om 7264 df-1st 7366 df-2nd 7367 df-supp 7498 df-tpos 7555 df-wrecs 7610 df-recs 7672 df-rdg 7710 df-1o 7764 df-2o 7765 df-oadd 7768 df-er 7947 df-ec 7949 df-qs 7953 df-map 8062 df-en 8161 df-dom 8162 df-sdom 8163 df-fin 8164 df-fsupp 8483 df-sup 8555 df-inf 8556 df-oi 8622 df-card 9016 df-cda 9243 df-pnf 10330 df-mnf 10331 df-xr 10332 df-ltxr 10333 df-le 10334 df-sub 10522 df-neg 10523 df-div 10939 df-nn 11275 df-2 11335 df-3 11336 df-4 11337 df-5 11338 df-6 11339 df-7 11340 df-8 11341 df-9 11342 df-n0 11539 df-xnn0 11611 df-z 11625 df-dec 11741 df-uz 11887 df-q 11990 df-rp 12029 df-fz 12534 df-fzo 12674 df-fl 12801 df-mod 12877 df-seq 13009 df-exp 13068 df-hash 13322 df-cj 14126 df-re 14127 df-im 14128 df-sqrt 14262 df-abs 14263 df-clim 14506 df-sum 14704 df-dvds 15268 df-gcd 15500 df-prm 15668 df-phi 15752 df-pc 15823 df-struct 16134 df-ndx 16135 df-slot 16136 df-base 16138 df-sets 16139 df-ress 16140 df-plusg 16229 df-mulr 16230 df-starv 16231 df-sca 16232 df-vsca 16233 df-ip 16234 df-tset 16235 df-ple 16236 df-ds 16238 df-unif 16239 df-0g 16370 df-gsum 16371 df-imas 16436 df-qus 16437 df-mgm 17510 df-sgrp 17552 df-mnd 17563 df-mhm 17603 df-submnd 17604 df-grp 17694 df-minusg 17695 df-sbg 17696 df-mulg 17810 df-subg 17857 df-nsg 17858 df-eqg 17859 df-ghm 17924 df-cntz 18015 df-cmn 18461 df-abl 18462 df-mgp 18757 df-ur 18769 df-ring 18816 df-cring 18817 df-oppr 18890 df-dvdsr 18908 df-unit 18909 df-invr 18939 df-dvr 18950 df-rnghom 18984 df-drng 19018 df-field 19019 df-subrg 19047 df-lmod 19134 df-lss 19202 df-lsp 19244 df-sra 19446 df-rgmod 19447 df-lidl 19448 df-rsp 19449 df-2idl 19506 df-nzr 19532 df-rlreg 19557 df-domn 19558 df-idom 19559 df-cnfld 20020 df-zring 20092 df-zrh 20125 df-zn 20128 df-lgs 25311

This theorem is referenced by: lgsquad 25399

W3C validator