Theorem sqrt2irr 12528

Description: The square root of 2 is not rational. That is, for any rational number, (√‘2) does not equal it. However, if we were to say "the square root of 2 is irrational" that would mean something stronger: "for any rational number, (√‘2) is apart from it" (the two statements are equivalent given excluded middle). See sqrt2irrap 12546 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12527, which shows that if 𝐴 / 𝐵 = √(2), then 𝐴 and 𝐵 are even, so 𝐴 / 2 and 𝐵 / 2 are smaller representatives, which is absurd. (Contributed by NM, 8-Jan-2002.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
sqrt2irr	⊢ (√‘2) ∉ ℚ

Proof of Theorem sqrt2irr

Dummy variables 𝑥 𝑛 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		peano2nn 9055	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
2		breq2 4051	. . . . . . . . 9 ⊢ (𝑛 = 1 → (𝑧 < 𝑛 ↔ 𝑧 < 1))
3	2	imbi1d 231	. . . . . . . 8 ⊢ (𝑛 = 1 → ((𝑧 < 𝑛 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ (𝑧 < 1 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
4	3	ralbidv 2507	. . . . . . 7 ⊢ (𝑛 = 1 → (∀𝑧 ∈ ℕ (𝑧 < 𝑛 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ∀𝑧 ∈ ℕ (𝑧 < 1 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
5		breq2 4051	. . . . . . . . 9 ⊢ (𝑛 = 𝑦 → (𝑧 < 𝑛 ↔ 𝑧 < 𝑦))
6	5	imbi1d 231	. . . . . . . 8 ⊢ (𝑛 = 𝑦 → ((𝑧 < 𝑛 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
7	6	ralbidv 2507	. . . . . . 7 ⊢ (𝑛 = 𝑦 → (∀𝑧 ∈ ℕ (𝑧 < 𝑛 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
8		breq2 4051	. . . . . . . . 9 ⊢ (𝑛 = (𝑦 + 1) → (𝑧 < 𝑛 ↔ 𝑧 < (𝑦 + 1)))
9	8	imbi1d 231	. . . . . . . 8 ⊢ (𝑛 = (𝑦 + 1) → ((𝑧 < 𝑛 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ (𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
10	9	ralbidv 2507	. . . . . . 7 ⊢ (𝑛 = (𝑦 + 1) → (∀𝑧 ∈ ℕ (𝑧 < 𝑛 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
11		nnnlt1 9069	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → ¬ 𝑧 < 1)
12	11	pm2.21d 620	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ → (𝑧 < 1 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)))
13	12	rgen 2560	. . . . . . 7 ⊢ ∀𝑧 ∈ ℕ (𝑧 < 1 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))
14		nnrp 9792	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ+)
15		rphalflt 9812	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
16	14, 15	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → (𝑦 / 2) < 𝑦)
17		breq1 4050	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑦 / 2) → (𝑧 < 𝑦 ↔ (𝑦 / 2) < 𝑦))
18		oveq2 5959	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦 / 2) → (𝑥 / 𝑧) = (𝑥 / (𝑦 / 2)))
19	18	neeq2d 2396	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 / 2) → ((√‘2) ≠ (𝑥 / 𝑧) ↔ (√‘2) ≠ (𝑥 / (𝑦 / 2))))
20	19	ralbidv 2507	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑦 / 2) → (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧) ↔ ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2))))
21	17, 20	imbi12d 234	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = (𝑦 / 2) → ((𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ((𝑦 / 2) < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2)))))
22	21	rspcv 2874	. . . . . . . . . . . . . 14 ⊢ ((𝑦 / 2) ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → ((𝑦 / 2) < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2)))))
23	22	com13 80	. . . . . . . . . . . . 13 ⊢ ((𝑦 / 2) < 𝑦 → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → ((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2)))))
24	16, 23	syl 14	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → ((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2)))))
25		simpr 110	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → (√‘2) = (𝑧 / 𝑦))
26		zcn 9384	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
27	26	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 𝑧 ∈ ℂ)
28		nncn 9051	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
29	28	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 𝑦 ∈ ℂ)
30		2cnd 9116	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 2 ∈ ℂ)
31		nnap0 9072	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0)
32	31	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 𝑦 # 0)
33		2ap0 9136	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 # 0
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 2 # 0)
35	27, 29, 30, 32, 34	divcanap7d 8899	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → ((𝑧 / 2) / (𝑦 / 2)) = (𝑧 / 𝑦))
36	25, 35	eqtr4d 2242	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → (√‘2) = ((𝑧 / 2) / (𝑦 / 2)))
37		simplr 528	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 𝑧 ∈ ℤ)
38		simpll 527	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 𝑦 ∈ ℕ)
39	37, 38, 25	sqrt2irrlem 12527	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → ((𝑧 / 2) ∈ ℤ ∧ (𝑦 / 2) ∈ ℕ))
40	39	simprd 114	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → (𝑦 / 2) ∈ ℕ)
41	39	simpld 112	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → (𝑧 / 2) ∈ ℤ)
42		oveq1 5958	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑧 / 2) → (𝑥 / (𝑦 / 2)) = ((𝑧 / 2) / (𝑦 / 2)))
43	42	neeq2d 2396	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑧 / 2) → ((√‘2) ≠ (𝑥 / (𝑦 / 2)) ↔ (√‘2) ≠ ((𝑧 / 2) / (𝑦 / 2))))
44	43	rspcv 2874	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑧 / 2) ∈ ℤ → (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2)) → (√‘2) ≠ ((𝑧 / 2) / (𝑦 / 2))))
45	41, 44	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2)) → (√‘2) ≠ ((𝑧 / 2) / (𝑦 / 2))))
46	40, 45	embantd 56	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → (((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2))) → (√‘2) ≠ ((𝑧 / 2) / (𝑦 / 2))))
47	46	necon2bd 2435	. . . . . . . . . . . . . . . 16 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → ((√‘2) = ((𝑧 / 2) / (𝑦 / 2)) → ¬ ((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2)))))
48	36, 47	mpd 13	. . . . . . . . . . . . . . 15 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → ¬ ((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2))))
49	48	ex 115	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) → ((√‘2) = (𝑧 / 𝑦) → ¬ ((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2)))))
50	49	necon2ad 2434	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) → (((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2))) → (√‘2) ≠ (𝑧 / 𝑦)))
51	50	ralrimdva 2587	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → (((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2))) → ∀𝑧 ∈ ℤ (√‘2) ≠ (𝑧 / 𝑦)))
52	24, 51	syld 45	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → ∀𝑧 ∈ ℤ (√‘2) ≠ (𝑧 / 𝑦)))
53		oveq1 5958	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 / 𝑦) = (𝑧 / 𝑦))
54	53	neeq2d 2396	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((√‘2) ≠ (𝑥 / 𝑦) ↔ (√‘2) ≠ (𝑧 / 𝑦)))
55	54	cbvralv 2739	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑦) ↔ ∀𝑧 ∈ ℤ (√‘2) ≠ (𝑧 / 𝑦))
56	52, 55	imbitrrdi 162	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑦)))
57		oveq2 5959	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝑥 / 𝑧) = (𝑥 / 𝑦))
58	57	neeq2d 2396	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((√‘2) ≠ (𝑥 / 𝑧) ↔ (√‘2) ≠ (𝑥 / 𝑦)))
59	58	ralbidv 2507	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧) ↔ ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑦)))
60	59	ceqsralv 2804	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑦)))
61	56, 60	sylibrd 169	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → ∀𝑧 ∈ ℕ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
62	61	ancld 325	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ∧ ∀𝑧 ∈ ℕ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)))))
63		nnleltp1 9439	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑧 ≤ 𝑦 ↔ 𝑧 < (𝑦 + 1)))
64		nnz 9398	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
65		nnz 9398	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
66		zleloe 9426	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ ℤ ∧ 𝑦 ∈ ℤ) → (𝑧 ≤ 𝑦 ↔ (𝑧 < 𝑦 ∨ 𝑧 = 𝑦)))
67	64, 65, 66	syl2an 289	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑧 ≤ 𝑦 ↔ (𝑧 < 𝑦 ∨ 𝑧 = 𝑦)))
68	63, 67	bitr3d 190	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑧 < (𝑦 + 1) ↔ (𝑧 < 𝑦 ∨ 𝑧 = 𝑦)))
69	68	ancoms 268	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑧 < (𝑦 + 1) ↔ (𝑧 < 𝑦 ∨ 𝑧 = 𝑦)))
70	69	imbi1d 231	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ((𝑧 < 𝑦 ∨ 𝑧 = 𝑦) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
71		jaob 712	. . . . . . . . . . 11 ⊢ (((𝑧 < 𝑦 ∨ 𝑧 = 𝑦) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ((𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ∧ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
72	70, 71	bitrdi 196	. . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ((𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ∧ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)))))
73	72	ralbidva 2503	. . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ∀𝑧 ∈ ℕ ((𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ∧ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)))))
74		r19.26 2633	. . . . . . . . 9 ⊢ (∀𝑧 ∈ ℕ ((𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ∧ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))) ↔ (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ∧ ∀𝑧 ∈ ℕ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
75	73, 74	bitrdi 196	. . . . . . . 8 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ∧ ∀𝑧 ∈ ℕ (𝑧 = 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)))))
76	62, 75	sylibrd 169	. . . . . . 7 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → ∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
77	4, 7, 10, 10, 13, 76	nnind 9059	. . . . . 6 ⊢ ((𝑦 + 1) ∈ ℕ → ∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)))
78	1, 77	syl 14	. . . . 5 ⊢ (𝑦 ∈ ℕ → ∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)))
79		nnre 9050	. . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
80	79	ltp1d 9010	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 < (𝑦 + 1))
81		breq1 4050	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 < (𝑦 + 1) ↔ 𝑦 < (𝑦 + 1)))
82		df-ne 2378	. . . . . . . . . 10 ⊢ ((√‘2) ≠ (𝑥 / 𝑦) ↔ ¬ (√‘2) = (𝑥 / 𝑦))
83	58, 82	bitrdi 196	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((√‘2) ≠ (𝑥 / 𝑧) ↔ ¬ (√‘2) = (𝑥 / 𝑦)))
84	83	ralbidv 2507	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧) ↔ ∀𝑥 ∈ ℤ ¬ (√‘2) = (𝑥 / 𝑦)))
85		ralnex 2495	. . . . . . . 8 ⊢ (∀𝑥 ∈ ℤ ¬ (√‘2) = (𝑥 / 𝑦) ↔ ¬ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦))
86	84, 85	bitrdi 196	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧) ↔ ¬ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦)))
87	81, 86	imbi12d 234	. . . . . 6 ⊢ (𝑧 = 𝑦 → ((𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ (𝑦 < (𝑦 + 1) → ¬ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦))))
88	87	rspcv 2874	. . . . 5 ⊢ (𝑦 ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 < (𝑦 + 1) → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) → (𝑦 < (𝑦 + 1) → ¬ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦))))
89	78, 80, 88	mp2d 47	. . . 4 ⊢ (𝑦 ∈ ℕ → ¬ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦))
90	89	nrex 2599	. . 3 ⊢ ¬ ∃𝑦 ∈ ℕ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦)
91		elq 9750	. . . 4 ⊢ ((√‘2) ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (√‘2) = (𝑥 / 𝑦))
92		rexcom 2671	. . . 4 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (√‘2) = (𝑥 / 𝑦) ↔ ∃𝑦 ∈ ℕ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦))
93	91, 92	bitri 184	. . 3 ⊢ ((√‘2) ∈ ℚ ↔ ∃𝑦 ∈ ℕ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦))
94	90, 93	mtbir 673	. 2 ⊢ ¬ (√‘2) ∈ ℚ
95	94	nelir 2475	1 ⊢ (√‘2) ∉ ℚ

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 710   = wceq 1373   ∈ wcel 2177   ≠ wne 2377   ∉ wnel 2472  ∀wral 2485  ∃wrex 2486   class class class wbr 4047  ‘cfv 5276  (class class class)co 5951  ℂcc 7930  0cc0 7932  1c1 7933   + caddc 7935   < clt 8114   ≤ cle 8115   # cap 8661   / cdiv 8752  ℕcn 9043  2c2 9094  ℤcz 9379  ℚcq 9747  ℝ⁺crp 9782  √csqrt 11351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4163  ax-sep 4166  ax-nul 4174  ax-pow 4222  ax-pr 4257  ax-un 4484  ax-setind 4589  ax-iinf 4640  ax-cnex 8023  ax-resscn 8024  ax-1cn 8025  ax-1re 8026  ax-icn 8027  ax-addcl 8028  ax-addrcl 8029  ax-mulcl 8030  ax-mulrcl 8031  ax-addcom 8032  ax-mulcom 8033  ax-addass 8034  ax-mulass 8035  ax-distr 8036  ax-i2m1 8037  ax-0lt1 8038  ax-1rid 8039  ax-0id 8040  ax-rnegex 8041  ax-precex 8042  ax-cnre 8043  ax-pre-ltirr 8044  ax-pre-ltwlin 8045  ax-pre-lttrn 8046  ax-pre-apti 8047  ax-pre-ltadd 8048  ax-pre-mulgt0 8049  ax-pre-mulext 8050  ax-arch 8051  ax-caucvg 8052

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3000  df-csb 3095  df-dif 3169  df-un 3171  df-in 3173  df-ss 3180  df-nul 3462  df-if 3573  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-uni 3853  df-int 3888  df-iun 3931  df-br 4048  df-opab 4110  df-mpt 4111  df-tr 4147  df-id 4344  df-po 4347  df-iso 4348  df-iord 4417  df-on 4419  df-ilim 4420  df-suc 4422  df-iom 4643  df-xp 4685  df-rel 4686  df-cnv 4687  df-co 4688  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692  df-iota 5237  df-fun 5278  df-fn 5279  df-f 5280  df-f1 5281  df-fo 5282  df-f1o 5283  df-fv 5284  df-riota 5906  df-ov 5954  df-oprab 5955  df-mpo 5956  df-1st 6233  df-2nd 6234  df-recs 6398  df-frec 6484  df-pnf 8116  df-mnf 8117  df-xr 8118  df-ltxr 8119  df-le 8120  df-sub 8252  df-neg 8253  df-reap 8655  df-ap 8662  df-div 8753  df-inn 9044  df-2 9102  df-3 9103  df-4 9104  df-n0 9303  df-z 9380  df-uz 9656  df-q 9748  df-rp 9783  df-seqfrec 10600  df-exp 10691  df-rsqrt 11353

This theorem is referenced by:  sqrt2irr0  12530