Theorem sqrt2irr 12727

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 12745 for the proof that the square root of two is irrational.

The proof's core is proven in sqrt2irrlem 12726, 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 9148	. . . . . 6 ⊢ (𝑦 ∈ ℕ → (𝑦 + 1) ∈ ℕ)
2		breq2 4090	. . . . . . . . 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 2530	. . . . . . 7 ⊢ (𝑛 = 1 → (∀𝑧 ∈ ℕ (𝑧 < 𝑛 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ∀𝑧 ∈ ℕ (𝑧 < 1 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
5		breq2 4090	. . . . . . . . 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 2530	. . . . . . 7 ⊢ (𝑛 = 𝑦 → (∀𝑧 ∈ ℕ (𝑧 < 𝑛 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)) ↔ ∀𝑧 ∈ ℕ (𝑧 < 𝑦 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))))
8		breq2 4090	. . . . . . . . 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 2530	. . . . . . 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 9162	. . . . . . . . 9 ⊢ (𝑧 ∈ ℕ → ¬ 𝑧 < 1)
12	11	pm2.21d 622	. . . . . . . 8 ⊢ (𝑧 ∈ ℕ → (𝑧 < 1 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧)))
13	12	rgen 2583	. . . . . . 7 ⊢ ∀𝑧 ∈ ℕ (𝑧 < 1 → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧))
14		nnrp 9891	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ+)
15		rphalflt 9911	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ℝ+ → (𝑦 / 2) < 𝑦)
16	14, 15	syl 14	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ ℕ → (𝑦 / 2) < 𝑦)
17		breq1 4089	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = (𝑦 / 2) → (𝑧 < 𝑦 ↔ (𝑦 / 2) < 𝑦))
18		oveq2 6021	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦 / 2) → (𝑥 / 𝑧) = (𝑥 / (𝑦 / 2)))
19	18	neeq2d 2419	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦 / 2) → ((√‘2) ≠ (𝑥 / 𝑧) ↔ (√‘2) ≠ (𝑥 / (𝑦 / 2))))
20	19	ralbidv 2530	. . . . . . . . . . . . . . . 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 2904	. . . . . . . . . . . . . 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 9477	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℂ)
27	26	ad2antlr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 𝑧 ∈ ℂ)
28		nncn 9144	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ)
29	28	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 𝑦 ∈ ℂ)
30		2cnd 9209	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 2 ∈ ℂ)
31		nnap0 9165	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℕ → 𝑦 # 0)
32	31	ad2antrr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 𝑦 # 0)
33		2ap0 9229	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 # 0
34	33	a1i 9	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → 2 # 0)
35	27, 29, 30, 32, 34	divcanap7d 8992	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) ∧ (√‘2) = (𝑧 / 𝑦)) → ((𝑧 / 2) / (𝑦 / 2)) = (𝑧 / 𝑦))
36	25, 35	eqtr4d 2265	. . . . . . . . . . . . . . . 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 12726	. . . . . . . . . . . . . . . . . . 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 6020	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = (𝑧 / 2) → (𝑥 / (𝑦 / 2)) = ((𝑧 / 2) / (𝑦 / 2)))
43	42	neeq2d 2419	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = (𝑧 / 2) → ((√‘2) ≠ (𝑥 / (𝑦 / 2)) ↔ (√‘2) ≠ ((𝑧 / 2) / (𝑦 / 2))))
44	43	rspcv 2904	. . . . . . . . . . . . . . . . . . 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 2458	. . . . . . . . . . . . . . . 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 2457	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ ℕ ∧ 𝑧 ∈ ℤ) → (((𝑦 / 2) ∈ ℕ → ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / (𝑦 / 2))) → (√‘2) ≠ (𝑧 / 𝑦)))
51	50	ralrimdva 2610	. . . . . . . . . . . 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 6020	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 / 𝑦) = (𝑧 / 𝑦))
54	53	neeq2d 2419	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((√‘2) ≠ (𝑥 / 𝑦) ↔ (√‘2) ≠ (𝑧 / 𝑦)))
55	54	cbvralv 2765	. . . . . . . . . . 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 6021	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑦 → (𝑥 / 𝑧) = (𝑥 / 𝑦))
58	57	neeq2d 2419	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑦 → ((√‘2) ≠ (𝑥 / 𝑧) ↔ (√‘2) ≠ (𝑥 / 𝑦)))
59	58	ralbidv 2530	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧) ↔ ∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑦)))
60	59	ceqsralv 2832	. . . . . . . . . 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 9532	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ ℕ ∧ 𝑦 ∈ ℕ) → (𝑧 ≤ 𝑦 ↔ 𝑧 < (𝑦 + 1)))
64		nnz 9491	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℤ)
65		nnz 9491	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℤ)
66		zleloe 9519	. . . . . . . . . . . . . . 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 715	. . . . . . . . . . 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 2526	. . . . . . . . 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 2657	. . . . . . . . 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 9152	. . . . . 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 9143	. . . . . 6 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℝ)
80	79	ltp1d 9103	. . . . 5 ⊢ (𝑦 ∈ ℕ → 𝑦 < (𝑦 + 1))
81		breq1 4089	. . . . . . 7 ⊢ (𝑧 = 𝑦 → (𝑧 < (𝑦 + 1) ↔ 𝑦 < (𝑦 + 1)))
82		df-ne 2401	. . . . . . . . . 10 ⊢ ((√‘2) ≠ (𝑥 / 𝑦) ↔ ¬ (√‘2) = (𝑥 / 𝑦))
83	58, 82	bitrdi 196	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → ((√‘2) ≠ (𝑥 / 𝑧) ↔ ¬ (√‘2) = (𝑥 / 𝑦)))
84	83	ralbidv 2530	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → (∀𝑥 ∈ ℤ (√‘2) ≠ (𝑥 / 𝑧) ↔ ∀𝑥 ∈ ℤ ¬ (√‘2) = (𝑥 / 𝑦)))
85		ralnex 2518	. . . . . . . 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 2904	. . . . 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 2622	. . 3 ⊢ ¬ ∃𝑦 ∈ ℕ ∃𝑥 ∈ ℤ (√‘2) = (𝑥 / 𝑦)
91		elq 9849	. . . 4 ⊢ ((√‘2) ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ (√‘2) = (𝑥 / 𝑦))
92		rexcom 2695	. . . 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 675	. 2 ⊢ ¬ (√‘2) ∈ ℚ
95	94	nelir 2498	1 ⊢ (√‘2) ∉ ℚ

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200   ≠ wne 2400   ∉ wnel 2495  ∀wral 2508  ∃wrex 2509   class class class wbr 4086  ‘cfv 5324  (class class class)co 6013  ℂcc 8023  0cc0 8025  1c1 8026   + caddc 8028   < clt 8207   ≤ cle 8208   # cap 8754   / cdiv 8845  ℕcn 9136  2c2 9187  ℤcz 9472  ℚcq 9846  ℝ⁺crp 9881  √csqrt 11550

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-seqfrec 10703  df-exp 10794  df-rsqrt 11552

This theorem is referenced by:  sqrt2irr0  12729