Theorem sqrt2irrap 12775

Description: The square root of 2 is irrational. That is, for any rational number, (√‘2) is apart from it. In the absence of excluded middle, we can distinguish between this and "the square root of 2 is not rational" which is sqrt2irr 12757. (Contributed by Jim Kingdon, 2-Oct-2021.)

Assertion

Ref	Expression
sqrt2irrap	⊢ (𝑄 ∈ ℚ → (√‘2) # 𝑄)

Proof of Theorem sqrt2irrap

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elq 9861	. . 3 ⊢ (𝑄 ∈ ℚ ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℕ 𝑄 = (𝑎 / 𝑏))
2	1	biimpi 120	. 2 ⊢ (𝑄 ∈ ℚ → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℕ 𝑄 = (𝑎 / 𝑏))
3		simplrl 537	. . . . . . . . 9 ⊢ (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → 𝑎 ∈ ℤ)
4	3	adantr 276	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ∈ ℤ)
5		simplrr 538	. . . . . . . . 9 ⊢ (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → 𝑏 ∈ ℕ)
6	5	adantr 276	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 ∈ ℕ)
7		znq 9863	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ) → (𝑎 / 𝑏) ∈ ℚ)
8		qre 9864	. . . . . . . . 9 ⊢ ((𝑎 / 𝑏) ∈ ℚ → (𝑎 / 𝑏) ∈ ℝ)
9	7, 8	syl 14	. . . . . . . 8 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ) → (𝑎 / 𝑏) ∈ ℝ)
10	4, 6, 9	syl2anc 411	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) ∈ ℝ)
11		sqrt2re 12758	. . . . . . . 8 ⊢ (√‘2) ∈ ℝ
12	11	a1i 9	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (√‘2) ∈ ℝ)
13		0red 8185	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 ∈ ℝ)
14	4	zcnd 9608	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ∈ ℂ)
15	6	nncnd 9162	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 ∈ ℂ)
16	6	nnap0d 9194	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 # 0)
17	14, 15, 16	divrecapd 8978	. . . . . . . . 9 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) = (𝑎 · (1 / 𝑏)))
18	4	zred 9607	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ∈ ℝ)
19	6	nnrecred 9195	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (1 / 𝑏) ∈ ℝ)
20		simpr 110	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ≤ 0)
21		1red 8199	. . . . . . . . . . 11 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 1 ∈ ℝ)
22	6	nnrpd 9934	. . . . . . . . . . 11 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 ∈ ℝ+)
23		0le1 8666	. . . . . . . . . . . 12 ⊢ 0 ≤ 1
24	23	a1i 9	. . . . . . . . . . 11 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 ≤ 1)
25	21, 22, 24	divge0d 9977	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 ≤ (1 / 𝑏))
26		mulle0r 9129	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℝ ∧ (1 / 𝑏) ∈ ℝ) ∧ (𝑎 ≤ 0 ∧ 0 ≤ (1 / 𝑏))) → (𝑎 · (1 / 𝑏)) ≤ 0)
27	18, 19, 20, 25, 26	syl22anc 1274	. . . . . . . . 9 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 · (1 / 𝑏)) ≤ 0)
28	17, 27	eqbrtrd 4111	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) ≤ 0)
29		2re 9218	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
30		2pos 9239	. . . . . . . . . 10 ⊢ 0 < 2
31	29, 30	sqrtgt0ii 11714	. . . . . . . . 9 ⊢ 0 < (√‘2)
32	31	a1i 9	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 < (√‘2))
33	10, 13, 12, 28, 32	lelttrd 8309	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) < (√‘2))
34	10, 12, 33	gtapd 8822	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (√‘2) # (𝑎 / 𝑏))
35	3	adantr 276	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → 𝑎 ∈ ℤ)
36		simpr 110	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → 0 < 𝑎)
37		elnnz 9494	. . . . . . . 8 ⊢ (𝑎 ∈ ℕ ↔ (𝑎 ∈ ℤ ∧ 0 < 𝑎))
38	35, 36, 37	sylanbrc 417	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → 𝑎 ∈ ℕ)
39	5	adantr 276	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → 𝑏 ∈ ℕ)
40		sqrt2irraplemnn 12774	. . . . . . 7 ⊢ ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (√‘2) # (𝑎 / 𝑏))
41	38, 39, 40	syl2anc 411	. . . . . 6 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → (√‘2) # (𝑎 / 𝑏))
42		0z 9495	. . . . . . . . 9 ⊢ 0 ∈ ℤ
43		zlelttric 9529	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑎 ≤ 0 ∨ 0 < 𝑎))
44	42, 43	mpan2 425	. . . . . . . 8 ⊢ (𝑎 ∈ ℤ → (𝑎 ≤ 0 ∨ 0 < 𝑎))
45	44	ad2antrl 490	. . . . . . 7 ⊢ ((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) → (𝑎 ≤ 0 ∨ 0 < 𝑎))
46	45	adantr 276	. . . . . 6 ⊢ (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → (𝑎 ≤ 0 ∨ 0 < 𝑎))
47	34, 41, 46	mpjaodan 805	. . . . 5 ⊢ (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → (√‘2) # (𝑎 / 𝑏))
48		simpr 110	. . . . 5 ⊢ (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → 𝑄 = (𝑎 / 𝑏))
49	47, 48	breqtrrd 4117	. . . 4 ⊢ (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → (√‘2) # 𝑄)
50	49	ex 115	. . 3 ⊢ ((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) → (𝑄 = (𝑎 / 𝑏) → (√‘2) # 𝑄))
51	50	rexlimdvva 2657	. 2 ⊢ (𝑄 ∈ ℚ → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℕ 𝑄 = (𝑎 / 𝑏) → (√‘2) # 𝑄))
52	2, 51	mpd 13	1 ⊢ (𝑄 ∈ ℚ → (√‘2) # 𝑄)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 715   = wceq 1397   ∈ wcel 2201  ∃wrex 2510   class class class wbr 4089  ‘cfv 5328  (class class class)co 6023  ℝcr 8036  0cc0 8037  1c1 8038   · cmul 8042   < clt 8219   ≤ cle 8220   # cap 8766   / cdiv 8857  ℕcn 9148  2c2 9199  ℤcz 9484  ℚcq 9858  √csqrt 11579

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155  ax-arch 8156  ax-caucvg 8157

This theorem depends on definitions:  df-bi 117  df-stab 838  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-xor 1420  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-1o 6587  df-2o 6588  df-er 6707  df-en 6915  df-sup 7188  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-n0 9408  df-z 9485  df-uz 9761  df-q 9859  df-rp 9894  df-fz 10249  df-fzo 10383  df-fl 10536  df-mod 10591  df-seqfrec 10716  df-exp 10807  df-cj 11425  df-re 11426  df-im 11427  df-rsqrt 11581  df-abs 11582  df-dvds 12372  df-gcd 12548  df-prm 12703

This theorem is referenced by:  2irrexpqap  15731