Theorem sqrt2irrap 12710

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 12692. (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 9825	. . 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 535	. . . . . . . . 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 536	. . . . . . . . 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 9827	. . . . . . . . 9 ⊢ ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ) → (𝑎 / 𝑏) ∈ ℚ)
8		qre 9828	. . . . . . . . 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 12693	. . . . . . . 8 ⊢ (√‘2) ∈ ℝ
12	11	a1i 9	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (√‘2) ∈ ℝ)
13		0red 8155	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 ∈ ℝ)
14	4	zcnd 9578	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ∈ ℂ)
15	6	nncnd 9132	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 ∈ ℂ)
16	6	nnap0d 9164	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 # 0)
17	14, 15, 16	divrecapd 8948	. . . . . . . . 9 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) = (𝑎 · (1 / 𝑏)))
18	4	zred 9577	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ∈ ℝ)
19	6	nnrecred 9165	. . . . . . . . . 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 8169	. . . . . . . . . . 11 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 1 ∈ ℝ)
22	6	nnrpd 9898	. . . . . . . . . . 11 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 ∈ ℝ+)
23		0le1 8636	. . . . . . . . . . . 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 9941	. . . . . . . . . 10 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 ≤ (1 / 𝑏))
26		mulle0r 9099	. . . . . . . . . 10 ⊢ (((𝑎 ∈ ℝ ∧ (1 / 𝑏) ∈ ℝ) ∧ (𝑎 ≤ 0 ∧ 0 ≤ (1 / 𝑏))) → (𝑎 · (1 / 𝑏)) ≤ 0)
27	18, 19, 20, 25, 26	syl22anc 1272	. . . . . . . . 9 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 · (1 / 𝑏)) ≤ 0)
28	17, 27	eqbrtrd 4105	. . . . . . . 8 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) ≤ 0)
29		2re 9188	. . . . . . . . . 10 ⊢ 2 ∈ ℝ
30		2pos 9209	. . . . . . . . . 10 ⊢ 0 < 2
31	29, 30	sqrtgt0ii 11650	. . . . . . . . 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 8279	. . . . . . 7 ⊢ ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) < (√‘2))
34	10, 12, 33	gtapd 8792	. . . . . 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 9464	. . . . . . . 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 12709	. . . . . . 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 9465	. . . . . . . . 9 ⊢ 0 ∈ ℤ
43		zlelttric 9499	. . . . . . . . 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 803	. . . . 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 4111	. . . 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 2656	. 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 713   = wceq 1395   ∈ wcel 2200  ∃wrex 2509   class class class wbr 4083  ‘cfv 5318  (class class class)co 6007  ℝcr 8006  0cc0 8007  1c1 8008   · cmul 8012   < clt 8189   ≤ cle 8190   # cap 8736   / cdiv 8827  ℕcn 9118  2c2 9169  ℤcz 9454  ℚcq 9822  √csqrt 11515

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-mulrcl 8106  ax-addcom 8107  ax-mulcom 8108  ax-addass 8109  ax-mulass 8110  ax-distr 8111  ax-i2m1 8112  ax-0lt1 8113  ax-1rid 8114  ax-0id 8115  ax-rnegex 8116  ax-precex 8117  ax-cnre 8118  ax-pre-ltirr 8119  ax-pre-ltwlin 8120  ax-pre-lttrn 8121  ax-pre-apti 8122  ax-pre-ltadd 8123  ax-pre-mulgt0 8124  ax-pre-mulext 8125  ax-arch 8126  ax-caucvg 8127

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-xor 1418  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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-po 4387  df-iso 4388  df-iord 4457  df-on 4459  df-ilim 4460  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-recs 6457  df-frec 6543  df-1o 6568  df-2o 6569  df-er 6688  df-en 6896  df-sup 7159  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194  df-le 8195  df-sub 8327  df-neg 8328  df-reap 8730  df-ap 8737  df-div 8828  df-inn 9119  df-2 9177  df-3 9178  df-4 9179  df-n0 9378  df-z 9455  df-uz 9731  df-q 9823  df-rp 9858  df-fz 10213  df-fzo 10347  df-fl 10498  df-mod 10553  df-seqfrec 10678  df-exp 10769  df-cj 11361  df-re 11362  df-im 11363  df-rsqrt 11517  df-abs 11518  df-dvds 12307  df-gcd 12483  df-prm 12638

This theorem is referenced by:  2irrexpqap  15660