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Theorem sqrt2irrap 10952
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 10935. (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.
StepHypRef Expression
1 elq 9016 . . 3 (𝑄 ∈ ℚ ↔ ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℕ 𝑄 = (𝑎 / 𝑏))
21biimpi 118 . 2 (𝑄 ∈ ℚ → ∃𝑎 ∈ ℤ ∃𝑏 ∈ ℕ 𝑄 = (𝑎 / 𝑏))
3 simplrl 502 . . . . . . . . 9 (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → 𝑎 ∈ ℤ)
43adantr 270 . . . . . . . 8 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ∈ ℤ)
5 simplrr 503 . . . . . . . . 9 (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → 𝑏 ∈ ℕ)
65adantr 270 . . . . . . . 8 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 ∈ ℕ)
7 znq 9018 . . . . . . . . 9 ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ) → (𝑎 / 𝑏) ∈ ℚ)
8 qre 9019 . . . . . . . . 9 ((𝑎 / 𝑏) ∈ ℚ → (𝑎 / 𝑏) ∈ ℝ)
97, 8syl 14 . . . . . . . 8 ((𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ) → (𝑎 / 𝑏) ∈ ℝ)
104, 6, 9syl2anc 403 . . . . . . 7 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) ∈ ℝ)
11 sqrt2re 10936 . . . . . . . 8 (√‘2) ∈ ℝ
1211a1i 9 . . . . . . 7 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (√‘2) ∈ ℝ)
13 0red 7410 . . . . . . . 8 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 ∈ ℝ)
144zcnd 8779 . . . . . . . . . 10 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ∈ ℂ)
156nncnd 8348 . . . . . . . . . 10 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 ∈ ℂ)
166nnap0d 8379 . . . . . . . . . 10 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 # 0)
1714, 15, 16divrecapd 8175 . . . . . . . . 9 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) = (𝑎 · (1 / 𝑏)))
184zred 8778 . . . . . . . . . 10 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ∈ ℝ)
196nnrecred 8380 . . . . . . . . . 10 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (1 / 𝑏) ∈ ℝ)
20 simpr 108 . . . . . . . . . 10 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑎 ≤ 0)
21 1red 7424 . . . . . . . . . . 11 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 1 ∈ ℝ)
226nnrpd 9081 . . . . . . . . . . 11 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 𝑏 ∈ ℝ+)
23 0le1 7880 . . . . . . . . . . . 12 0 ≤ 1
2423a1i 9 . . . . . . . . . . 11 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 ≤ 1)
2521, 22, 24divge0d 9123 . . . . . . . . . 10 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 ≤ (1 / 𝑏))
26 mulle0r 8317 . . . . . . . . . 10 (((𝑎 ∈ ℝ ∧ (1 / 𝑏) ∈ ℝ) ∧ (𝑎 ≤ 0 ∧ 0 ≤ (1 / 𝑏))) → (𝑎 · (1 / 𝑏)) ≤ 0)
2718, 19, 20, 25, 26syl22anc 1173 . . . . . . . . 9 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 · (1 / 𝑏)) ≤ 0)
2817, 27eqbrtrd 3834 . . . . . . . 8 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) ≤ 0)
29 2re 8404 . . . . . . . . . 10 2 ∈ ℝ
30 2pos 8425 . . . . . . . . . 10 0 < 2
3129, 30sqrtgt0ii 10405 . . . . . . . . 9 0 < (√‘2)
3231a1i 9 . . . . . . . 8 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → 0 < (√‘2))
3310, 13, 12, 28, 32lelttrd 7529 . . . . . . 7 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (𝑎 / 𝑏) < (√‘2))
3410, 12, 33gtapd 8030 . . . . . 6 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 𝑎 ≤ 0) → (√‘2) # (𝑎 / 𝑏))
353adantr 270 . . . . . . . 8 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → 𝑎 ∈ ℤ)
36 simpr 108 . . . . . . . 8 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → 0 < 𝑎)
37 elnnz 8670 . . . . . . . 8 (𝑎 ∈ ℕ ↔ (𝑎 ∈ ℤ ∧ 0 < 𝑎))
3835, 36, 37sylanbrc 408 . . . . . . 7 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → 𝑎 ∈ ℕ)
395adantr 270 . . . . . . 7 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → 𝑏 ∈ ℕ)
40 sqrt2irraplemnn 10951 . . . . . . 7 ((𝑎 ∈ ℕ ∧ 𝑏 ∈ ℕ) → (√‘2) # (𝑎 / 𝑏))
4138, 39, 40syl2anc 403 . . . . . 6 ((((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) ∧ 0 < 𝑎) → (√‘2) # (𝑎 / 𝑏))
42 0z 8671 . . . . . . . . 9 0 ∈ ℤ
43 zlelttric 8705 . . . . . . . . 9 ((𝑎 ∈ ℤ ∧ 0 ∈ ℤ) → (𝑎 ≤ 0 ∨ 0 < 𝑎))
4442, 43mpan2 416 . . . . . . . 8 (𝑎 ∈ ℤ → (𝑎 ≤ 0 ∨ 0 < 𝑎))
4544ad2antrl 474 . . . . . . 7 ((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) → (𝑎 ≤ 0 ∨ 0 < 𝑎))
4645adantr 270 . . . . . 6 (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → (𝑎 ≤ 0 ∨ 0 < 𝑎))
4734, 41, 46mpjaodan 745 . . . . 5 (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → (√‘2) # (𝑎 / 𝑏))
48 simpr 108 . . . . 5 (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → 𝑄 = (𝑎 / 𝑏))
4947, 48breqtrrd 3840 . . . 4 (((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) ∧ 𝑄 = (𝑎 / 𝑏)) → (√‘2) # 𝑄)
5049ex 113 . . 3 ((𝑄 ∈ ℚ ∧ (𝑎 ∈ ℤ ∧ 𝑏 ∈ ℕ)) → (𝑄 = (𝑎 / 𝑏) → (√‘2) # 𝑄))
5150rexlimdvva 2492 . 2 (𝑄 ∈ ℚ → (∃𝑎 ∈ ℤ ∃𝑏 ∈ ℕ 𝑄 = (𝑎 / 𝑏) → (√‘2) # 𝑄))
522, 51mpd 13 1 (𝑄 ∈ ℚ → (√‘2) # 𝑄)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wo 662   = wceq 1287  wcel 1436  wrex 2356   class class class wbr 3814  cfv 4972  (class class class)co 5594  cr 7270  0cc0 7271  1c1 7272   · cmul 7276   < clt 7443  cle 7444   # cap 7976   / cdiv 8055  cn 8334  2c2 8384  cz 8660  cq 9013  csqrt 10270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3922  ax-sep 3925  ax-nul 3933  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-iinf 4369  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-mulrcl 7365  ax-addcom 7366  ax-mulcom 7367  ax-addass 7368  ax-mulass 7369  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-1rid 7373  ax-0id 7374  ax-rnegex 7375  ax-precex 7376  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-apti 7381  ax-pre-ltadd 7382  ax-pre-mulgt0 7383  ax-pre-mulext 7384  ax-arch 7385  ax-caucvg 7386
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-xor 1310  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rmo 2363  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-nul 3273  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-iun 3709  df-br 3815  df-opab 3869  df-mpt 3870  df-tr 3905  df-id 4087  df-po 4090  df-iso 4091  df-iord 4160  df-on 4162  df-ilim 4163  df-suc 4165  df-iom 4372  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-fo 4978  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-1st 5849  df-2nd 5850  df-recs 6005  df-frec 6091  df-1o 6116  df-2o 6117  df-er 6225  df-en 6391  df-sup 6600  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-reap 7970  df-ap 7977  df-div 8056  df-inn 8335  df-2 8393  df-3 8394  df-4 8395  df-n0 8584  df-z 8661  df-uz 8929  df-q 9014  df-rp 9044  df-fz 9334  df-fzo 9458  df-fl 9580  df-mod 9633  df-iseq 9755  df-iexp 9806  df-cj 10117  df-re 10118  df-im 10119  df-rsqrt 10272  df-abs 10273  df-dvds 10591  df-gcd 10733  df-prm 10884
This theorem is referenced by: (None)
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