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Mirrors > Home > ILE Home > Th. List > 2irrexpqap | GIF version |
Description: There exist real numbers 𝑎 and 𝑏 which are irrational (in the sense of being apart from any rational number) such that (𝑎↑𝑏) is rational. Statement in the Metamath book, section 1.1.5, footnote 27 on page 17, and the "constructive proof" for theorem 1.2 of [Bauer], p. 483. This is a constructive proof because it is based on two explicitly named irrational numbers (√‘2) and (2 logb 9), see sqrt2irrap 12025, 2logb9irrap 13233 and sqrt2cxp2logb9e3 13231. Therefore, this proof is acceptable/usable in intuitionistic logic. (Contributed by Jim Kingdon, 12-Jul-2024.) |
Ref | Expression |
---|---|
2irrexpqap | ⊢ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ (∀𝑝 ∈ ℚ 𝑎 # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ (𝑎↑𝑐𝑏) ∈ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqrt2re 12008 | . 2 ⊢ (√‘2) ∈ ℝ | |
2 | 2logb9irr 13227 | . . 3 ⊢ (2 logb 9) ∈ (ℝ ∖ ℚ) | |
3 | eldifi 3225 | . . 3 ⊢ ((2 logb 9) ∈ (ℝ ∖ ℚ) → (2 logb 9) ∈ ℝ) | |
4 | 2, 3 | ax-mp 5 | . 2 ⊢ (2 logb 9) ∈ ℝ |
5 | sqrt2irrap 12025 | . . . 4 ⊢ (𝑝 ∈ ℚ → (√‘2) # 𝑝) | |
6 | 5 | rgen 2507 | . . 3 ⊢ ∀𝑝 ∈ ℚ (√‘2) # 𝑝 |
7 | 2logb9irrap 13233 | . . . 4 ⊢ (𝑞 ∈ ℚ → (2 logb 9) # 𝑞) | |
8 | 7 | rgen 2507 | . . 3 ⊢ ∀𝑞 ∈ ℚ (2 logb 9) # 𝑞 |
9 | sqrt2cxp2logb9e3 13231 | . . . 4 ⊢ ((√‘2)↑𝑐(2 logb 9)) = 3 | |
10 | 3z 9175 | . . . . 5 ⊢ 3 ∈ ℤ | |
11 | zq 9513 | . . . . 5 ⊢ (3 ∈ ℤ → 3 ∈ ℚ) | |
12 | 10, 11 | ax-mp 5 | . . . 4 ⊢ 3 ∈ ℚ |
13 | 9, 12 | eqeltri 2227 | . . 3 ⊢ ((√‘2)↑𝑐(2 logb 9)) ∈ ℚ |
14 | 6, 8, 13 | 3pm3.2i 1160 | . 2 ⊢ (∀𝑝 ∈ ℚ (√‘2) # 𝑝 ∧ ∀𝑞 ∈ ℚ (2 logb 9) # 𝑞 ∧ ((√‘2)↑𝑐(2 logb 9)) ∈ ℚ) |
15 | breq1 3964 | . . . . 5 ⊢ (𝑎 = (√‘2) → (𝑎 # 𝑝 ↔ (√‘2) # 𝑝)) | |
16 | 15 | ralbidv 2454 | . . . 4 ⊢ (𝑎 = (√‘2) → (∀𝑝 ∈ ℚ 𝑎 # 𝑝 ↔ ∀𝑝 ∈ ℚ (√‘2) # 𝑝)) |
17 | biidd 171 | . . . 4 ⊢ (𝑎 = (√‘2) → (∀𝑞 ∈ ℚ 𝑏 # 𝑞 ↔ ∀𝑞 ∈ ℚ 𝑏 # 𝑞)) | |
18 | oveq1 5821 | . . . . 5 ⊢ (𝑎 = (√‘2) → (𝑎↑𝑐𝑏) = ((√‘2)↑𝑐𝑏)) | |
19 | 18 | eleq1d 2223 | . . . 4 ⊢ (𝑎 = (√‘2) → ((𝑎↑𝑐𝑏) ∈ ℚ ↔ ((√‘2)↑𝑐𝑏) ∈ ℚ)) |
20 | 16, 17, 19 | 3anbi123d 1291 | . . 3 ⊢ (𝑎 = (√‘2) → ((∀𝑝 ∈ ℚ 𝑎 # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ (𝑎↑𝑐𝑏) ∈ ℚ) ↔ (∀𝑝 ∈ ℚ (√‘2) # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ ((√‘2)↑𝑐𝑏) ∈ ℚ))) |
21 | biidd 171 | . . . 4 ⊢ (𝑏 = (2 logb 9) → (∀𝑝 ∈ ℚ (√‘2) # 𝑝 ↔ ∀𝑝 ∈ ℚ (√‘2) # 𝑝)) | |
22 | breq1 3964 | . . . . 5 ⊢ (𝑏 = (2 logb 9) → (𝑏 # 𝑞 ↔ (2 logb 9) # 𝑞)) | |
23 | 22 | ralbidv 2454 | . . . 4 ⊢ (𝑏 = (2 logb 9) → (∀𝑞 ∈ ℚ 𝑏 # 𝑞 ↔ ∀𝑞 ∈ ℚ (2 logb 9) # 𝑞)) |
24 | oveq2 5822 | . . . . 5 ⊢ (𝑏 = (2 logb 9) → ((√‘2)↑𝑐𝑏) = ((√‘2)↑𝑐(2 logb 9))) | |
25 | 24 | eleq1d 2223 | . . . 4 ⊢ (𝑏 = (2 logb 9) → (((√‘2)↑𝑐𝑏) ∈ ℚ ↔ ((√‘2)↑𝑐(2 logb 9)) ∈ ℚ)) |
26 | 21, 23, 25 | 3anbi123d 1291 | . . 3 ⊢ (𝑏 = (2 logb 9) → ((∀𝑝 ∈ ℚ (√‘2) # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ ((√‘2)↑𝑐𝑏) ∈ ℚ) ↔ (∀𝑝 ∈ ℚ (√‘2) # 𝑝 ∧ ∀𝑞 ∈ ℚ (2 logb 9) # 𝑞 ∧ ((√‘2)↑𝑐(2 logb 9)) ∈ ℚ))) |
27 | 20, 26 | rspc2ev 2828 | . 2 ⊢ (((√‘2) ∈ ℝ ∧ (2 logb 9) ∈ ℝ ∧ (∀𝑝 ∈ ℚ (√‘2) # 𝑝 ∧ ∀𝑞 ∈ ℚ (2 logb 9) # 𝑞 ∧ ((√‘2)↑𝑐(2 logb 9)) ∈ ℚ)) → ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ (∀𝑝 ∈ ℚ 𝑎 # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ (𝑎↑𝑐𝑏) ∈ ℚ)) |
28 | 1, 4, 14, 27 | mp3an 1316 | 1 ⊢ ∃𝑎 ∈ ℝ ∃𝑏 ∈ ℝ (∀𝑝 ∈ ℚ 𝑎 # 𝑝 ∧ ∀𝑞 ∈ ℚ 𝑏 # 𝑞 ∧ (𝑎↑𝑐𝑏) ∈ ℚ) |
Colors of variables: wff set class |
Syntax hints: ∧ w3a 963 = wceq 1332 ∈ wcel 2125 ∀wral 2432 ∃wrex 2433 ∖ cdif 3095 class class class wbr 3961 ‘cfv 5163 (class class class)co 5814 ℝcr 7710 # cap 8435 2c2 8863 3c3 8864 9c9 8870 ℤcz 9146 ℚcq 9506 √csqrt 10873 ↑𝑐ccxp 13117 logb clogb 13199 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-13 2127 ax-14 2128 ax-ext 2136 ax-coll 4075 ax-sep 4078 ax-nul 4086 ax-pow 4130 ax-pr 4164 ax-un 4388 ax-setind 4490 ax-iinf 4541 ax-cnex 7802 ax-resscn 7803 ax-1cn 7804 ax-1re 7805 ax-icn 7806 ax-addcl 7807 ax-addrcl 7808 ax-mulcl 7809 ax-mulrcl 7810 ax-addcom 7811 ax-mulcom 7812 ax-addass 7813 ax-mulass 7814 ax-distr 7815 ax-i2m1 7816 ax-0lt1 7817 ax-1rid 7818 ax-0id 7819 ax-rnegex 7820 ax-precex 7821 ax-cnre 7822 ax-pre-ltirr 7823 ax-pre-ltwlin 7824 ax-pre-lttrn 7825 ax-pre-apti 7826 ax-pre-ltadd 7827 ax-pre-mulgt0 7828 ax-pre-mulext 7829 ax-arch 7830 ax-caucvg 7831 ax-pre-suploc 7832 ax-addf 7833 ax-mulf 7834 |
This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-xor 1355 df-nf 1438 df-sb 1740 df-eu 2006 df-mo 2007 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ne 2325 df-nel 2420 df-ral 2437 df-rex 2438 df-reu 2439 df-rmo 2440 df-rab 2441 df-v 2711 df-sbc 2934 df-csb 3028 df-dif 3100 df-un 3102 df-in 3104 df-ss 3111 df-nul 3391 df-if 3502 df-pw 3541 df-sn 3562 df-pr 3563 df-op 3565 df-uni 3769 df-int 3804 df-iun 3847 df-disj 3939 df-br 3962 df-opab 4022 df-mpt 4023 df-tr 4059 df-id 4248 df-po 4251 df-iso 4252 df-iord 4321 df-on 4323 df-ilim 4324 df-suc 4326 df-iom 4544 df-xp 4585 df-rel 4586 df-cnv 4587 df-co 4588 df-dm 4589 df-rn 4590 df-res 4591 df-ima 4592 df-iota 5128 df-fun 5165 df-fn 5166 df-f 5167 df-f1 5168 df-fo 5169 df-f1o 5170 df-fv 5171 df-isom 5172 df-riota 5770 df-ov 5817 df-oprab 5818 df-mpo 5819 df-of 6022 df-1st 6078 df-2nd 6079 df-recs 6242 df-irdg 6307 df-frec 6328 df-1o 6353 df-2o 6354 df-oadd 6357 df-er 6469 df-map 6584 df-pm 6585 df-en 6675 df-dom 6676 df-fin 6677 df-sup 6916 df-inf 6917 df-pnf 7893 df-mnf 7894 df-xr 7895 df-ltxr 7896 df-le 7897 df-sub 8027 df-neg 8028 df-reap 8429 df-ap 8436 df-div 8525 df-inn 8813 df-2 8871 df-3 8872 df-4 8873 df-5 8874 df-6 8875 df-7 8876 df-8 8877 df-9 8878 df-n0 9070 df-z 9147 df-uz 9419 df-q 9507 df-rp 9539 df-xneg 9657 df-xadd 9658 df-ioo 9774 df-ico 9776 df-icc 9777 df-fz 9891 df-fzo 10020 df-fl 10147 df-mod 10200 df-seqfrec 10323 df-exp 10397 df-fac 10577 df-bc 10599 df-ihash 10627 df-shft 10692 df-cj 10719 df-re 10720 df-im 10721 df-rsqrt 10875 df-abs 10876 df-clim 11153 df-sumdc 11228 df-ef 11522 df-e 11523 df-dvds 11661 df-gcd 11803 df-prm 11956 df-rest 12292 df-topgen 12311 df-psmet 12326 df-xmet 12327 df-met 12328 df-bl 12329 df-mopn 12330 df-top 12335 df-topon 12348 df-bases 12380 df-ntr 12435 df-cn 12527 df-cnp 12528 df-tx 12592 df-cncf 12897 df-limced 12964 df-dvap 12965 df-relog 13118 df-rpcxp 13119 df-logb 13200 |
This theorem is referenced by: (None) |
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