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Mirrors > Home > ILE Home > Th. List > 2logb9irrap | GIF version |
Description: Example for logbgcd1irrap 14021. The logarithm of nine to base two is irrational (in the sense of being apart from any rational number). (Contributed by Jim Kingdon, 12-Jul-2024.) |
Ref | Expression |
---|---|
2logb9irrap | ⊢ (𝑄 ∈ ℚ → (2 logb 9) # 𝑄) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sq3 10589 | . . . . 5 ⊢ (3↑2) = 9 | |
2 | 1 | eqcomi 2181 | . . . 4 ⊢ 9 = (3↑2) |
3 | 2 | oveq1i 5878 | . . 3 ⊢ (9 gcd 2) = ((3↑2) gcd 2) |
4 | 2re 8965 | . . . . . 6 ⊢ 2 ∈ ℝ | |
5 | 2lt3 9065 | . . . . . 6 ⊢ 2 < 3 | |
6 | 4, 5 | gtneii 8030 | . . . . 5 ⊢ 3 ≠ 2 |
7 | 3prm 12098 | . . . . . 6 ⊢ 3 ∈ ℙ | |
8 | 2prm 12097 | . . . . . 6 ⊢ 2 ∈ ℙ | |
9 | prmrp 12115 | . . . . . 6 ⊢ ((3 ∈ ℙ ∧ 2 ∈ ℙ) → ((3 gcd 2) = 1 ↔ 3 ≠ 2)) | |
10 | 7, 8, 9 | mp2an 426 | . . . . 5 ⊢ ((3 gcd 2) = 1 ↔ 3 ≠ 2) |
11 | 6, 10 | mpbir 146 | . . . 4 ⊢ (3 gcd 2) = 1 |
12 | 3z 9258 | . . . . 5 ⊢ 3 ∈ ℤ | |
13 | 2z 9257 | . . . . 5 ⊢ 2 ∈ ℤ | |
14 | 2nn0 9169 | . . . . 5 ⊢ 2 ∈ ℕ0 | |
15 | rpexp1i 12124 | . . . . 5 ⊢ ((3 ∈ ℤ ∧ 2 ∈ ℤ ∧ 2 ∈ ℕ0) → ((3 gcd 2) = 1 → ((3↑2) gcd 2) = 1)) | |
16 | 12, 13, 14, 15 | mp3an 1337 | . . . 4 ⊢ ((3 gcd 2) = 1 → ((3↑2) gcd 2) = 1) |
17 | 11, 16 | ax-mp 5 | . . 3 ⊢ ((3↑2) gcd 2) = 1 |
18 | 3, 17 | eqtri 2198 | . 2 ⊢ (9 gcd 2) = 1 |
19 | 9nn 9063 | . . . . 5 ⊢ 9 ∈ ℕ | |
20 | 19 | nnzi 9250 | . . . 4 ⊢ 9 ∈ ℤ |
21 | 9re 8982 | . . . . 5 ⊢ 9 ∈ ℝ | |
22 | 2lt9 9098 | . . . . 5 ⊢ 2 < 9 | |
23 | 4, 21, 22 | ltleii 8037 | . . . 4 ⊢ 2 ≤ 9 |
24 | eluz2 9510 | . . . 4 ⊢ (9 ∈ (ℤ≥‘2) ↔ (2 ∈ ℤ ∧ 9 ∈ ℤ ∧ 2 ≤ 9)) | |
25 | 13, 20, 23, 24 | mpbir3an 1179 | . . 3 ⊢ 9 ∈ (ℤ≥‘2) |
26 | uzid 9518 | . . . 4 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
27 | 13, 26 | ax-mp 5 | . . 3 ⊢ 2 ∈ (ℤ≥‘2) |
28 | logbgcd1irrap 14021 | . . 3 ⊢ (((9 ∈ (ℤ≥‘2) ∧ 2 ∈ (ℤ≥‘2)) ∧ ((9 gcd 2) = 1 ∧ 𝑄 ∈ ℚ)) → (2 logb 9) # 𝑄) | |
29 | 25, 27, 28 | mpanl12 436 | . 2 ⊢ (((9 gcd 2) = 1 ∧ 𝑄 ∈ ℚ) → (2 logb 9) # 𝑄) |
30 | 18, 29 | mpan 424 | 1 ⊢ (𝑄 ∈ ℚ → (2 logb 9) # 𝑄) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 class class class wbr 4000 ‘cfv 5211 (class class class)co 5868 1c1 7790 ≤ cle 7970 # cap 8515 2c2 8946 3c3 8947 9c9 8953 ℕ0cn0 9152 ℤcz 9229 ℤ≥cuz 9504 ℚcq 9595 ↑cexp 10492 gcd cgcd 11913 ℙcprime 12077 logb clogb 13994 |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-iinf 4583 ax-cnex 7880 ax-resscn 7881 ax-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-apti 7904 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 ax-pre-mulext 7907 ax-arch 7908 ax-caucvg 7909 ax-pre-suploc 7910 ax-addf 7911 ax-mulf 7912 |
This theorem depends on definitions: df-bi 117 df-stab 831 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-disj 3978 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4289 df-po 4292 df-iso 4293 df-iord 4362 df-on 4364 df-ilim 4365 df-suc 4367 df-iom 4586 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-isom 5220 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-of 6076 df-1st 6134 df-2nd 6135 df-recs 6299 df-irdg 6364 df-frec 6385 df-1o 6410 df-2o 6411 df-oadd 6414 df-er 6528 df-map 6643 df-pm 6644 df-en 6734 df-dom 6735 df-fin 6736 df-sup 6976 df-inf 6977 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-reap 8509 df-ap 8516 df-div 8606 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-5 8957 df-6 8958 df-7 8959 df-8 8960 df-9 8961 df-n0 9153 df-z 9230 df-uz 9505 df-q 9596 df-rp 9628 df-xneg 9746 df-xadd 9747 df-ioo 9866 df-ico 9868 df-icc 9869 df-fz 9983 df-fzo 10116 df-fl 10243 df-mod 10296 df-seqfrec 10419 df-exp 10493 df-fac 10677 df-bc 10699 df-ihash 10727 df-shft 10795 df-cj 10822 df-re 10823 df-im 10824 df-rsqrt 10978 df-abs 10979 df-clim 11258 df-sumdc 11333 df-ef 11627 df-e 11628 df-dvds 11766 df-gcd 11914 df-prm 12078 df-rest 12625 df-topgen 12644 df-psmet 13120 df-xmet 13121 df-met 13122 df-bl 13123 df-mopn 13124 df-top 13129 df-topon 13142 df-bases 13174 df-ntr 13229 df-cn 13321 df-cnp 13322 df-tx 13386 df-cncf 13691 df-limced 13758 df-dvap 13759 df-relog 13912 df-rpcxp 13913 df-logb 13995 |
This theorem is referenced by: 2irrexpqap 14029 |
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