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Mirrors > Home > ILE Home > Th. List > 2logb9irrALT | GIF version |
Description: Alternate proof of 2logb9irr 14260: The logarithm of nine to base two is not rational. (Contributed by AV, 31-Dec-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
2logb9irrALT | ⊢ (2 logb 9) ∈ (ℝ ∖ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sq3 10611 | . . . . 5 ⊢ (3↑2) = 9 | |
2 | 1 | eqcomi 2181 | . . . 4 ⊢ 9 = (3↑2) |
3 | 2 | oveq2i 5883 | . . 3 ⊢ (2 logb 9) = (2 logb (3↑2)) |
4 | 2rp 9654 | . . . . 5 ⊢ 2 ∈ ℝ+ | |
5 | 1re 7953 | . . . . . 6 ⊢ 1 ∈ ℝ | |
6 | 2re 8985 | . . . . . 6 ⊢ 2 ∈ ℝ | |
7 | 1lt2 9084 | . . . . . 6 ⊢ 1 < 2 | |
8 | 5, 6, 7 | gtapii 8587 | . . . . 5 ⊢ 2 # 1 |
9 | 4, 8 | pm3.2i 272 | . . . 4 ⊢ (2 ∈ ℝ+ ∧ 2 # 1) |
10 | 3rp 9655 | . . . 4 ⊢ 3 ∈ ℝ+ | |
11 | 2z 9277 | . . . 4 ⊢ 2 ∈ ℤ | |
12 | rplogbzexp 14243 | . . . 4 ⊢ (((2 ∈ ℝ+ ∧ 2 # 1) ∧ 3 ∈ ℝ+ ∧ 2 ∈ ℤ) → (2 logb (3↑2)) = (2 · (2 logb 3))) | |
13 | 9, 10, 11, 12 | mp3an 1337 | . . 3 ⊢ (2 logb (3↑2)) = (2 · (2 logb 3)) |
14 | 3, 13 | eqtri 2198 | . 2 ⊢ (2 logb 9) = (2 · (2 logb 3)) |
15 | 2cn 8986 | . . . 4 ⊢ 2 ∈ ℂ | |
16 | rplogbcl 14235 | . . . . . 6 ⊢ ((2 ∈ ℝ+ ∧ 2 # 1 ∧ 3 ∈ ℝ+) → (2 logb 3) ∈ ℝ) | |
17 | 4, 8, 10, 16 | mp3an 1337 | . . . . 5 ⊢ (2 logb 3) ∈ ℝ |
18 | 17 | recni 7966 | . . . 4 ⊢ (2 logb 3) ∈ ℂ |
19 | 15, 18 | mulcomi 7960 | . . 3 ⊢ (2 · (2 logb 3)) = ((2 logb 3) · 2) |
20 | 2logb3irr 14262 | . . . 4 ⊢ (2 logb 3) ∈ (ℝ ∖ ℚ) | |
21 | zq 9622 | . . . . 5 ⊢ (2 ∈ ℤ → 2 ∈ ℚ) | |
22 | 11, 21 | ax-mp 5 | . . . 4 ⊢ 2 ∈ ℚ |
23 | 2ne0 9007 | . . . 4 ⊢ 2 ≠ 0 | |
24 | irrmul 9643 | . . . 4 ⊢ (((2 logb 3) ∈ (ℝ ∖ ℚ) ∧ 2 ∈ ℚ ∧ 2 ≠ 0) → ((2 logb 3) · 2) ∈ (ℝ ∖ ℚ)) | |
25 | 20, 22, 23, 24 | mp3an 1337 | . . 3 ⊢ ((2 logb 3) · 2) ∈ (ℝ ∖ ℚ) |
26 | 19, 25 | eqeltri 2250 | . 2 ⊢ (2 · (2 logb 3)) ∈ (ℝ ∖ ℚ) |
27 | 14, 26 | eqeltri 2250 | 1 ⊢ (2 logb 9) ∈ (ℝ ∖ ℚ) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1353 ∈ wcel 2148 ≠ wne 2347 ∖ cdif 3126 class class class wbr 4002 (class class class)co 5872 ℝcr 7807 0cc0 7808 1c1 7809 · cmul 7813 # cap 8534 2c2 8966 3c3 8967 9c9 8973 ℤcz 9249 ℚcq 9615 ℝ+crp 9649 ↑cexp 10514 logb clogb 14232 |
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 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7899 ax-resscn 7900 ax-1cn 7901 ax-1re 7902 ax-icn 7903 ax-addcl 7904 ax-addrcl 7905 ax-mulcl 7906 ax-mulrcl 7907 ax-addcom 7908 ax-mulcom 7909 ax-addass 7910 ax-mulass 7911 ax-distr 7912 ax-i2m1 7913 ax-0lt1 7914 ax-1rid 7915 ax-0id 7916 ax-rnegex 7917 ax-precex 7918 ax-cnre 7919 ax-pre-ltirr 7920 ax-pre-ltwlin 7921 ax-pre-lttrn 7922 ax-pre-apti 7923 ax-pre-ltadd 7924 ax-pre-mulgt0 7925 ax-pre-mulext 7926 ax-arch 7927 ax-caucvg 7928 ax-pre-suploc 7929 ax-addf 7930 ax-mulf 7931 |
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 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-disj 3980 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-isom 5224 df-riota 5828 df-ov 5875 df-oprab 5876 df-mpo 5877 df-of 6080 df-1st 6138 df-2nd 6139 df-recs 6303 df-irdg 6368 df-frec 6389 df-1o 6414 df-2o 6415 df-oadd 6418 df-er 6532 df-map 6647 df-pm 6648 df-en 6738 df-dom 6739 df-fin 6740 df-sup 6980 df-inf 6981 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 df-sub 8126 df-neg 8127 df-reap 8528 df-ap 8535 df-div 8626 df-inn 8916 df-2 8974 df-3 8975 df-4 8976 df-5 8977 df-6 8978 df-7 8979 df-8 8980 df-9 8981 df-n0 9173 df-z 9250 df-uz 9525 df-q 9616 df-rp 9650 df-xneg 9768 df-xadd 9769 df-ioo 9888 df-ico 9890 df-icc 9891 df-fz 10005 df-fzo 10138 df-fl 10265 df-mod 10318 df-seqfrec 10441 df-exp 10515 df-fac 10699 df-bc 10721 df-ihash 10749 df-shft 10817 df-cj 10844 df-re 10845 df-im 10846 df-rsqrt 11000 df-abs 11001 df-clim 11280 df-sumdc 11355 df-ef 11649 df-e 11650 df-dvds 11788 df-gcd 11936 df-prm 12100 df-rest 12678 df-topgen 12697 df-psmet 13316 df-xmet 13317 df-met 13318 df-bl 13319 df-mopn 13320 df-top 13367 df-topon 13380 df-bases 13412 df-ntr 13467 df-cn 13559 df-cnp 13560 df-tx 13624 df-cncf 13929 df-limced 13996 df-dvap 13997 df-relog 14150 df-rpcxp 14151 df-logb 14233 |
This theorem is referenced by: (None) |
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