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Mirrors > Home > ILE Home > Th. List > logbgcd1irraplemap | GIF version |
Description: Lemma for logbgcd1irrap 13968. The result, with the rational number expressed as numerator and denominator. (Contributed by Jim Kingdon, 9-Jul-2024.) |
Ref | Expression |
---|---|
logbgcd1irraplem.x | ⊢ (𝜑 → 𝑋 ∈ (ℤ≥‘2)) |
logbgcd1irraplem.b | ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) |
logbgcd1irraplem.rp | ⊢ (𝜑 → (𝑋 gcd 𝐵) = 1) |
logbgcd1irraplem.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
logbgcd1irraplem.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Ref | Expression |
---|---|
logbgcd1irraplemap | ⊢ (𝜑 → (𝐵 logb 𝑋) # (𝑀 / 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | logbgcd1irraplem.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (ℤ≥‘2)) | |
2 | logbgcd1irraplem.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (ℤ≥‘2)) | |
3 | logbgcd1irraplem.rp | . . . . 5 ⊢ (𝜑 → (𝑋 gcd 𝐵) = 1) | |
4 | logbgcd1irraplem.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | logbgcd1irraplem.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 1, 2, 3, 4, 5 | logbgcd1irraplemexp 13966 | . . . 4 ⊢ (𝜑 → (𝑋↑𝑁) # (𝐵↑𝑀)) |
7 | eluz2nn 9539 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℕ) | |
8 | 2, 7 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℕ) |
9 | 8 | nnrpd 9665 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
10 | 1red 7947 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
11 | 8 | nnred 8905 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
12 | eluz2gt1 9575 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘2) → 1 < 𝐵) | |
13 | 2, 12 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 1 < 𝐵) |
14 | 10, 11, 13 | gtapd 8568 | . . . . . 6 ⊢ (𝜑 → 𝐵 # 1) |
15 | eluz2nn 9539 | . . . . . . . 8 ⊢ (𝑋 ∈ (ℤ≥‘2) → 𝑋 ∈ ℕ) | |
16 | 1, 15 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℕ) |
17 | 16 | nnrpd 9665 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
18 | rpcxplogb 13962 | . . . . . 6 ⊢ ((𝐵 ∈ ℝ+ ∧ 𝐵 # 1 ∧ 𝑋 ∈ ℝ+) → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) | |
19 | 9, 14, 17, 18 | syl3anc 1238 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐(𝐵 logb 𝑋)) = 𝑋) |
20 | 19 | oveq1d 5880 | . . . 4 ⊢ (𝜑 → ((𝐵↑𝑐(𝐵 logb 𝑋))↑𝑁) = (𝑋↑𝑁)) |
21 | znq 9597 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 / 𝑁) ∈ ℚ) | |
22 | 4, 5, 21 | syl2anc 411 | . . . . . . 7 ⊢ (𝜑 → (𝑀 / 𝑁) ∈ ℚ) |
23 | qre 9598 | . . . . . . 7 ⊢ ((𝑀 / 𝑁) ∈ ℚ → (𝑀 / 𝑁) ∈ ℝ) | |
24 | 22, 23 | syl 14 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 𝑁) ∈ ℝ) |
25 | 5 | nncnd 8906 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
26 | 9, 24, 25 | cxpmuld 13936 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐((𝑀 / 𝑁) · 𝑁)) = ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑐𝑁)) |
27 | 4 | zcnd 9349 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
28 | 5 | nnap0d 8938 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 # 0) |
29 | 27, 25, 28 | divcanap1d 8721 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 / 𝑁) · 𝑁) = 𝑀) |
30 | 29 | oveq2d 5881 | . . . . . 6 ⊢ (𝜑 → (𝐵↑𝑐((𝑀 / 𝑁) · 𝑁)) = (𝐵↑𝑐𝑀)) |
31 | cxpexpnn 13897 | . . . . . . 7 ⊢ ((𝐵 ∈ ℕ ∧ 𝑀 ∈ ℤ) → (𝐵↑𝑐𝑀) = (𝐵↑𝑀)) | |
32 | 8, 4, 31 | syl2anc 411 | . . . . . 6 ⊢ (𝜑 → (𝐵↑𝑐𝑀) = (𝐵↑𝑀)) |
33 | 30, 32 | eqtrd 2208 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐((𝑀 / 𝑁) · 𝑁)) = (𝐵↑𝑀)) |
34 | 9, 24 | rpcxpcld 13932 | . . . . . 6 ⊢ (𝜑 → (𝐵↑𝑐(𝑀 / 𝑁)) ∈ ℝ+) |
35 | 5 | nnzd 9347 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
36 | cxpexprp 13896 | . . . . . 6 ⊢ (((𝐵↑𝑐(𝑀 / 𝑁)) ∈ ℝ+ ∧ 𝑁 ∈ ℤ) → ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑐𝑁) = ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑁)) | |
37 | 34, 35, 36 | syl2anc 411 | . . . . 5 ⊢ (𝜑 → ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑐𝑁) = ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑁)) |
38 | 26, 33, 37 | 3eqtr3rd 2217 | . . . 4 ⊢ (𝜑 → ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑁) = (𝐵↑𝑀)) |
39 | 6, 20, 38 | 3brtr4d 4030 | . . 3 ⊢ (𝜑 → ((𝐵↑𝑐(𝐵 logb 𝑋))↑𝑁) # ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑁)) |
40 | relogbzcl 13950 | . . . . . . 7 ⊢ ((𝐵 ∈ (ℤ≥‘2) ∧ 𝑋 ∈ ℝ+) → (𝐵 logb 𝑋) ∈ ℝ) | |
41 | 2, 17, 40 | syl2anc 411 | . . . . . 6 ⊢ (𝜑 → (𝐵 logb 𝑋) ∈ ℝ) |
42 | 41 | recnd 7960 | . . . . 5 ⊢ (𝜑 → (𝐵 logb 𝑋) ∈ ℂ) |
43 | 9, 42 | rpcncxpcld 13927 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐(𝐵 logb 𝑋)) ∈ ℂ) |
44 | qcn 9607 | . . . . . 6 ⊢ ((𝑀 / 𝑁) ∈ ℚ → (𝑀 / 𝑁) ∈ ℂ) | |
45 | 22, 44 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝑀 / 𝑁) ∈ ℂ) |
46 | 9, 45 | rpcncxpcld 13927 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐(𝑀 / 𝑁)) ∈ ℂ) |
47 | apexp1 10666 | . . . 4 ⊢ (((𝐵↑𝑐(𝐵 logb 𝑋)) ∈ ℂ ∧ (𝐵↑𝑐(𝑀 / 𝑁)) ∈ ℂ ∧ 𝑁 ∈ ℕ) → (((𝐵↑𝑐(𝐵 logb 𝑋))↑𝑁) # ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑁) → (𝐵↑𝑐(𝐵 logb 𝑋)) # (𝐵↑𝑐(𝑀 / 𝑁)))) | |
48 | 43, 46, 5, 47 | syl3anc 1238 | . . 3 ⊢ (𝜑 → (((𝐵↑𝑐(𝐵 logb 𝑋))↑𝑁) # ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑁) → (𝐵↑𝑐(𝐵 logb 𝑋)) # (𝐵↑𝑐(𝑀 / 𝑁)))) |
49 | 39, 48 | mpd 13 | . 2 ⊢ (𝜑 → (𝐵↑𝑐(𝐵 logb 𝑋)) # (𝐵↑𝑐(𝑀 / 𝑁))) |
50 | apcxp2 13938 | . . 3 ⊢ (((𝐵 ∈ ℝ+ ∧ 𝐵 # 1) ∧ ((𝐵 logb 𝑋) ∈ ℝ ∧ (𝑀 / 𝑁) ∈ ℝ)) → ((𝐵 logb 𝑋) # (𝑀 / 𝑁) ↔ (𝐵↑𝑐(𝐵 logb 𝑋)) # (𝐵↑𝑐(𝑀 / 𝑁)))) | |
51 | 9, 14, 41, 24, 50 | syl22anc 1239 | . 2 ⊢ (𝜑 → ((𝐵 logb 𝑋) # (𝑀 / 𝑁) ↔ (𝐵↑𝑐(𝐵 logb 𝑋)) # (𝐵↑𝑐(𝑀 / 𝑁)))) |
52 | 49, 51 | mpbird 167 | 1 ⊢ (𝜑 → (𝐵 logb 𝑋) # (𝑀 / 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 = wceq 1353 ∈ wcel 2146 class class class wbr 3998 ‘cfv 5208 (class class class)co 5865 ℂcc 7784 ℝcr 7785 1c1 7787 · cmul 7791 < clt 7966 # cap 8512 / cdiv 8602 ℕcn 8892 2c2 8943 ℤcz 9226 ℤ≥cuz 9501 ℚcq 9592 ℝ+crp 9624 ↑cexp 10489 gcd cgcd 11910 ↑𝑐ccxp 13858 logb clogb 13941 |
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 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904 ax-arch 7905 ax-caucvg 7906 ax-pre-suploc 7907 ax-addf 7908 ax-mulf 7909 |
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 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-disj 3976 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-isom 5217 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-of 6073 df-1st 6131 df-2nd 6132 df-recs 6296 df-irdg 6361 df-frec 6382 df-1o 6407 df-2o 6408 df-oadd 6411 df-er 6525 df-map 6640 df-pm 6641 df-en 6731 df-dom 6732 df-fin 6733 df-sup 6973 df-inf 6974 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 df-inn 8893 df-2 8951 df-3 8952 df-4 8953 df-n0 9150 df-z 9227 df-uz 9502 df-q 9593 df-rp 9625 df-xneg 9743 df-xadd 9744 df-ioo 9863 df-ico 9865 df-icc 9866 df-fz 9980 df-fzo 10113 df-fl 10240 df-mod 10293 df-seqfrec 10416 df-exp 10490 df-fac 10674 df-bc 10696 df-ihash 10724 df-shft 10792 df-cj 10819 df-re 10820 df-im 10821 df-rsqrt 10975 df-abs 10976 df-clim 11255 df-sumdc 11330 df-ef 11624 df-e 11625 df-dvds 11763 df-gcd 11911 df-prm 12075 df-rest 12621 df-topgen 12640 df-psmet 13067 df-xmet 13068 df-met 13069 df-bl 13070 df-mopn 13071 df-top 13076 df-topon 13089 df-bases 13121 df-ntr 13176 df-cn 13268 df-cnp 13269 df-tx 13333 df-cncf 13638 df-limced 13705 df-dvap 13706 df-relog 13859 df-rpcxp 13860 df-logb 13942 |
This theorem is referenced by: logbgcd1irrap 13968 |
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