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Mirrors > Home > ILE Home > Th. List > logbgcd1irraplemap | GIF version |
Description: Lemma for logbgcd1irrap 14055. 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 14053 | . . . 4 ⊢ (𝜑 → (𝑋↑𝑁) # (𝐵↑𝑀)) |
7 | eluz2nn 9555 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘2) → 𝐵 ∈ ℕ) | |
8 | 2, 7 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℕ) |
9 | 8 | nnrpd 9681 | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
10 | 1red 7963 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
11 | 8 | nnred 8921 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
12 | eluz2gt1 9591 | . . . . . . . 8 ⊢ (𝐵 ∈ (ℤ≥‘2) → 1 < 𝐵) | |
13 | 2, 12 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 1 < 𝐵) |
14 | 10, 11, 13 | gtapd 8584 | . . . . . 6 ⊢ (𝜑 → 𝐵 # 1) |
15 | eluz2nn 9555 | . . . . . . . 8 ⊢ (𝑋 ∈ (ℤ≥‘2) → 𝑋 ∈ ℕ) | |
16 | 1, 15 | syl 14 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℕ) |
17 | 16 | nnrpd 9681 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
18 | rpcxplogb 14049 | . . . . . 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 5884 | . . . 4 ⊢ (𝜑 → ((𝐵↑𝑐(𝐵 logb 𝑋))↑𝑁) = (𝑋↑𝑁)) |
21 | znq 9613 | . . . . . . . 8 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 / 𝑁) ∈ ℚ) | |
22 | 4, 5, 21 | syl2anc 411 | . . . . . . 7 ⊢ (𝜑 → (𝑀 / 𝑁) ∈ ℚ) |
23 | qre 9614 | . . . . . . 7 ⊢ ((𝑀 / 𝑁) ∈ ℚ → (𝑀 / 𝑁) ∈ ℝ) | |
24 | 22, 23 | syl 14 | . . . . . 6 ⊢ (𝜑 → (𝑀 / 𝑁) ∈ ℝ) |
25 | 5 | nncnd 8922 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
26 | 9, 24, 25 | cxpmuld 14023 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐((𝑀 / 𝑁) · 𝑁)) = ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑐𝑁)) |
27 | 4 | zcnd 9365 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
28 | 5 | nnap0d 8954 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 # 0) |
29 | 27, 25, 28 | divcanap1d 8737 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 / 𝑁) · 𝑁) = 𝑀) |
30 | 29 | oveq2d 5885 | . . . . . 6 ⊢ (𝜑 → (𝐵↑𝑐((𝑀 / 𝑁) · 𝑁)) = (𝐵↑𝑐𝑀)) |
31 | cxpexpnn 13984 | . . . . . . 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 2210 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑐((𝑀 / 𝑁) · 𝑁)) = (𝐵↑𝑀)) |
34 | 9, 24 | rpcxpcld 14019 | . . . . . 6 ⊢ (𝜑 → (𝐵↑𝑐(𝑀 / 𝑁)) ∈ ℝ+) |
35 | 5 | nnzd 9363 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
36 | cxpexprp 13983 | . . . . . 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 2219 | . . . 4 ⊢ (𝜑 → ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑁) = (𝐵↑𝑀)) |
39 | 6, 20, 38 | 3brtr4d 4032 | . . 3 ⊢ (𝜑 → ((𝐵↑𝑐(𝐵 logb 𝑋))↑𝑁) # ((𝐵↑𝑐(𝑀 / 𝑁))↑𝑁)) |
40 | relogbzcl 14037 | . . . . . . 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 7976 | . . . . 5 ⊢ (𝜑 → (𝐵 logb 𝑋) ∈ ℂ) |
43 | 9, 42 | rpcncxpcld 14014 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐(𝐵 logb 𝑋)) ∈ ℂ) |
44 | qcn 9623 | . . . . . 6 ⊢ ((𝑀 / 𝑁) ∈ ℚ → (𝑀 / 𝑁) ∈ ℂ) | |
45 | 22, 44 | syl 14 | . . . . 5 ⊢ (𝜑 → (𝑀 / 𝑁) ∈ ℂ) |
46 | 9, 45 | rpcncxpcld 14014 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑐(𝑀 / 𝑁)) ∈ ℂ) |
47 | apexp1 10682 | . . . 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 14025 | . . 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 2148 class class class wbr 4000 ‘cfv 5212 (class class class)co 5869 ℂcc 7800 ℝcr 7801 1c1 7803 · cmul 7807 < clt 7982 # cap 8528 / cdiv 8618 ℕcn 8908 2c2 8959 ℤcz 9242 ℤ≥cuz 9517 ℚcq 9608 ℝ+crp 9640 ↑cexp 10505 gcd cgcd 11926 ↑𝑐ccxp 13945 logb clogb 14028 |
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 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 ax-pre-suploc 7923 ax-addf 7924 ax-mulf 7925 |
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 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-isom 5221 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-of 6077 df-1st 6135 df-2nd 6136 df-recs 6300 df-irdg 6365 df-frec 6386 df-1o 6411 df-2o 6412 df-oadd 6415 df-er 6529 df-map 6644 df-pm 6645 df-en 6735 df-dom 6736 df-fin 6737 df-sup 6977 df-inf 6978 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-q 9609 df-rp 9641 df-xneg 9759 df-xadd 9760 df-ioo 9879 df-ico 9881 df-icc 9882 df-fz 9996 df-fzo 10129 df-fl 10256 df-mod 10309 df-seqfrec 10432 df-exp 10506 df-fac 10690 df-bc 10712 df-ihash 10740 df-shft 10808 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 df-clim 11271 df-sumdc 11346 df-ef 11640 df-e 11641 df-dvds 11779 df-gcd 11927 df-prm 12091 df-rest 12638 df-topgen 12657 df-psmet 13154 df-xmet 13155 df-met 13156 df-bl 13157 df-mopn 13158 df-top 13163 df-topon 13176 df-bases 13208 df-ntr 13263 df-cn 13355 df-cnp 13356 df-tx 13420 df-cncf 13725 df-limced 13792 df-dvap 13793 df-relog 13946 df-rpcxp 13947 df-logb 14029 |
This theorem is referenced by: logbgcd1irrap 14055 |
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