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Mirrors > Home > ILE Home > Th. List > expcanlem | GIF version |
Description: Lemma for expcan 10691. Proving the order in one direction. (Contributed by Jim Kingdon, 29-Jan-2022.) |
Ref | Expression |
---|---|
expcanlem.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
expcanlem.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
expcanlem.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
expcanlem.gt1 | ⊢ (𝜑 → 1 < 𝐴) |
Ref | Expression |
---|---|
expcanlem | ⊢ (𝜑 → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) → 𝑀 ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | expcanlem.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | expcanlem.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
3 | expcanlem.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
4 | expcanlem.gt1 | . . . 4 ⊢ (𝜑 → 1 < 𝐴) | |
5 | ltexp2a 10569 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (1 < 𝐴 ∧ 𝑁 < 𝑀)) → (𝐴↑𝑁) < (𝐴↑𝑀)) | |
6 | 5 | expr 375 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 1 < 𝐴) → (𝑁 < 𝑀 → (𝐴↑𝑁) < (𝐴↑𝑀))) |
7 | 1, 2, 3, 4, 6 | syl31anc 1241 | . . 3 ⊢ (𝜑 → (𝑁 < 𝑀 → (𝐴↑𝑁) < (𝐴↑𝑀))) |
8 | 7 | con3d 631 | . 2 ⊢ (𝜑 → (¬ (𝐴↑𝑁) < (𝐴↑𝑀) → ¬ 𝑁 < 𝑀)) |
9 | 0red 7957 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
10 | 1red 7971 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
11 | 0lt1 8082 | . . . . . . 7 ⊢ 0 < 1 | |
12 | 11 | a1i 9 | . . . . . 6 ⊢ (𝜑 → 0 < 1) |
13 | 9, 10, 1, 12, 4 | lttrd 8081 | . . . . 5 ⊢ (𝜑 → 0 < 𝐴) |
14 | 1, 13 | gt0ap0d 8584 | . . . 4 ⊢ (𝜑 → 𝐴 # 0) |
15 | 1, 14, 3 | reexpclzapd 10675 | . . 3 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℝ) |
16 | 1, 14, 2 | reexpclzapd 10675 | . . 3 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) |
17 | 15, 16 | lenltd 8073 | . 2 ⊢ (𝜑 → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) ↔ ¬ (𝐴↑𝑁) < (𝐴↑𝑀))) |
18 | 3 | zred 9373 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
19 | 2 | zred 9373 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
20 | 18, 19 | lenltd 8073 | . 2 ⊢ (𝜑 → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) |
21 | 8, 17, 20 | 3imtr4d 203 | 1 ⊢ (𝜑 → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) → 𝑀 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 978 ∈ wcel 2148 class class class wbr 4003 (class class class)co 5874 ℝcr 7809 0cc0 7810 1c1 7811 < clt 7990 ≤ cle 7991 ℤcz 9251 ↑cexp 10516 |
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 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 |
This theorem depends on definitions: df-bi 117 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-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7992 df-mnf 7993 df-xr 7994 df-ltxr 7995 df-le 7996 df-sub 8128 df-neg 8129 df-reap 8530 df-ap 8537 df-div 8628 df-inn 8918 df-n0 9175 df-z 9252 df-uz 9527 df-rp 9652 df-seqfrec 10443 df-exp 10517 |
This theorem is referenced by: expcan 10691 |
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