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Mirrors > Home > ILE Home > Th. List > expcanlem | GIF version |
Description: Lemma for expcan 9960. 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 9844 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (1 < 𝐴 ∧ 𝑁 < 𝑀)) → (𝐴↑𝑁) < (𝐴↑𝑀)) | |
6 | 5 | expr 367 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 1 < 𝐴) → (𝑁 < 𝑀 → (𝐴↑𝑁) < (𝐴↑𝑀))) |
7 | 1, 2, 3, 4, 6 | syl31anc 1173 | . . 3 ⊢ (𝜑 → (𝑁 < 𝑀 → (𝐴↑𝑁) < (𝐴↑𝑀))) |
8 | 7 | con3d 594 | . 2 ⊢ (𝜑 → (¬ (𝐴↑𝑁) < (𝐴↑𝑀) → ¬ 𝑁 < 𝑀)) |
9 | 0red 7392 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℝ) | |
10 | 1red 7406 | . . . . . 6 ⊢ (𝜑 → 1 ∈ ℝ) | |
11 | 0lt1 7513 | . . . . . . 7 ⊢ 0 < 1 | |
12 | 11 | a1i 9 | . . . . . 6 ⊢ (𝜑 → 0 < 1) |
13 | 9, 10, 1, 12, 4 | lttrd 7512 | . . . . 5 ⊢ (𝜑 → 0 < 𝐴) |
14 | 1, 13 | gt0ap0d 8005 | . . . 4 ⊢ (𝜑 → 𝐴 # 0) |
15 | 1, 14, 3 | reexpclzapd 9946 | . . 3 ⊢ (𝜑 → (𝐴↑𝑀) ∈ ℝ) |
16 | 1, 14, 2 | reexpclzapd 9946 | . . 3 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℝ) |
17 | 15, 16 | lenltd 7504 | . 2 ⊢ (𝜑 → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) ↔ ¬ (𝐴↑𝑁) < (𝐴↑𝑀))) |
18 | 3 | zred 8764 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ) |
19 | 2 | zred 8764 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ) |
20 | 18, 19 | lenltd 7504 | . 2 ⊢ (𝜑 → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀)) |
21 | 8, 17, 20 | 3imtr4d 201 | 1 ⊢ (𝜑 → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) → 𝑀 ≤ 𝑁)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ w3a 920 ∈ wcel 1434 class class class wbr 3811 (class class class)co 5591 ℝcr 7252 0cc0 7253 1c1 7254 < clt 7425 ≤ cle 7426 ℤcz 8646 ↑cexp 9791 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-po 4087 df-iso 4088 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-n0 8566 df-z 8647 df-uz 8915 df-rp 9030 df-iseq 9741 df-iexp 9792 |
This theorem is referenced by: expcan 9960 |
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