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Mirrors > Home > ILE Home > Th. List > expcan | GIF version |
Description: Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.) |
Ref | Expression |
---|---|
expcan | ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) ↔ 𝑀 = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 942 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝐴 ∈ ℝ) | |
2 | simpl2 943 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑀 ∈ ℤ) | |
3 | simpl3 944 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑁 ∈ ℤ) | |
4 | simpr 108 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 1 < 𝐴) | |
5 | 1, 2, 3, 4 | expcanlem 9957 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) → 𝑀 ≤ 𝑁)) |
6 | 1, 3, 2, 4 | expcanlem 9957 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑁) ≤ (𝐴↑𝑀) → 𝑁 ≤ 𝑀)) |
7 | 5, 6 | anim12d 328 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (((𝐴↑𝑀) ≤ (𝐴↑𝑁) ∧ (𝐴↑𝑁) ≤ (𝐴↑𝑀)) → (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))) |
8 | 0red 7390 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 ∈ ℝ) | |
9 | 1red 7404 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 1 ∈ ℝ) | |
10 | 0lt1 7511 | . . . . . . . 8 ⊢ 0 < 1 | |
11 | 10 | a1i 9 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 < 1) |
12 | 8, 9, 1, 11, 4 | lttrd 7510 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 < 𝐴) |
13 | 1, 12 | gt0ap0d 8003 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝐴 # 0) |
14 | 1, 13, 2 | reexpclzapd 9944 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝐴↑𝑀) ∈ ℝ) |
15 | 1, 13, 3 | reexpclzapd 9944 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝐴↑𝑁) ∈ ℝ) |
16 | 14, 15 | letri3d 7501 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) ↔ ((𝐴↑𝑀) ≤ (𝐴↑𝑁) ∧ (𝐴↑𝑁) ≤ (𝐴↑𝑀)))) |
17 | 2 | zred 8762 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑀 ∈ ℝ) |
18 | 3 | zred 8762 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑁 ∈ ℝ) |
19 | 17, 18 | letri3d 7501 | . . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 = 𝑁 ↔ (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀))) |
20 | 7, 16, 19 | 3imtr4d 201 | . 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) → 𝑀 = 𝑁)) |
21 | oveq2 5597 | . 2 ⊢ (𝑀 = 𝑁 → (𝐴↑𝑀) = (𝐴↑𝑁)) | |
22 | 20, 21 | impbid1 140 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) ↔ 𝑀 = 𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 (class class class)co 5589 ℝcr 7250 0cc0 7251 1c1 7252 < clt 7423 ≤ cle 7424 ℤcz 8644 ↑cexp 9789 |
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 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-mulrcl 7345 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-precex 7356 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 ax-pre-mulgt0 7363 ax-pre-mulext 7364 |
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 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-frec 6086 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-reap 7950 df-ap 7957 df-div 8036 df-inn 8315 df-n0 8564 df-z 8645 df-uz 8913 df-rp 9028 df-iseq 9739 df-iexp 9790 |
This theorem is referenced by: expcand 9959 |
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