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Theorem expcan 11074

Description: Cancellation law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

**Assertion**
Ref	Expression
expcan	⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) ↔ 𝑀 = 𝑁))

**Proof of Theorem expcan**
Step	Hyp	Ref	Expression
1		simpl1 1027	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝐴 ∈ ℝ)
2		simpl2 1028	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑀 ∈ ℤ)
3		simpl3 1029	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑁 ∈ ℤ)
4		simpr 110	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 1 < 𝐴)
5	1, 2, 3, 4	expcanlem 11073	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) → 𝑀 ≤ 𝑁))
6	1, 3, 2, 4	expcanlem 11073	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑁) ≤ (𝐴↑𝑀) → 𝑁 ≤ 𝑀))
7	5, 6	anim12d 335	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (((𝐴↑𝑀) ≤ (𝐴↑𝑁) ∧ (𝐴↑𝑁) ≤ (𝐴↑𝑀)) → (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)))
8		0red 8271	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 ∈ ℝ)
9		1red 8285	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 1 ∈ ℝ)
10		0lt1 8396	. . . . . . . 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 8395	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 < 𝐴)
13	1, 12	gt0ap0d 8899	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝐴 # 0)
14	1, 13, 2	reexpclzapd 11056	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝐴↑𝑀) ∈ ℝ)
15	1, 13, 3	reexpclzapd 11056	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝐴↑𝑁) ∈ ℝ)
16	14, 15	letri3d 8385	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) ↔ ((𝐴↑𝑀) ≤ (𝐴↑𝑁) ∧ (𝐴↑𝑁) ≤ (𝐴↑𝑀))))
17	2	zred 9696	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑀 ∈ ℝ)
18	3	zred 9696	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑁 ∈ ℝ)
19	17, 18	letri3d 8385	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 = 𝑁 ↔ (𝑀 ≤ 𝑁 ∧ 𝑁 ≤ 𝑀)))
20	7, 16, 19	3imtr4d 203	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) = (𝐴↑𝑁) → 𝑀 = 𝑁))
21		oveq2 6057	. 2 ⊢ (𝑀 = 𝑁 → (𝐴↑𝑀) = (𝐴↑𝑁))
22	20, 21	impbid1 142	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 104 ↔ wb 105 ∧ w3a 1005 = wceq 1398 ∈ wcel 2203 class class class wbr 4108 (class class class)co 6049 ℝcr 8122 0cc0 8123 1c1 8124 < clt 8304 ≤ cle 8305 ℤcz 9573 ↑cexp 10896

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4224 ax-sep 4227 ax-nul 4235 ax-pow 4286 ax-pr 4321 ax-un 4553 ax-setind 4658 ax-iinf 4709 ax-cnex 8214 ax-resscn 8215 ax-1cn 8216 ax-1re 8217 ax-icn 8218 ax-addcl 8219 ax-addrcl 8220 ax-mulcl 8221 ax-mulrcl 8222 ax-addcom 8223 ax-mulcom 8224 ax-addass 8225 ax-mulass 8226 ax-distr 8227 ax-i2m1 8228 ax-0lt1 8229 ax-1rid 8230 ax-0id 8231 ax-rnegex 8232 ax-precex 8233 ax-cnre 8234 ax-pre-ltirr 8235 ax-pre-ltwlin 8236 ax-pre-lttrn 8237 ax-pre-apti 8238 ax-pre-ltadd 8239 ax-pre-mulgt0 8240 ax-pre-mulext 8241

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2814 df-sbc 3042 df-csb 3138 df-dif 3212 df-un 3214 df-in 3216 df-ss 3223 df-nul 3508 df-if 3620 df-pw 3670 df-sn 3694 df-pr 3695 df-op 3697 df-uni 3914 df-int 3949 df-iun 3992 df-br 4109 df-opab 4171 df-mpt 4172 df-tr 4208 df-id 4413 df-po 4416 df-iso 4417 df-iord 4486 df-on 4488 df-ilim 4489 df-suc 4491 df-iom 4712 df-xp 4754 df-rel 4755 df-cnv 4756 df-co 4757 df-dm 4758 df-rn 4759 df-res 4760 df-ima 4761 df-iota 5311 df-fun 5353 df-fn 5354 df-f 5355 df-f1 5356 df-fo 5357 df-f1o 5358 df-fv 5359 df-riota 6002 df-ov 6052 df-oprab 6053 df-mpo 6054 df-1st 6333 df-2nd 6334 df-recs 6535 df-frec 6621 df-pnf 8306 df-mnf 8307 df-xr 8308 df-ltxr 8309 df-le 8310 df-sub 8442 df-neg 8443 df-reap 8845 df-ap 8852 df-div 8943 df-inn 9234 df-n0 9493 df-z 9574 df-uz 9850 df-rp 9983 df-seqfrec 10806 df-exp 10897

This theorem is referenced by: expcand 11075

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