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| Mirrors > Home > MPE Home > Th. List > Mathboxes > expeqidd | Structured version Visualization version GIF version | ||
| Description: A nonnegative real number is zero or one if and only if it is itself when raised to an integer greater than one. (Contributed by SN, 3-Jul-2025.) | 
| Ref | Expression | 
|---|---|
| expeqidd.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) | 
| expeqidd.n | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) | 
| expeqidd.0 | ⊢ (𝜑 → 0 ≤ 𝐴) | 
| Ref | Expression | 
|---|---|
| expeqidd | ⊢ (𝜑 → ((𝐴↑𝑁) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-ne 2941 | . . . . 5 ⊢ (𝐴 ≠ 0 ↔ ¬ 𝐴 = 0) | |
| 2 | expeqidd.a | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 3 | 2 | recnd 11289 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℂ) | 
| 4 | 3 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → 𝐴 ∈ ℂ) | 
| 5 | simplr 769 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → 𝐴 ≠ 0) | |
| 6 | expeqidd.n | . . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘2)) | |
| 7 | eluz2nn 12924 | . . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (ℤ≥‘2) → 𝑁 ∈ ℕ) | |
| 8 | 6, 7 | syl 17 | . . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ) | 
| 9 | 8 | nnzd 12640 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ) | 
| 10 | 9 | ad2antrr 726 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → 𝑁 ∈ ℤ) | 
| 11 | 4, 5, 10 | expm1d 14196 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → (𝐴↑(𝑁 − 1)) = ((𝐴↑𝑁) / 𝐴)) | 
| 12 | simpr 484 | . . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → (𝐴↑𝑁) = 𝐴) | |
| 13 | 12 | oveq1d 7446 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → ((𝐴↑𝑁) / 𝐴) = (𝐴 / 𝐴)) | 
| 14 | 4, 5 | dividd 12041 | . . . . . . . . 9 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → (𝐴 / 𝐴) = 1) | 
| 15 | 11, 13, 14 | 3eqtrd 2781 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → (𝐴↑(𝑁 − 1)) = 1) | 
| 16 | 2 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℝ) | 
| 17 | uz2m1nn 12965 | . . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘2) → (𝑁 − 1) ∈ ℕ) | |
| 18 | 6, 17 | syl 17 | . . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ) | 
| 19 | 18 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → (𝑁 − 1) ∈ ℕ) | 
| 20 | expeqidd.0 | . . . . . . . . . . 11 ⊢ (𝜑 → 0 ≤ 𝐴) | |
| 21 | 20 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → 0 ≤ 𝐴) | 
| 22 | 16, 19, 21 | expeq1d 42359 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝐴 ≠ 0) → ((𝐴↑(𝑁 − 1)) = 1 ↔ 𝐴 = 1)) | 
| 23 | 22 | biimpa 476 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑(𝑁 − 1)) = 1) → 𝐴 = 1) | 
| 24 | 15, 23 | syldan 591 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝐴 ≠ 0) ∧ (𝐴↑𝑁) = 𝐴) → 𝐴 = 1) | 
| 25 | 24 | an32s 652 | . . . . . 6 ⊢ (((𝜑 ∧ (𝐴↑𝑁) = 𝐴) ∧ 𝐴 ≠ 0) → 𝐴 = 1) | 
| 26 | 25 | ex 412 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴↑𝑁) = 𝐴) → (𝐴 ≠ 0 → 𝐴 = 1)) | 
| 27 | 1, 26 | biimtrrid 243 | . . . 4 ⊢ ((𝜑 ∧ (𝐴↑𝑁) = 𝐴) → (¬ 𝐴 = 0 → 𝐴 = 1)) | 
| 28 | 27 | orrd 864 | . . 3 ⊢ ((𝜑 ∧ (𝐴↑𝑁) = 𝐴) → (𝐴 = 0 ∨ 𝐴 = 1)) | 
| 29 | 28 | ex 412 | . 2 ⊢ (𝜑 → ((𝐴↑𝑁) = 𝐴 → (𝐴 = 0 ∨ 𝐴 = 1))) | 
| 30 | 8 | 0expd 14179 | . . . 4 ⊢ (𝜑 → (0↑𝑁) = 0) | 
| 31 | oveq1 7438 | . . . . 5 ⊢ (𝐴 = 0 → (𝐴↑𝑁) = (0↑𝑁)) | |
| 32 | id 22 | . . . . 5 ⊢ (𝐴 = 0 → 𝐴 = 0) | |
| 33 | 31, 32 | eqeq12d 2753 | . . . 4 ⊢ (𝐴 = 0 → ((𝐴↑𝑁) = 𝐴 ↔ (0↑𝑁) = 0)) | 
| 34 | 30, 33 | syl5ibrcom 247 | . . 3 ⊢ (𝜑 → (𝐴 = 0 → (𝐴↑𝑁) = 𝐴)) | 
| 35 | 1exp 14132 | . . . . 5 ⊢ (𝑁 ∈ ℤ → (1↑𝑁) = 1) | |
| 36 | 9, 35 | syl 17 | . . . 4 ⊢ (𝜑 → (1↑𝑁) = 1) | 
| 37 | oveq1 7438 | . . . . 5 ⊢ (𝐴 = 1 → (𝐴↑𝑁) = (1↑𝑁)) | |
| 38 | id 22 | . . . . 5 ⊢ (𝐴 = 1 → 𝐴 = 1) | |
| 39 | 37, 38 | eqeq12d 2753 | . . . 4 ⊢ (𝐴 = 1 → ((𝐴↑𝑁) = 𝐴 ↔ (1↑𝑁) = 1)) | 
| 40 | 36, 39 | syl5ibrcom 247 | . . 3 ⊢ (𝜑 → (𝐴 = 1 → (𝐴↑𝑁) = 𝐴)) | 
| 41 | 34, 40 | jaod 860 | . 2 ⊢ (𝜑 → ((𝐴 = 0 ∨ 𝐴 = 1) → (𝐴↑𝑁) = 𝐴)) | 
| 42 | 29, 41 | impbid 212 | 1 ⊢ (𝜑 → ((𝐴↑𝑁) = 𝐴 ↔ (𝐴 = 0 ∨ 𝐴 = 1))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 class class class wbr 5143 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 2c2 12321 ℤcz 12613 ℤ≥cuz 12878 ↑cexp 14102 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-seq 14043 df-exp 14103 | 
| This theorem is referenced by: (None) | 
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