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| Mirrors > Home > MPE Home > Th. List > Mathboxes > exp11d | Structured version Visualization version GIF version | ||
| Description: exp11nnd 14300 for nonzero integer exponents. (Contributed by SN, 14-Sep-2023.) |
| Ref | Expression |
|---|---|
| exp11d.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
| exp11d.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
| exp11d.3 | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
| exp11d.4 | ⊢ (𝜑 → 𝑁 ≠ 0) |
| exp11d.5 | ⊢ (𝜑 → (𝐴↑𝑁) = (𝐵↑𝑁)) |
| Ref | Expression |
|---|---|
| exp11d | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 = 0) | |
| 2 | exp11d.4 | . . . 4 ⊢ (𝜑 → 𝑁 ≠ 0) | |
| 3 | 2 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝑁 ≠ 0) |
| 4 | 1, 3 | pm2.21ddne 3026 | . 2 ⊢ ((𝜑 ∧ 𝑁 = 0) → 𝐴 = 𝐵) |
| 5 | exp11d.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
| 6 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 ∈ ℝ+) |
| 7 | exp11d.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
| 8 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐵 ∈ ℝ+) |
| 9 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝑁 ∈ ℕ) | |
| 10 | exp11d.5 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) = (𝐵↑𝑁)) | |
| 11 | 10 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → (𝐴↑𝑁) = (𝐵↑𝑁)) |
| 12 | 6, 8, 9, 11 | exp11nnd 14300 | . 2 ⊢ ((𝜑 ∧ 𝑁 ∈ ℕ) → 𝐴 = 𝐵) |
| 13 | 5 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → 𝐴 ∈ ℝ+) |
| 14 | 7 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → 𝐵 ∈ ℝ+) |
| 15 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ) | |
| 16 | 13 | rpcnd 13079 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → 𝐴 ∈ ℂ) |
| 17 | 15 | nnnn0d 12587 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℕ0) |
| 18 | 16, 17 | expcld 14186 | . . . 4 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (𝐴↑-𝑁) ∈ ℂ) |
| 19 | 14 | rpcnd 13079 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → 𝐵 ∈ ℂ) |
| 20 | 19, 17 | expcld 14186 | . . . 4 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (𝐵↑-𝑁) ∈ ℂ) |
| 21 | 13 | rpne0d 13082 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → 𝐴 ≠ 0) |
| 22 | 15 | nnzd 12640 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → -𝑁 ∈ ℤ) |
| 23 | 16, 21, 22 | expne0d 14192 | . . . 4 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (𝐴↑-𝑁) ≠ 0) |
| 24 | 14 | rpne0d 13082 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → 𝐵 ≠ 0) |
| 25 | 19, 24, 22 | expne0d 14192 | . . . 4 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (𝐵↑-𝑁) ≠ 0) |
| 26 | 10 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (𝐴↑𝑁) = (𝐵↑𝑁)) |
| 27 | exp11d.3 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
| 28 | 27 | zcnd 12723 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 29 | 28 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → 𝑁 ∈ ℂ) |
| 30 | expneg2 14111 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁))) | |
| 31 | 16, 29, 17, 30 | syl3anc 1373 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (𝐴↑𝑁) = (1 / (𝐴↑-𝑁))) |
| 32 | expneg2 14111 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 𝑁 ∈ ℂ ∧ -𝑁 ∈ ℕ0) → (𝐵↑𝑁) = (1 / (𝐵↑-𝑁))) | |
| 33 | 19, 29, 17, 32 | syl3anc 1373 | . . . . 5 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (𝐵↑𝑁) = (1 / (𝐵↑-𝑁))) |
| 34 | 26, 31, 33 | 3eqtr3d 2785 | . . . 4 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (1 / (𝐴↑-𝑁)) = (1 / (𝐵↑-𝑁))) |
| 35 | 18, 20, 23, 25, 34 | rec11d 12064 | . . 3 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → (𝐴↑-𝑁) = (𝐵↑-𝑁)) |
| 36 | 13, 14, 15, 35 | exp11nnd 14300 | . 2 ⊢ ((𝜑 ∧ -𝑁 ∈ ℕ) → 𝐴 = 𝐵) |
| 37 | elz 12615 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
| 38 | 27, 37 | sylib 218 | . . 3 ⊢ (𝜑 → (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) |
| 39 | 38 | simprd 495 | . 2 ⊢ (𝜑 → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
| 40 | 4, 12, 36, 39 | mpjao3dan 1434 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2940 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 -cneg 11493 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 ℤcz 12613 ℝ+crp 13034 ↑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-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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