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Mirrors > Home > MPE Home > Th. List > ltexp2 | Structured version Visualization version GIF version |
Description: Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.) |
Ref | Expression |
---|---|
ltexp2 | ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 7199 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝐴↑𝑥) = (𝐴↑𝑦)) | |
2 | oveq2 7199 | . . . . . 6 ⊢ (𝑥 = 𝑀 → (𝐴↑𝑥) = (𝐴↑𝑀)) | |
3 | oveq2 7199 | . . . . . 6 ⊢ (𝑥 = 𝑁 → (𝐴↑𝑥) = (𝐴↑𝑁)) | |
4 | zssre 12148 | . . . . . 6 ⊢ ℤ ⊆ ℝ | |
5 | simpl 486 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 𝐴 ∈ ℝ) | |
6 | 0red 10801 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 0 ∈ ℝ) | |
7 | 1red 10799 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 1 ∈ ℝ) | |
8 | 0lt1 11319 | . . . . . . . . . . 11 ⊢ 0 < 1 | |
9 | 8 | a1i 11 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 0 < 1) |
10 | simpr 488 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 1 < 𝐴) | |
11 | 6, 7, 5, 9, 10 | lttrd 10958 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 0 < 𝐴) |
12 | 5, 11 | elrpd 12590 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 1 < 𝐴) → 𝐴 ∈ ℝ+) |
13 | rpexpcl 13619 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℝ+ ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ ℝ+) | |
14 | 12, 13 | sylan 583 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ ℝ+) |
15 | 14 | rpred 12593 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ 𝑥 ∈ ℤ) → (𝐴↑𝑥) ∈ ℝ) |
16 | simpll 767 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝐴 ∈ ℝ) | |
17 | simprl 771 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑥 ∈ ℤ) | |
18 | simprr 773 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 𝑦 ∈ ℤ) | |
19 | simplr 769 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → 1 < 𝐴) | |
20 | ltexp2a 13701 | . . . . . . . 8 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ (1 < 𝐴 ∧ 𝑥 < 𝑦)) → (𝐴↑𝑥) < (𝐴↑𝑦)) | |
21 | 20 | expr 460 | . . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ) ∧ 1 < 𝐴) → (𝑥 < 𝑦 → (𝐴↑𝑥) < (𝐴↑𝑦))) |
22 | 16, 17, 18, 19, 21 | syl31anc 1375 | . . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝑥 ∈ ℤ ∧ 𝑦 ∈ ℤ)) → (𝑥 < 𝑦 → (𝐴↑𝑥) < (𝐴↑𝑦))) |
23 | 1, 2, 3, 4, 15, 22 | ltord1 11323 | . . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ)) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁))) |
24 | 23 | ancom2s 650 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 1 < 𝐴) ∧ (𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ)) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁))) |
25 | 24 | exp43 440 | . . 3 ⊢ (𝐴 ∈ ℝ → (1 < 𝐴 → (𝑁 ∈ ℤ → (𝑀 ∈ ℤ → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))))) |
26 | 25 | com24 95 | . 2 ⊢ (𝐴 ∈ ℝ → (𝑀 ∈ ℤ → (𝑁 ∈ ℤ → (1 < 𝐴 → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))))) |
27 | 26 | 3imp1 1349 | 1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 ∈ wcel 2112 class class class wbr 5039 (class class class)co 7191 ℝcr 10693 0cc0 10694 1c1 10695 < clt 10832 ℤcz 12141 ℝ+crp 12551 ↑cexp 13600 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-n0 12056 df-z 12142 df-uz 12404 df-rp 12552 df-seq 13540 df-exp 13601 |
This theorem is referenced by: leexp2 13706 ltexp2r 13708 ltexp2d 13785 |
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