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Mirrors > Home > MPE Home > Th. List > leexp2ad | Structured version Visualization version GIF version |
Description: Ordering relationship for exponentiation. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
resqcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
leexp2ad.2 | ⊢ (𝜑 → 1 ≤ 𝐴) |
leexp2ad.3 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
Ref | Expression |
---|---|
leexp2ad | ⊢ (𝜑 → (𝐴↑𝑀) ≤ (𝐴↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resqcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | leexp2ad.2 | . 2 ⊢ (𝜑 → 1 ≤ 𝐴) | |
3 | leexp2ad.3 | . 2 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
4 | leexp2a 13888 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 1 ≤ 𝐴 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝐴↑𝑀) ≤ (𝐴↑𝑁)) | |
5 | 1, 2, 3, 4 | syl3anc 1370 | 1 ⊢ (𝜑 → (𝐴↑𝑀) ≤ (𝐴↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 class class class wbr 5076 ‘cfv 6435 (class class class)co 7277 ℝcr 10868 1c1 10870 ≤ cle 11008 ℤ≥cuz 12580 ↑cexp 13780 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-n0 12232 df-z 12318 df-uz 12581 df-rp 12729 df-seq 13720 df-exp 13781 |
This theorem is referenced by: faclbnd 14002 faclbnd3 14004 bitsmod 16141 fislw 19228 plyeq0lem 25369 aaliou3lem8 25503 ftalem1 26220 ftalem2 26221 vma1 26313 fsumvma2 26360 chpchtsum 26365 bposlem2 26431 chebbnd1lem1 26615 pntlemf 26751 ostth2lem2 26780 aks4d1p3 40083 jm3.1lem2 40837 etransclem23 43768 |
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