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Mirrors > Home > MPE Home > Th. List > dmlogdmgm | Structured version Visualization version GIF version |
Description: If 𝐴 is in the continuous domain of the logarithm, then it is in the domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.) |
Ref | Expression |
---|---|
dmlogdmgm | ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 4105 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ∈ ℂ) | |
2 | simpr 487 | . . . 4 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → -𝐴 ∈ ℕ0) | |
3 | 2 | nn0ge0d 11961 | . . 3 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → 0 ≤ -𝐴) |
4 | 1 | adantr 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → 𝐴 ∈ ℂ) |
5 | 2 | nn0red 11959 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → -𝐴 ∈ ℝ) |
6 | 4, 5 | negrebd 10998 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → 𝐴 ∈ ℝ) |
7 | eqid 2823 | . . . . . . . . . 10 ⊢ (ℂ ∖ (-∞(,]0)) = (ℂ ∖ (-∞(,]0)) | |
8 | 7 | ellogdm 25224 | . . . . . . . . 9 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) ↔ (𝐴 ∈ ℂ ∧ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+))) |
9 | 8 | simprbi 499 | . . . . . . . 8 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → (𝐴 ∈ ℝ → 𝐴 ∈ ℝ+)) |
10 | 9 | imp 409 | . . . . . . 7 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ 𝐴 ∈ ℝ) → 𝐴 ∈ ℝ+) |
11 | 6, 10 | syldan 593 | . . . . . 6 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → 𝐴 ∈ ℝ+) |
12 | 11 | rpgt0d 12437 | . . . . 5 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → 0 < 𝐴) |
13 | 6 | lt0neg2d 11212 | . . . . 5 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → (0 < 𝐴 ↔ -𝐴 < 0)) |
14 | 12, 13 | mpbid 234 | . . . 4 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → -𝐴 < 0) |
15 | 0red 10646 | . . . . 5 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → 0 ∈ ℝ) | |
16 | 5, 15 | ltnled 10789 | . . . 4 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → (-𝐴 < 0 ↔ ¬ 0 ≤ -𝐴)) |
17 | 14, 16 | mpbid 234 | . . 3 ⊢ ((𝐴 ∈ (ℂ ∖ (-∞(,]0)) ∧ -𝐴 ∈ ℕ0) → ¬ 0 ≤ -𝐴) |
18 | 3, 17 | pm2.65da 815 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → ¬ -𝐴 ∈ ℕ0) |
19 | eldmgm 25601 | . 2 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) | |
20 | 1, 18, 19 | sylanbrc 585 | 1 ⊢ (𝐴 ∈ (ℂ ∖ (-∞(,]0)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∈ wcel 2114 ∖ cdif 3935 class class class wbr 5068 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 -∞cmnf 10675 < clt 10677 ≤ cle 10678 -cneg 10873 ℕcn 11640 ℕ0cn0 11900 ℤcz 11984 ℝ+crp 12392 (,]cioc 12742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-rp 12393 df-ioc 12746 |
This theorem is referenced by: rpdmgm 25604 |
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