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Mirrors > Home > MPE Home > Th. List > regamcl | Structured version Visualization version GIF version |
Description: The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.) |
Ref | Expression |
---|---|
regamcl | ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifi 4141 | . . . . . 6 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ ℝ) | |
2 | 1 | recnd 11287 | . . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ ℂ) |
3 | eldifn 4142 | . . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → ¬ 𝐴 ∈ (ℤ ∖ ℕ)) | |
4 | 2, 3 | eldifd 3974 | . . . 4 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
5 | gamcl 27102 | . . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℂ) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℂ) |
7 | 4 | dmgmn0 27084 | . . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ≠ 0) |
8 | 6, 2, 7 | divcan4d 12047 | . 2 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (((Γ‘𝐴) · 𝐴) / 𝐴) = (Γ‘𝐴)) |
9 | nnuz 12919 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
10 | 1zzd 12646 | . . . 4 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 1 ∈ ℤ) | |
11 | eqid 2735 | . . . . 5 ⊢ (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1))) = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1))) | |
12 | 11, 4 | gamcvg2 27118 | . . . 4 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → seq1( · , (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)))) ⇝ ((Γ‘𝐴) · 𝐴)) |
13 | simpr 484 | . . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ) | |
14 | 13 | peano2nnd 12281 | . . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ) |
15 | 14 | nnrpd 13073 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℝ+) |
16 | 13 | nnrpd 13073 | . . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ+) |
17 | 15, 16 | rpdivcld 13092 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ+) |
18 | 17 | rpred 13075 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ) |
19 | 17 | rpge0d 13079 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 0 ≤ ((𝑚 + 1) / 𝑚)) |
20 | 1 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℝ) |
21 | 18, 19, 20 | recxpcld 26780 | . . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (((𝑚 + 1) / 𝑚)↑𝑐𝐴) ∈ ℝ) |
22 | 20, 13 | nndivred 12318 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝐴 / 𝑚) ∈ ℝ) |
23 | 1red 11260 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 1 ∈ ℝ) | |
24 | 22, 23 | readdcld 11288 | . . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ∈ ℝ) |
25 | 4 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
26 | 25, 13 | dmgmdivn0 27086 | . . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ≠ 0) |
27 | 21, 24, 26 | redivcld 12093 | . . . . . . . 8 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)) ∈ ℝ) |
28 | 27 | fmpttd 7135 | . . . . . . 7 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1))):ℕ⟶ℝ) |
29 | 28 | ffvelcdmda 7104 | . . . . . 6 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)))‘𝑛) ∈ ℝ) |
30 | remulcl 11238 | . . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑛 · 𝑥) ∈ ℝ) | |
31 | 30 | adantl 481 | . . . . . 6 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ (𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ)) → (𝑛 · 𝑥) ∈ ℝ) |
32 | 9, 10, 29, 31 | seqf 14061 | . . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → seq1( · , (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1)))):ℕ⟶ℝ) |
33 | 32 | ffvelcdmda 7104 | . . . 4 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑛 ∈ ℕ) → (seq1( · , (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑𝑐𝐴) / ((𝐴 / 𝑚) + 1))))‘𝑛) ∈ ℝ) |
34 | 9, 10, 12, 33 | climrecl 15616 | . . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → ((Γ‘𝐴) · 𝐴) ∈ ℝ) |
35 | 34, 1, 7 | redivcld 12093 | . 2 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (((Γ‘𝐴) · 𝐴) / 𝐴) ∈ ℝ) |
36 | 8, 35 | eqeltrrd 2840 | 1 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2106 ∖ cdif 3960 ↦ cmpt 5231 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 1c1 11154 + caddc 11156 · cmul 11158 / cdiv 11918 ℕcn 12264 ℤcz 12611 seqcseq 14039 ↑𝑐ccxp 26612 Γcgam 27075 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-oadd 8509 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-dju 9939 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-tan 16104 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-cmp 23411 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-ulm 26435 df-log 26613 df-cxp 26614 df-lgam 27077 df-gam 27078 |
This theorem is referenced by: (None) |
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