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Theorem regamcl 26206

Description: The Gamma function is real for real input. (Contributed by Mario Carneiro, 9-Jul-2017.)

**Assertion**
Ref	Expression
regamcl	⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ)

Proof of Theorem regamcl Dummy variables 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eldifi 4066	. . . . . 6 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ ℝ)
2	1	recnd 11002	. . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ ℂ)
3		eldifn 4067	. . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → ¬ 𝐴 ∈ (ℤ ∖ ℕ))
4	2, 3	eldifd 3903	. . . 4 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
5		gamcl 26189	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℂ)
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℂ)
7	4	dmgmn0 26171	. . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ≠ 0)
8	6, 2, 7	divcan4d 11755	. 2 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (((Γ‘𝐴) · 𝐴) / 𝐴) = (Γ‘𝐴))
9		nnuz 12618	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
10		1zzd 12349	. . . 4 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 1 ∈ ℤ)
11		eqid 2740	. . . . 5 ⊢ (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1))) = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)))
12	11, 4	gamcvg2 26205	. . . 4 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → seq1( · , (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)))) ⇝ ((Γ‘𝐴) · 𝐴))
13		simpr 485	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ)
14	13	peano2nnd 11988	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
15	14	nnrpd 12767	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℝ⁺)
16	13	nnrpd 12767	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
17	15, 16	rpdivcld 12786	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ⁺)
18	17	rpred 12769	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ)
19	17	rpge0d 12773	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 0 ≤ ((𝑚 + 1) / 𝑚))
20	1	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ ℝ)
21	18, 19, 20	recxpcld 25874	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (((𝑚 + 1) / 𝑚)↑_𝑐𝐴) ∈ ℝ)
22	20, 13	nndivred 12025	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝐴 / 𝑚) ∈ ℝ)
23		1red 10975	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 1 ∈ ℝ)
24	22, 23	readdcld 11003	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ∈ ℝ)
25	4	adantr 481	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
26	25, 13	dmgmdivn0 26173	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ≠ 0)
27	21, 24, 26	redivcld 11801	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)) ∈ ℝ)
28	27	fmpttd 6984	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1))):ℕ⟶ℝ)
29	28	ffvelrnda 6956	. . . . . 6 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)))‘𝑛) ∈ ℝ)
30		remulcl 10955	. . . . . . 7 ⊢ ((𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑛 · 𝑥) ∈ ℝ)
31	30	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ (𝑛 ∈ ℝ ∧ 𝑥 ∈ ℝ)) → (𝑛 · 𝑥) ∈ ℝ)
32	9, 10, 29, 31	seqf 13740	. . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → seq1( · , (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)))):ℕ⟶ℝ)
33	32	ffvelrnda 6956	. . . 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 15288	. . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → ((Γ‘𝐴) · 𝐴) ∈ ℝ)
35	34, 1, 7	redivcld 11801	. 2 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (((Γ‘𝐴) · 𝐴) / 𝐴) ∈ ℝ)
36	8, 35	eqeltrrd 2842	1 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2110 ∖ cdif 3889 ↦ cmpt 5162 ‘cfv 6431 (class class class)co 7269 ℂcc 10868 ℝcr 10869 1c1 10871 + caddc 10873 · cmul 10875 / cdiv 11630 ℕcn 11971 ℤcz 12317 seqcseq 13717 ↑_𝑐ccxp 25707 Γcgam 26162

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-inf2 9375 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-pre-sup 10948 ax-addf 10949 ax-mulf 10950

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-isom 6440 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-supp 7967 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-2o 8287 df-oadd 8290 df-er 8479 df-map 8598 df-pm 8599 df-ixp 8667 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-fsupp 9105 df-fi 9146 df-sup 9177 df-inf 9178 df-oi 9245 df-dju 9658 df-card 9696 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-div 11631 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-7 12039 df-8 12040 df-9 12041 df-n0 12232 df-z 12318 df-dec 12435 df-uz 12580 df-q 12686 df-rp 12728 df-xneg 12845 df-xadd 12846 df-xmul 12847 df-ioo 13080 df-ioc 13081 df-ico 13082 df-icc 13083 df-fz 13237 df-fzo 13380 df-fl 13508 df-mod 13586 df-seq 13718 df-exp 13779 df-fac 13984 df-bc 14013 df-hash 14041 df-shft 14774 df-cj 14806 df-re 14807 df-im 14808 df-sqrt 14942 df-abs 14943 df-limsup 15176 df-clim 15193 df-rlim 15194 df-sum 15394 df-ef 15773 df-sin 15775 df-cos 15776 df-tan 15777 df-pi 15778 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-starv 16973 df-sca 16974 df-vsca 16975 df-ip 16976 df-tset 16977 df-ple 16978 df-ds 16980 df-unif 16981 df-hom 16982 df-cco 16983 df-rest 17129 df-topn 17130 df-0g 17148 df-gsum 17149 df-topgen 17150 df-pt 17151 df-prds 17154 df-xrs 17209 df-qtop 17214 df-imas 17215 df-xps 17217 df-mre 17291 df-mrc 17292 df-acs 17294 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-mulg 18697 df-cntz 18919 df-cmn 19384 df-psmet 20585 df-xmet 20586 df-met 20587 df-bl 20588 df-mopn 20589 df-fbas 20590 df-fg 20591 df-cnfld 20594 df-top 22039 df-topon 22056 df-topsp 22078 df-bases 22092 df-cld 22166 df-ntr 22167 df-cls 22168 df-nei 22245 df-lp 22283 df-perf 22284 df-cn 22374 df-cnp 22375 df-haus 22462 df-cmp 22534 df-tx 22709 df-hmeo 22902 df-fil 22993 df-fm 23085 df-flim 23086 df-flf 23087 df-xms 23469 df-ms 23470 df-tms 23471 df-cncf 24037 df-limc 25026 df-dv 25027 df-ulm 25532 df-log 25708 df-cxp 25709 df-lgam 26164 df-gam 26165

This theorem is referenced by: (None)

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