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Theorem regamcl 26802

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 4126	. . . . . 6 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ ℝ)
2	1	recnd 11247	. . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ ℂ)
3		eldifn 4127	. . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → ¬ 𝐴 ∈ (ℤ ∖ ℕ))
4	2, 3	eldifd 3959	. . . 4 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
5		gamcl 26785	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℂ)
6	4, 5	syl 17	. . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℂ)
7	4	dmgmn0 26767	. . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 𝐴 ≠ 0)
8	6, 2, 7	divcan4d 12001	. 2 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (((Γ‘𝐴) · 𝐴) / 𝐴) = (Γ‘𝐴))
9		nnuz 12870	. . . 4 ⊢ ℕ = (ℤ_≥‘1)
10		1zzd 12598	. . . 4 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → 1 ∈ ℤ)
11		eqid 2731	. . . . 5 ⊢ (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1))) = (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)))
12	11, 4	gamcvg2 26801	. . . 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 12234	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℕ)
15	14	nnrpd 13019	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝑚 + 1) ∈ ℝ⁺)
16	13	nnrpd 13019	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℝ⁺)
17	15, 16	rpdivcld 13038	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ⁺)
18	17	rpred 13021	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝑚 + 1) / 𝑚) ∈ ℝ)
19	17	rpge0d 13025	. . . . . . . . . 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 26468	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (((𝑚 + 1) / 𝑚)↑_𝑐𝐴) ∈ ℝ)
22	20, 13	nndivred 12271	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → (𝐴 / 𝑚) ∈ ℝ)
23		1red 11220	. . . . . . . . . 10 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → 1 ∈ ℝ)
24	22, 23	readdcld 11248	. . . . . . . . 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 26769	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((𝐴 / 𝑚) + 1) ≠ 0)
27	21, 24, 26	redivcld 12047	. . . . . . . 8 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑚 ∈ ℕ) → ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)) ∈ ℝ)
28	27	fmpttd 7116	. . . . . . 7 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1))):ℕ⟶ℝ)
29	28	ffvelcdmda 7086	. . . . . 6 ⊢ ((𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) ∧ 𝑛 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)))‘𝑛) ∈ ℝ)
30		remulcl 11199	. . . . . . 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 13994	. . . . 5 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → seq1( · , (𝑚 ∈ ℕ ↦ ((((𝑚 + 1) / 𝑚)↑_𝑐𝐴) / ((𝐴 / 𝑚) + 1)))):ℕ⟶ℝ)
33	32	ffvelcdmda 7086	. . . 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 15532	. . 3 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → ((Γ‘𝐴) · 𝐴) ∈ ℝ)
35	34, 1, 7	redivcld 12047	. 2 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (((Γ‘𝐴) · 𝐴) / 𝐴) ∈ ℝ)
36	8, 35	eqeltrrd 2833	1 ⊢ (𝐴 ∈ (ℝ ∖ (ℤ ∖ ℕ)) → (Γ‘𝐴) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2105 ∖ cdif 3945 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7412 ℂcc 11112 ℝcr 11113 1c1 11115 + caddc 11117 · cmul 11119 / cdiv 11876 ℕcn 12217 ℤcz 12563 seqcseq 13971 ↑_𝑐ccxp 26301 Γcgam 26758

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-inf2 9640 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191 ax-pre-sup 11192 ax-addf 11193 ax-mulf 11194

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7674 df-om 7860 df-1st 7979 df-2nd 7980 df-supp 8151 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-oadd 8474 df-er 8707 df-map 8826 df-pm 8827 df-ixp 8896 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fsupp 9366 df-fi 9410 df-sup 9441 df-inf 9442 df-oi 9509 df-dju 9900 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-ioc 13334 df-ico 13335 df-icc 13336 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15019 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-limsup 15420 df-clim 15437 df-rlim 15438 df-sum 15638 df-ef 16016 df-sin 16018 df-cos 16019 df-tan 16020 df-pi 16021 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-mulg 18988 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-fbas 21142 df-fg 21143 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-ntr 22745 df-cls 22746 df-nei 22823 df-lp 22861 df-perf 22862 df-cn 22952 df-cnp 22953 df-haus 23040 df-cmp 23112 df-tx 23287 df-hmeo 23480 df-fil 23571 df-fm 23663 df-flim 23664 df-flf 23665 df-xms 24047 df-ms 24048 df-tms 24049 df-cncf 24619 df-limc 25616 df-dv 25617 df-ulm 26126 df-log 26302 df-cxp 26303 df-lgam 26760 df-gam 26761

This theorem is referenced by: (None)

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