gamp1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > gamp1		Structured version Visualization version GIF version

Theorem gamp1 27326

Description: The functional equation of the Gamma function. (Contributed by Mario Carneiro, 9-Jul-2017.)

**Assertion**
Ref	Expression
gamp1	⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘(𝐴 + 1)) = ((Γ‘𝐴) · 𝐴))

**Proof of Theorem gamp1**
Step	Hyp	Ref	Expression
1		lgamp1 27325	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘(𝐴 + 1)) = ((log Γ‘𝐴) + (log‘𝐴)))
2	1	fveq2d 6885	. . 3 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘(𝐴 + 1))) = (exp‘((log Γ‘𝐴) + (log‘𝐴))))
3		lgamcl 27309	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log Γ‘𝐴) ∈ ℂ)
4		eldifi 4078	. . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ ℂ)
5		id 23	. . . . . 6 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))
6	5	dmgmn0 27294	. . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → 𝐴 ≠ 0)
7	4, 6	logcld 26839	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (log‘𝐴) ∈ ℂ)
8		efadd 16205	. . . 4 ⊢ (((log Γ‘𝐴) ∈ ℂ ∧ (log‘𝐴) ∈ ℂ) → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))))
9	3, 7, 8	syl2anc 596	. . 3 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘((log Γ‘𝐴) + (log‘𝐴))) = ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))))
10		eflog 26845	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (exp‘(log‘𝐴)) = 𝐴)
11	4, 6, 10	syl2anc 596	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log‘𝐴)) = 𝐴)
12	11	oveq2d 7432	. . 3 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → ((exp‘(log Γ‘𝐴)) · (exp‘(log‘𝐴))) = ((exp‘(log Γ‘𝐴)) · 𝐴))
13	2, 9, 12	3eqtrd 2799	. 2 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘(𝐴 + 1))) = ((exp‘(log Γ‘𝐴)) · 𝐴))
14		1nn0 12569	. . . . 5 ⊢ 1 ∈ ℕ₀
15	14	a1i 11	. . . 4 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → 1 ∈ ℕ₀)
16	5, 15	dmgmaddnn0 27295	. . 3 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (𝐴 + 1) ∈ (ℂ ∖ (ℤ ∖ ℕ)))
17		eflgam 27313	. . 3 ⊢ ((𝐴 + 1) ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘(𝐴 + 1))) = (Γ‘(𝐴 + 1)))
18	16, 17	syl 18	. 2 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘(𝐴 + 1))) = (Γ‘(𝐴 + 1)))
19		eflgam 27313	. . 3 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (exp‘(log Γ‘𝐴)) = (Γ‘𝐴))
20	19	oveq1d 7431	. 2 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → ((exp‘(log Γ‘𝐴)) · 𝐴) = ((Γ‘𝐴) · 𝐴))
21	13, 18, 20	3eqtr3d 2803	1 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) → (Γ‘(𝐴 + 1)) = ((Γ‘𝐴) · 𝐴))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∖ cdif 3896 ‘cfv 6535 (class class class)co 7416 ℂcc 11147 0cc0 11149 1c1 11150 + caddc 11152 · cmul 11154 ℕcn 12282 ℕ₀cn0 12553 ℤcz 12640 expce 16172 logclog 26823 log Γclgam 27284 Γcgam 27285

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-inf2 9627 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-oadd 8466 df-er 8703 df-map 8835 df-pm 8836 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9489 df-dju 9931 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-ioo 13427 df-ioc 13428 df-ico 13429 df-icc 13430 df-fz 13587 df-fzo 13735 df-fl 13878 df-mod 13956 df-seq 14091 df-exp 14151 df-fac 14363 df-bc 14392 df-hash 14420 df-shft 15165 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-limsup 15583 df-clim 15600 df-rlim 15601 df-sum 15799 df-ef 16178 df-sin 16180 df-cos 16181 df-tan 16182 df-pi 16183 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-rest 17532 df-topn 17533 df-0g 17551 df-gsum 17552 df-topgen 17553 df-pt 17554 df-prds 17557 df-xrs 17613 df-qtop 17618 df-imas 17619 df-xps 17621 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-mulg 19217 df-cntz 19470 df-cmn 19935 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-fbas 21614 df-fg 21615 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-ntr 23277 df-cls 23278 df-nei 23355 df-lp 23393 df-perf 23394 df-cn 23484 df-cnp 23485 df-haus 23572 df-cmp 23644 df-tx 23820 df-hmeo 24013 df-fil 24104 df-fm 24196 df-flim 24197 df-flf 24198 df-xms 24578 df-ms 24579 df-tms 24580 df-cncf 25138 df-limc 26125 df-dv 26126 df-ulm 26645 df-log 26825 df-cxp 26826 df-lgam 27287 df-gam 27288

This theorem is used by: facgam 27334

W3C validator