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Mirrors > Home > MPE Home > Th. List > lgamucov2 | Structured version Visualization version GIF version |
Description: The 𝑈 regions used in the proof of lgamgulm 25329 have interiors which cover the entire domain of the Gamma function. (Contributed by Mario Carneiro, 8-Jul-2017.) |
Ref | Expression |
---|---|
lgamucov.u | ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))} |
lgamucov.a | ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
Ref | Expression |
---|---|
lgamucov2 | ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lgamucov.u | . . 3 ⊢ 𝑈 = {𝑥 ∈ ℂ ∣ ((abs‘𝑥) ≤ 𝑟 ∧ ∀𝑘 ∈ ℕ0 (1 / 𝑟) ≤ (abs‘(𝑥 + 𝑘)))} | |
2 | lgamucov.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) | |
3 | eqid 2771 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
4 | 1, 2, 3 | lgamucov 25332 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘(TopOpen‘ℂfld))‘𝑈)) |
5 | 3 | cnfldtop 23110 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
6 | 1 | ssrab3 3940 | . . . . 5 ⊢ 𝑈 ⊆ ℂ |
7 | unicntop 23112 | . . . . . 6 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
8 | 7 | ntrss2 21384 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑈 ⊆ ℂ) → ((int‘(TopOpen‘ℂfld))‘𝑈) ⊆ 𝑈) |
9 | 5, 6, 8 | mp2an 680 | . . . 4 ⊢ ((int‘(TopOpen‘ℂfld))‘𝑈) ⊆ 𝑈 |
10 | 9 | sseli 3847 | . . 3 ⊢ (𝐴 ∈ ((int‘(TopOpen‘ℂfld))‘𝑈) → 𝐴 ∈ 𝑈) |
11 | 10 | reximi 3183 | . 2 ⊢ (∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘(TopOpen‘ℂfld))‘𝑈) → ∃𝑟 ∈ ℕ 𝐴 ∈ 𝑈) |
12 | 4, 11 | syl 17 | 1 ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∈ wcel 2051 ∀wral 3081 ∃wrex 3082 {crab 3085 ∖ cdif 3819 ⊆ wss 3822 class class class wbr 4925 ‘cfv 6185 (class class class)co 6974 ℂcc 10331 1c1 10334 + caddc 10336 ≤ cle 10473 / cdiv 11096 ℕcn 11437 ℕ0cn0 11705 ℤcz 11791 abscabs 14452 TopOpenctopn 16549 ℂfldccnfld 20262 Topctop 21220 intcnt 21344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-cnex 10389 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 ax-pre-mulgt0 10410 ax-pre-sup 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-iin 4791 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-lim 6031 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-om 7395 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-rdg 7848 df-1o 7903 df-oadd 7907 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-sdom 8307 df-fin 8308 df-fi 8668 df-sup 8699 df-inf 8700 df-pnf 10474 df-mnf 10475 df-xr 10476 df-ltxr 10477 df-le 10478 df-sub 10670 df-neg 10671 df-div 11097 df-nn 11438 df-2 11501 df-3 11502 df-4 11503 df-5 11504 df-6 11505 df-7 11506 df-8 11507 df-9 11508 df-n0 11706 df-z 11792 df-dec 11910 df-uz 12057 df-q 12161 df-rp 12203 df-xneg 12322 df-xadd 12323 df-xmul 12324 df-ioo 12556 df-fz 12707 df-fl 12975 df-seq 13183 df-exp 13243 df-cj 14317 df-re 14318 df-im 14319 df-sqrt 14453 df-abs 14454 df-struct 16339 df-ndx 16340 df-slot 16341 df-base 16343 df-plusg 16432 df-mulr 16433 df-starv 16434 df-tset 16438 df-ple 16439 df-ds 16441 df-unif 16442 df-rest 16550 df-topn 16551 df-topgen 16571 df-psmet 20254 df-xmet 20255 df-met 20256 df-bl 20257 df-mopn 20258 df-cnfld 20263 df-top 21221 df-topon 21238 df-topsp 21260 df-bases 21273 df-cld 21346 df-ntr 21347 df-xms 22648 df-ms 22649 |
This theorem is referenced by: lgamcvglem 25334 |
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