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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 26165 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 2739 | . . 3 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
4 | 1, 2, 3 | lgamucov 26168 | . 2 ⊢ (𝜑 → ∃𝑟 ∈ ℕ 𝐴 ∈ ((int‘(TopOpen‘ℂfld))‘𝑈)) |
5 | 3 | cnfldtop 23928 | . . . . 5 ⊢ (TopOpen‘ℂfld) ∈ Top |
6 | 1 | ssrab3 4019 | . . . . 5 ⊢ 𝑈 ⊆ ℂ |
7 | unicntop 23930 | . . . . . 6 ⊢ ℂ = ∪ (TopOpen‘ℂfld) | |
8 | 7 | ntrss2 22189 | . . . . 5 ⊢ (((TopOpen‘ℂfld) ∈ Top ∧ 𝑈 ⊆ ℂ) → ((int‘(TopOpen‘ℂfld))‘𝑈) ⊆ 𝑈) |
9 | 5, 6, 8 | mp2an 688 | . . . 4 ⊢ ((int‘(TopOpen‘ℂfld))‘𝑈) ⊆ 𝑈 |
10 | 9 | sseli 3921 | . . 3 ⊢ (𝐴 ∈ ((int‘(TopOpen‘ℂfld))‘𝑈) → 𝐴 ∈ 𝑈) |
11 | 10 | reximi 3176 | . 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 395 = wceq 1541 ∈ wcel 2109 ∀wral 3065 ∃wrex 3066 {crab 3069 ∖ cdif 3888 ⊆ wss 3891 class class class wbr 5078 ‘cfv 6430 (class class class)co 7268 ℂcc 10853 1c1 10856 + caddc 10858 ≤ cle 10994 / cdiv 11615 ℕcn 11956 ℕ0cn0 12216 ℤcz 12302 abscabs 14926 TopOpenctopn 17113 ℂfldccnfld 20578 Topctop 22023 intcnt 22149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fi 9131 df-sup 9162 df-inf 9163 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-fz 13222 df-fl 13493 df-seq 13703 df-exp 13764 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-struct 16829 df-slot 16864 df-ndx 16876 df-base 16894 df-plusg 16956 df-mulr 16957 df-starv 16958 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-rest 17114 df-topn 17115 df-topgen 17135 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-cnfld 20579 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-cld 22151 df-ntr 22152 df-xms 23454 df-ms 23455 |
This theorem is referenced by: lgamcvglem 26170 |
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