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Theorem cnllycmp 24703
Description: The topology on the complex numbers is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
Hypothesis
Ref Expression
cnllycmp.1 𝐽 = (TopOpenβ€˜β„‚fld)
Assertion
Ref Expression
cnllycmp 𝐽 ∈ 𝑛-Locally Comp

Proof of Theorem cnllycmp
Dummy variables 𝑠 π‘Ÿ 𝑒 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnllycmp.1 . . 3 𝐽 = (TopOpenβ€˜β„‚fld)
21cnfldtop 24521 . 2 𝐽 ∈ Top
3 cnxmet 24510 . . . . 5 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
41cnfldtopn 24519 . . . . . 6 𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))
54mopni2 24223 . . . . 5 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)
63, 5mp3an1 1447 . . . 4 ((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)
72a1i 11 . . . . . . 7 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ 𝐽 ∈ Top)
83a1i 11 . . . . . . . . 9 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚))
9 elssuni 4941 . . . . . . . . . . . 12 (π‘₯ ∈ 𝐽 β†’ π‘₯ βŠ† βˆͺ 𝐽)
109ad2antrr 723 . . . . . . . . . . 11 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ π‘₯ βŠ† βˆͺ 𝐽)
111cnfldtopon 24520 . . . . . . . . . . . 12 𝐽 ∈ (TopOnβ€˜β„‚)
1211toponunii 22639 . . . . . . . . . . 11 β„‚ = βˆͺ 𝐽
1310, 12sseqtrrdi 4033 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ π‘₯ βŠ† β„‚)
14 simplr 766 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ 𝑦 ∈ π‘₯)
1513, 14sseldd 3983 . . . . . . . . 9 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ 𝑦 ∈ β„‚)
16 rphalfcl 13006 . . . . . . . . . . 11 (π‘Ÿ ∈ ℝ+ β†’ (π‘Ÿ / 2) ∈ ℝ+)
1716ad2antrl 725 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (π‘Ÿ / 2) ∈ ℝ+)
1817rpxrd 13022 . . . . . . . . 9 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (π‘Ÿ / 2) ∈ ℝ*)
194blopn 24230 . . . . . . . . 9 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑦 ∈ β„‚ ∧ (π‘Ÿ / 2) ∈ ℝ*) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) ∈ 𝐽)
208, 15, 18, 19syl3anc 1370 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) ∈ 𝐽)
21 blcntr 24140 . . . . . . . . 9 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑦 ∈ β„‚ ∧ (π‘Ÿ / 2) ∈ ℝ+) β†’ 𝑦 ∈ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))
228, 15, 17, 21syl3anc 1370 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ 𝑦 ∈ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))
23 opnneip 22844 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) ∈ 𝐽 ∧ 𝑦 ∈ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) ∈ ((neiβ€˜π½)β€˜{𝑦}))
247, 20, 22, 23syl3anc 1370 . . . . . . 7 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) ∈ ((neiβ€˜π½)β€˜{𝑦}))
25 blssm 24145 . . . . . . . . 9 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑦 ∈ β„‚ ∧ (π‘Ÿ / 2) ∈ ℝ*) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) βŠ† β„‚)
268, 15, 18, 25syl3anc 1370 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) βŠ† β„‚)
2712sscls 22781 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) βŠ† β„‚) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) βŠ† ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))))
287, 26, 27syl2anc 583 . . . . . . 7 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) βŠ† ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))))
29 rpxr 12988 . . . . . . . . . . 11 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
3029ad2antrl 725 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ π‘Ÿ ∈ ℝ*)
31 rphalflt 13008 . . . . . . . . . . 11 (π‘Ÿ ∈ ℝ+ β†’ (π‘Ÿ / 2) < π‘Ÿ)
3231ad2antrl 725 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (π‘Ÿ / 2) < π‘Ÿ)
334blsscls 24237 . . . . . . . . . 10 ((((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑦 ∈ β„‚) ∧ ((π‘Ÿ / 2) ∈ ℝ* ∧ π‘Ÿ ∈ ℝ* ∧ (π‘Ÿ / 2) < π‘Ÿ)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ))
348, 15, 18, 30, 32, 33syl23anc 1376 . . . . . . . . 9 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ))
35 simprr 770 . . . . . . . . 9 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)
3634, 35sstrd 3992 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† π‘₯)
3736, 13sstrd 3992 . . . . . . 7 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† β„‚)
3812ssnei2 22841 . . . . . . 7 (((𝐽 ∈ Top ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) ∈ ((neiβ€˜π½)β€˜{𝑦})) ∧ ((𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) βŠ† ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† β„‚)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ ((neiβ€˜π½)β€˜{𝑦}))
397, 24, 28, 37, 38syl22anc 836 . . . . . 6 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ ((neiβ€˜π½)β€˜{𝑦}))
40 vex 3477 . . . . . . . 8 π‘₯ ∈ V
4140elpw2 5345 . . . . . . 7 (((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ 𝒫 π‘₯ ↔ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† π‘₯)
4236, 41sylibr 233 . . . . . 6 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ 𝒫 π‘₯)
4339, 42elind 4194 . . . . 5 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯))
4412clscld 22772 . . . . . . 7 ((𝐽 ∈ Top ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) βŠ† β„‚) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ (Clsdβ€˜π½))
457, 26, 44syl2anc 583 . . . . . 6 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ (Clsdβ€˜π½))
4615abscld 15388 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (absβ€˜π‘¦) ∈ ℝ)
4717rpred 13021 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (π‘Ÿ / 2) ∈ ℝ)
4846, 47readdcld 11248 . . . . . . 7 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((absβ€˜π‘¦) + (π‘Ÿ / 2)) ∈ ℝ)
49 eqid 2731 . . . . . . . . . 10 {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)} = {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)}
504, 49blcls 24236 . . . . . . . . 9 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑦 ∈ β„‚ ∧ (π‘Ÿ / 2) ∈ ℝ*) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)})
518, 15, 18, 50syl3anc 1370 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)})
52 simpr 484 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ 𝑧 ∈ β„‚)
5315adantr 480 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ 𝑦 ∈ β„‚)
5452, 53abs2difd 15409 . . . . . . . . . . . 12 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ ((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ≀ (absβ€˜(𝑧 βˆ’ 𝑦)))
5552abscld 15388 . . . . . . . . . . . . . 14 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (absβ€˜π‘§) ∈ ℝ)
5646adantr 480 . . . . . . . . . . . . . 14 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (absβ€˜π‘¦) ∈ ℝ)
5755, 56resubcld 11647 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ ((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ∈ ℝ)
5852, 53subcld 11576 . . . . . . . . . . . . . 14 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (𝑧 βˆ’ 𝑦) ∈ β„‚)
5958abscld 15388 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (absβ€˜(𝑧 βˆ’ 𝑦)) ∈ ℝ)
6047adantr 480 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (π‘Ÿ / 2) ∈ ℝ)
61 letr 11313 . . . . . . . . . . . . 13 ((((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ∈ ℝ ∧ (absβ€˜(𝑧 βˆ’ 𝑦)) ∈ ℝ ∧ (π‘Ÿ / 2) ∈ ℝ) β†’ ((((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ≀ (absβ€˜(𝑧 βˆ’ 𝑦)) ∧ (absβ€˜(𝑧 βˆ’ 𝑦)) ≀ (π‘Ÿ / 2)) β†’ ((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ≀ (π‘Ÿ / 2)))
6257, 59, 60, 61syl3anc 1370 . . . . . . . . . . . 12 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ ((((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ≀ (absβ€˜(𝑧 βˆ’ 𝑦)) ∧ (absβ€˜(𝑧 βˆ’ 𝑦)) ≀ (π‘Ÿ / 2)) β†’ ((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ≀ (π‘Ÿ / 2)))
6354, 62mpand 692 . . . . . . . . . . 11 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ ((absβ€˜(𝑧 βˆ’ 𝑦)) ≀ (π‘Ÿ / 2) β†’ ((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ≀ (π‘Ÿ / 2)))
6452, 53abssubd 15405 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (absβ€˜(𝑧 βˆ’ 𝑦)) = (absβ€˜(𝑦 βˆ’ 𝑧)))
65 eqid 2731 . . . . . . . . . . . . . . 15 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
6665cnmetdval 24508 . . . . . . . . . . . . . 14 ((𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ (𝑦(abs ∘ βˆ’ )𝑧) = (absβ€˜(𝑦 βˆ’ 𝑧)))
6715, 66sylan 579 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (𝑦(abs ∘ βˆ’ )𝑧) = (absβ€˜(𝑦 βˆ’ 𝑧)))
6864, 67eqtr4d 2774 . . . . . . . . . . . 12 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (absβ€˜(𝑧 βˆ’ 𝑦)) = (𝑦(abs ∘ βˆ’ )𝑧))
6968breq1d 5158 . . . . . . . . . . 11 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ ((absβ€˜(𝑧 βˆ’ 𝑦)) ≀ (π‘Ÿ / 2) ↔ (𝑦(abs ∘ βˆ’ )𝑧) ≀ (π‘Ÿ / 2)))
7055, 56, 60lesubadd2d 11818 . . . . . . . . . . 11 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (((absβ€˜π‘§) βˆ’ (absβ€˜π‘¦)) ≀ (π‘Ÿ / 2) ↔ (absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2))))
7163, 69, 703imtr3d 293 . . . . . . . . . 10 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ ((𝑦(abs ∘ βˆ’ )𝑧) ≀ (π‘Ÿ / 2) β†’ (absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2))))
7271ralrimiva 3145 . . . . . . . . 9 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ βˆ€π‘§ ∈ β„‚ ((𝑦(abs ∘ βˆ’ )𝑧) ≀ (π‘Ÿ / 2) β†’ (absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2))))
73 oveq2 7420 . . . . . . . . . . 11 (𝑀 = 𝑧 β†’ (𝑦(abs ∘ βˆ’ )𝑀) = (𝑦(abs ∘ βˆ’ )𝑧))
7473breq1d 5158 . . . . . . . . . 10 (𝑀 = 𝑧 β†’ ((𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2) ↔ (𝑦(abs ∘ βˆ’ )𝑧) ≀ (π‘Ÿ / 2)))
7574ralrab 3689 . . . . . . . . 9 (βˆ€π‘§ ∈ {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)} (absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2)) ↔ βˆ€π‘§ ∈ β„‚ ((𝑦(abs ∘ βˆ’ )𝑧) ≀ (π‘Ÿ / 2) β†’ (absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2))))
7672, 75sylibr 233 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ βˆ€π‘§ ∈ {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)} (absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2)))
77 ssralv 4050 . . . . . . . 8 (((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)} β†’ (βˆ€π‘§ ∈ {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)} (absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2)) β†’ βˆ€π‘§ ∈ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))(absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2))))
7851, 76, 77sylc 65 . . . . . . 7 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ βˆ€π‘§ ∈ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))(absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2)))
79 brralrspcev 5208 . . . . . . 7 ((((absβ€˜π‘¦) + (π‘Ÿ / 2)) ∈ ℝ ∧ βˆ€π‘§ ∈ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))(absβ€˜π‘§) ≀ ((absβ€˜π‘¦) + (π‘Ÿ / 2))) β†’ βˆƒπ‘  ∈ ℝ βˆ€π‘§ ∈ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))(absβ€˜π‘§) ≀ 𝑠)
8048, 78, 79syl2anc 583 . . . . . 6 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ βˆƒπ‘  ∈ ℝ βˆ€π‘§ ∈ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))(absβ€˜π‘§) ≀ 𝑠)
81 eqid 2731 . . . . . . . 8 (𝐽 β†Ύt ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))) = (𝐽 β†Ύt ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))))
821, 81cnheibor 24702 . . . . . . 7 (((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) βŠ† β„‚ β†’ ((𝐽 β†Ύt ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))) ∈ Comp ↔ (((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘  ∈ ℝ βˆ€π‘§ ∈ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))(absβ€˜π‘§) ≀ 𝑠)))
8337, 82syl 17 . . . . . 6 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ ((𝐽 β†Ύt ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))) ∈ Comp ↔ (((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ (Clsdβ€˜π½) ∧ βˆƒπ‘  ∈ ℝ βˆ€π‘§ ∈ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))(absβ€˜π‘§) ≀ 𝑠)))
8445, 80, 83mpbir2and 710 . . . . 5 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (𝐽 β†Ύt ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))) ∈ Comp)
85 oveq2 7420 . . . . . . 7 (𝑒 = ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) β†’ (𝐽 β†Ύt 𝑒) = (𝐽 β†Ύt ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))))
8685eleq1d 2817 . . . . . 6 (𝑒 = ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) β†’ ((𝐽 β†Ύt 𝑒) ∈ Comp ↔ (𝐽 β†Ύt ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))) ∈ Comp))
8786rspcev 3612 . . . . 5 ((((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2))) ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯) ∧ (𝐽 β†Ύt ((clsβ€˜π½)β€˜(𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))) ∈ Comp) β†’ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ Comp)
8843, 84, 87syl2anc 583 . . . 4 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ Comp)
896, 88rexlimddv 3160 . . 3 ((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ Comp)
9089rgen2 3196 . 2 βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ Comp
91 isnlly 23194 . 2 (𝐽 ∈ 𝑛-Locally Comp ↔ (𝐽 ∈ Top ∧ βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ Comp))
922, 90, 91mpbir2an 708 1 𝐽 ∈ 𝑛-Locally Comp
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  βˆƒwrex 3069  {crab 3431   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  {csn 4628  βˆͺ cuni 4908   class class class wbr 5148   ∘ ccom 5680  β€˜cfv 6543  (class class class)co 7412  β„‚cc 11111  β„cr 11112   + caddc 11116  β„*cxr 11252   < clt 11253   ≀ cle 11254   βˆ’ cmin 11449   / cdiv 11876  2c2 12272  β„+crp 12979  abscabs 15186   β†Ύt crest 17371  TopOpenctopn 17372  βˆžMetcxmet 21130  ballcbl 21132  β„‚fldccnfld 21145  Topctop 22616  Clsdccld 22741  clsccl 22743  neicnei 22822  Compccmp 23111  π‘›-Locally cnlly 23190
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 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191  ax-addf 11192
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 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-supp 8150  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-er 8706  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-fsupp 9365  df-fi 9409  df-sup 9440  df-inf 9441  df-oi 9508  df-card 9937  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-icc 13336  df-fz 13490  df-fzo 13633  df-seq 13972  df-exp 14033  df-hash 14296  df-cj 15051  df-re 15052  df-im 15053  df-sqrt 15187  df-abs 15188  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-cnfld 21146  df-top 22617  df-topon 22634  df-topsp 22656  df-bases 22670  df-cld 22744  df-cls 22746  df-nei 22823  df-cn 22952  df-cnp 22953  df-haus 23040  df-cmp 23112  df-nlly 23192  df-tx 23287  df-hmeo 23480  df-xms 24047  df-ms 24048  df-tms 24049  df-cncf 24619
This theorem is referenced by:  rellycmp  24704
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