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Theorem cnllycmp 24702
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 24520 . 2 𝐽 ∈ Top
3 cnxmet 24509 . . . . 5 (abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚)
41cnfldtopn 24518 . . . . . 6 𝐽 = (MetOpenβ€˜(abs ∘ βˆ’ ))
54mopni2 24222 . . . . 5 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)
63, 5mp3an1 1446 . . . 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 4940 . . . . . . . . . . . 12 (π‘₯ ∈ 𝐽 β†’ π‘₯ βŠ† βˆͺ 𝐽)
109ad2antrr 722 . . . . . . . . . . 11 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ π‘₯ βŠ† βˆͺ 𝐽)
111cnfldtopon 24519 . . . . . . . . . . . 12 𝐽 ∈ (TopOnβ€˜β„‚)
1211toponunii 22638 . . . . . . . . . . 11 β„‚ = βˆͺ 𝐽
1310, 12sseqtrrdi 4032 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ π‘₯ βŠ† β„‚)
14 simplr 765 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ 𝑦 ∈ π‘₯)
1513, 14sseldd 3982 . . . . . . . . 9 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ 𝑦 ∈ β„‚)
16 rphalfcl 13005 . . . . . . . . . . 11 (π‘Ÿ ∈ ℝ+ β†’ (π‘Ÿ / 2) ∈ ℝ+)
1716ad2antrl 724 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (π‘Ÿ / 2) ∈ ℝ+)
1817rpxrd 13021 . . . . . . . . 9 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (π‘Ÿ / 2) ∈ ℝ*)
194blopn 24229 . . . . . . . . 9 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑦 ∈ β„‚ ∧ (π‘Ÿ / 2) ∈ ℝ*) β†’ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)) ∈ 𝐽)
208, 15, 18, 19syl3anc 1369 . . . . . . . 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 24139 . . . . . . . . 9 (((abs ∘ βˆ’ ) ∈ (∞Metβ€˜β„‚) ∧ 𝑦 ∈ β„‚ ∧ (π‘Ÿ / 2) ∈ ℝ+) β†’ 𝑦 ∈ (𝑦(ballβ€˜(abs ∘ βˆ’ ))(π‘Ÿ / 2)))
228, 15, 17, 21syl3anc 1369 . . . . . . . 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 22843 . . . . . . . 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 1369 . . . . . . 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 24144 . . . . . . . . 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 1369 . . . . . . . 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 22780 . . . . . . . 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 582 . . . . . . 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 12987 . . . . . . . . . . 11 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
3029ad2antrl 724 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ π‘Ÿ ∈ ℝ*)
31 rphalflt 13007 . . . . . . . . . . 11 (π‘Ÿ ∈ ℝ+ β†’ (π‘Ÿ / 2) < π‘Ÿ)
3231ad2antrl 724 . . . . . . . . . 10 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (π‘Ÿ / 2) < π‘Ÿ)
334blsscls 24236 . . . . . . . . . 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 1375 . . . . . . . . 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 769 . . . . . . . . 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 3991 . . . . . . . 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 3991 . . . . . . 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 22840 . . . . . . 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 835 . . . . . 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 3476 . . . . . . . 8 π‘₯ ∈ V
4140elpw2 5344 . . . . . . 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 4193 . . . . 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 22771 . . . . . . 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 582 . . . . . 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 15387 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (absβ€˜π‘¦) ∈ ℝ)
4717rpred 13020 . . . . . . . 8 (((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) β†’ (π‘Ÿ / 2) ∈ ℝ)
4846, 47readdcld 11247 . . . . . . 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 2730 . . . . . . . . . 10 {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)} = {𝑀 ∈ β„‚ ∣ (𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2)}
504, 49blcls 24235 . . . . . . . . 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 1369 . . . . . . . 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 483 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ 𝑧 ∈ β„‚)
5315adantr 479 . . . . . . . . . . . . 13 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ 𝑦 ∈ β„‚)
5452, 53abs2difd 15408 . . . . . . . . . . . 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 15387 . . . . . . . . . . . . . 14 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (absβ€˜π‘§) ∈ ℝ)
5646adantr 479 . . . . . . . . . . . . . 14 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (absβ€˜π‘¦) ∈ ℝ)
5755, 56resubcld 11646 . . . . . . . . . . . . 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 11575 . . . . . . . . . . . . . 14 ((((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) ∧ (π‘Ÿ ∈ ℝ+ ∧ (𝑦(ballβ€˜(abs ∘ βˆ’ ))π‘Ÿ) βŠ† π‘₯)) ∧ 𝑧 ∈ β„‚) β†’ (𝑧 βˆ’ 𝑦) ∈ β„‚)
5958abscld 15387 . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 11312 . . . . . . . . . . . . 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 1369 . . . . . . . . . . . 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 691 . . . . . . . . . . 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 15404 . . . . . . . . . . . . 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 2730 . . . . . . . . . . . . . . 15 (abs ∘ βˆ’ ) = (abs ∘ βˆ’ )
6665cnmetdval 24507 . . . . . . . . . . . . . 14 ((𝑦 ∈ β„‚ ∧ 𝑧 ∈ β„‚) β†’ (𝑦(abs ∘ βˆ’ )𝑧) = (absβ€˜(𝑦 βˆ’ 𝑧)))
6715, 66sylan 578 . . . . . . . . . . . . 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 2773 . . . . . . . . . . . 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 5157 . . . . . . . . . . 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 11817 . . . . . . . . . . 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 292 . . . . . . . . . 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 3144 . . . . . . . . 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 7419 . . . . . . . . . . 11 (𝑀 = 𝑧 β†’ (𝑦(abs ∘ βˆ’ )𝑀) = (𝑦(abs ∘ βˆ’ )𝑧))
7473breq1d 5157 . . . . . . . . . 10 (𝑀 = 𝑧 β†’ ((𝑦(abs ∘ βˆ’ )𝑀) ≀ (π‘Ÿ / 2) ↔ (𝑦(abs ∘ βˆ’ )𝑧) ≀ (π‘Ÿ / 2)))
7574ralrab 3688 . . . . . . . . 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 4049 . . . . . . . 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 5207 . . . . . . 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 582 . . . . . 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 2730 . . . . . . . 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 24701 . . . . . . 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 709 . . . . 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 7419 . . . . . . 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 2816 . . . . . 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 3611 . . . . 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 582 . . . 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 3159 . . 3 ((π‘₯ ∈ 𝐽 ∧ 𝑦 ∈ π‘₯) β†’ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ Comp)
9089rgen2 3195 . 2 βˆ€π‘₯ ∈ 𝐽 βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘’ ∈ (((neiβ€˜π½)β€˜{𝑦}) ∩ 𝒫 π‘₯)(𝐽 β†Ύt 𝑒) ∈ Comp
91 isnlly 23193 . 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 707 1 𝐽 ∈ 𝑛-Locally Comp
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068  {crab 3430   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   class class class wbr 5147   ∘ ccom 5679  β€˜cfv 6542  (class class class)co 7411  β„‚cc 11110  β„cr 11111   + caddc 11115  β„*cxr 11251   < clt 11252   ≀ cle 11253   βˆ’ cmin 11448   / cdiv 11875  2c2 12271  β„+crp 12978  abscabs 15185   β†Ύt crest 17370  TopOpenctopn 17371  βˆžMetcxmet 21129  ballcbl 21131  β„‚fldccnfld 21144  Topctop 22615  Clsdccld 22740  clsccl 22742  neicnei 22821  Compccmp 23110  π‘›-Locally cnlly 23189
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-icc 13335  df-fz 13489  df-fzo 13632  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-hom 17225  df-cco 17226  df-rest 17372  df-topn 17373  df-0g 17391  df-gsum 17392  df-topgen 17393  df-pt 17394  df-prds 17397  df-xrs 17452  df-qtop 17457  df-imas 17458  df-xps 17460  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-mulg 18987  df-cntz 19222  df-cmn 19691  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-cls 22745  df-nei 22822  df-cn 22951  df-cnp 22952  df-haus 23039  df-cmp 23111  df-nlly 23191  df-tx 23286  df-hmeo 23479  df-xms 24046  df-ms 24047  df-tms 24048  df-cncf 24618
This theorem is referenced by:  rellycmp  24703
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