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Theorem lmmbr 24775
Description: Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 βŠ† (β„‚ Γ— 𝑋) allows to use objects more general than sequences when convenient; see the comment in df-lm 22733. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
lmmbr.2 𝐽 = (MetOpenβ€˜π·)
lmmbr.3 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
Assertion
Ref Expression
lmmbr (πœ‘ β†’ (𝐹(β‡π‘‘β€˜π½)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯))))
Distinct variable groups:   π‘₯,𝑦,𝐷   π‘₯,𝐹,𝑦   π‘₯,𝑃,𝑦   π‘₯,𝑋,𝑦   π‘₯,𝐽,𝑦   πœ‘,π‘₯
Allowed substitution hint:   πœ‘(𝑦)
Proof of Theorem lmmbr
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 lmmbr.3 . . . 4 (πœ‘ β†’ 𝐷 ∈ (∞Metβ€˜π‘‹))
2 lmmbr.2 . . . . 5 𝐽 = (MetOpenβ€˜π·)
32mopntopon 23945 . . . 4 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
41, 3syl 17 . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
54lmbr 22762 . 2 (πœ‘ β†’ (𝐹(β‡π‘‘β€˜π½)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))))
6 rpxr 12983 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ*)
72blopn 24009 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘₯ ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)π‘₯) ∈ 𝐽)
86, 7syl3an3 1166 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘₯ ∈ ℝ+) β†’ (𝑃(ballβ€˜π·)π‘₯) ∈ 𝐽)
9 blcntr 23919 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘₯ ∈ ℝ+) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘₯))
10 eleq2 2823 . . . . . . . . . . . . . 14 (𝑒 = (𝑃(ballβ€˜π·)π‘₯) β†’ (𝑃 ∈ 𝑒 ↔ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘₯)))
11 feq3 6701 . . . . . . . . . . . . . . 15 (𝑒 = (𝑃(ballβ€˜π·)π‘₯) β†’ ((𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’ ↔ (𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
1211rexbidv 3179 . . . . . . . . . . . . . 14 (𝑒 = (𝑃(ballβ€˜π·)π‘₯) β†’ (βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’ ↔ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
1310, 12imbi12d 345 . . . . . . . . . . . . 13 (𝑒 = (𝑃(ballβ€˜π·)π‘₯) β†’ ((𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’) ↔ (𝑃 ∈ (𝑃(ballβ€˜π·)π‘₯) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯))))
1413rspcva 3611 . . . . . . . . . . . 12 (((𝑃(ballβ€˜π·)π‘₯) ∈ 𝐽 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) β†’ (𝑃 ∈ (𝑃(ballβ€˜π·)π‘₯) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
1514impancom 453 . . . . . . . . . . 11 (((𝑃(ballβ€˜π·)π‘₯) ∈ 𝐽 ∧ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘₯)) β†’ (βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
168, 9, 15syl2anc 585 . . . . . . . . . 10 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘₯ ∈ ℝ+) β†’ (βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
17163expa 1119 . . . . . . . . 9 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋) ∧ π‘₯ ∈ ℝ+) β†’ (βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
1817adantlrl 719 . . . . . . . 8 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋)) ∧ π‘₯ ∈ ℝ+) β†’ (βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
1918impancom 453 . . . . . . 7 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋)) ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) β†’ (π‘₯ ∈ ℝ+ β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
2019ralrimiv 3146 . . . . . 6 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋)) ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) β†’ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯))
212mopni2 24002 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝐽 ∧ 𝑃 ∈ 𝑒) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒)
22 r19.29 3115 . . . . . . . . . . . 12 ((βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) ∧ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒) β†’ βˆƒπ‘₯ ∈ ℝ+ (βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) ∧ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒))
23 fss 6735 . . . . . . . . . . . . . . . 16 (((𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) ∧ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒) β†’ (𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)
2423expcom 415 . . . . . . . . . . . . . . 15 ((𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒 β†’ ((𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) β†’ (𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))
2524reximdv 3171 . . . . . . . . . . . . . 14 ((𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒 β†’ (βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))
2625impcom 409 . . . . . . . . . . . . 13 ((βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) ∧ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)
2726rexlimivw 3152 . . . . . . . . . . . 12 (βˆƒπ‘₯ ∈ ℝ+ (βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) ∧ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)
2822, 27syl 17 . . . . . . . . . . 11 ((βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) ∧ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)
2921, 28sylan2 594 . . . . . . . . . 10 ((βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) ∧ (𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝐽 ∧ 𝑃 ∈ 𝑒)) β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)
30293exp2 1355 . . . . . . . . 9 (βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯) β†’ (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (𝑒 ∈ 𝐽 β†’ (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))))
3130impcom 409 . . . . . . . 8 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)) β†’ (𝑒 ∈ 𝐽 β†’ (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
3231adantlr 714 . . . . . . 7 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋)) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)) β†’ (𝑒 ∈ 𝐽 β†’ (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
3332ralrimiv 3146 . . . . . 6 (((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋)) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)) β†’ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’))
3420, 33impbida 800 . . . . 5 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋)) β†’ (βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’) ↔ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
3534pm5.32da 580 . . . 4 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ (((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) ↔ ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯))))
36 df-3an 1090 . . . 4 ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) ↔ ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)))
37 df-3an 1090 . . . 4 ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)) ↔ ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋) ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
3835, 36, 373bitr4g 314 . . 3 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯))))
391, 38syl 17 . 2 (πœ‘ β†’ ((𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘’ ∈ 𝐽 (𝑃 ∈ 𝑒 β†’ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’)) ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯))))
405, 39bitrd 279 1 (πœ‘ β†’ (𝐹(β‡π‘‘β€˜π½)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm β„‚) ∧ 𝑃 ∈ 𝑋 ∧ βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ran β„€β‰₯(𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3062  βˆƒwrex 3071   βŠ† wss 3949   class class class wbr 5149  ran crn 5678   β†Ύ cres 5679  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ↑pm cpm 8821  β„‚cc 11108  β„*cxr 11247  β„€β‰₯cuz 12822  β„+crp 12974  βˆžMetcxmet 20929  ballcbl 20931  MetOpencmopn 20934  TopOnctopon 22412  β‡π‘‘clm 22730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-bl 20939  df-mopn 20940  df-top 22396  df-topon 22413  df-bases 22449  df-lm 22733
This theorem is referenced by:  lmmbr2  24776  lmcau  24830
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