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Theorem lmmbr 24774
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 22732. (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 23944 . . . 4 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
41, 3syl 17 . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
54lmbr 22761 . 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 12982 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ*)
72blopn 24008 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘₯ ∈ ℝ*) β†’ (𝑃(ballβ€˜π·)π‘₯) ∈ 𝐽)
86, 7syl3an3 1165 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘₯ ∈ ℝ+) β†’ (𝑃(ballβ€˜π·)π‘₯) ∈ 𝐽)
9 blcntr 23918 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑃 ∈ 𝑋 ∧ π‘₯ ∈ ℝ+) β†’ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘₯))
10 eleq2 2822 . . . . . . . . . . . . . 14 (𝑒 = (𝑃(ballβ€˜π·)π‘₯) β†’ (𝑃 ∈ 𝑒 ↔ 𝑃 ∈ (𝑃(ballβ€˜π·)π‘₯)))
11 feq3 6700 . . . . . . . . . . . . . . 15 (𝑒 = (𝑃(ballβ€˜π·)π‘₯) β†’ ((𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’ ↔ (𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
1211rexbidv 3178 . . . . . . . . . . . . . 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 344 . . . . . . . . . . . . 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 3610 . . . . . . . . . . . 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 452 . . . . . . . . . . 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 584 . . . . . . . . . 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 1118 . . . . . . . . 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 718 . . . . . . . 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 452 . . . . . . 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 3145 . . . . . 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 24001 . . . . . . . . . . 11 ((𝐷 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝐽 ∧ 𝑃 ∈ 𝑒) β†’ βˆƒπ‘₯ ∈ ℝ+ (𝑃(ballβ€˜π·)π‘₯) βŠ† 𝑒)
22 r19.29 3114 . . . . . . . . . . . 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 6734 . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . 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 3170 . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . 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 3151 . . . . . . . . . . . 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 593 . . . . . . . . . 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 1354 . . . . . . . . 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 408 . . . . . . . 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 713 . . . . . . 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 3145 . . . . . 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 799 . . . . 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 579 . . . 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 1089 . . . 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 1089 . . . 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 313 . . 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 278 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 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948   class class class wbr 5148  ran crn 5677   β†Ύ cres 5678  βŸΆwf 6539  β€˜cfv 6543  (class class class)co 7408   ↑pm cpm 8820  β„‚cc 11107  β„*cxr 11246  β„€β‰₯cuz 12821  β„+crp 12973  βˆžMetcxmet 20928  ballcbl 20930  MetOpencmopn 20933  TopOnctopon 22411  β‡π‘‘clm 22729
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186  ax-pre-sup 11187
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  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-op 4635  df-uni 4909  df-iun 4999  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-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-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-sup 9436  df-inf 9437  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-div 11871  df-nn 12212  df-2 12274  df-n0 12472  df-z 12558  df-uz 12822  df-q 12932  df-rp 12974  df-xneg 13091  df-xadd 13092  df-xmul 13093  df-topgen 17388  df-psmet 20935  df-xmet 20936  df-bl 20938  df-mopn 20939  df-top 22395  df-topon 22412  df-bases 22448  df-lm 22732
This theorem is referenced by:  lmmbr2  24775  lmcau  24829
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