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Theorem lmmbr 24645
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 22603. (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 23815 . . . 4 (𝐷 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
41, 3syl 17 . . 3 (πœ‘ β†’ 𝐽 ∈ (TopOnβ€˜π‘‹))
54lmbr 22632 . 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 12932 . . . . . . . . . . . 12 (π‘₯ ∈ ℝ+ β†’ π‘₯ ∈ ℝ*)
72blopn 23879 . . . . . . . . . . . 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 23789 . . . . . . . . . . 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 6655 . . . . . . . . . . . . . . 15 (𝑒 = (𝑃(ballβ€˜π·)π‘₯) β†’ ((𝐹 β†Ύ 𝑦):π‘¦βŸΆπ‘’ ↔ (𝐹 β†Ύ 𝑦):π‘¦βŸΆ(𝑃(ballβ€˜π·)π‘₯)))
1211rexbidv 3172 . . . . . . . . . . . . . 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 3581 . . . . . . . . . . . 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 3139 . . . . . 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 23872 . . . . . . . . . . 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 6689 . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . 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 3139 . . . . . 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 3061  βˆƒwrex 3070   βŠ† wss 3914   class class class wbr 5109  ran crn 5638   β†Ύ cres 5639  βŸΆwf 6496  β€˜cfv 6500  (class class class)co 7361   ↑pm cpm 8772  β„‚cc 11057  β„*cxr 11196  β„€β‰₯cuz 12771  β„+crp 12923  βˆžMetcxmet 20804  ballcbl 20806  MetOpencmopn 20809  TopOnctopon 22282  β‡π‘‘clm 22600
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136  ax-pre-sup 11137
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 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-sup 9386  df-inf 9387  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-div 11821  df-nn 12162  df-2 12224  df-n0 12422  df-z 12508  df-uz 12772  df-q 12882  df-rp 12924  df-xneg 13041  df-xadd 13042  df-xmul 13043  df-topgen 17333  df-psmet 20811  df-xmet 20812  df-bl 20814  df-mopn 20815  df-top 22266  df-topon 22283  df-bases 22319  df-lm 22603
This theorem is referenced by:  lmmbr2  24646  lmcau  24700
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