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Mirrors > Home > MPE Home > Th. List > rlimcn2 | Structured version Visualization version GIF version |
Description: Image of a limit under a continuous map, two-arg version. (Contributed by Mario Carneiro, 17-Sep-2014.) |
Ref | Expression |
---|---|
rlimcn2.1a | β’ ((π β§ π§ β π΄) β π΅ β π) |
rlimcn2.1b | β’ ((π β§ π§ β π΄) β πΆ β π) |
rlimcn2.2a | β’ (π β π β π) |
rlimcn2.2b | β’ (π β π β π) |
rlimcn2.3a | β’ (π β (π§ β π΄ β¦ π΅) βπ π ) |
rlimcn2.3b | β’ (π β (π§ β π΄ β¦ πΆ) βπ π) |
rlimcn2.4 | β’ (π β πΉ:(π Γ π)βΆβ) |
rlimcn2.5 | β’ ((π β§ π₯ β β+) β βπ β β+ βπ β β+ βπ’ β π βπ£ β π (((absβ(π’ β π )) < π β§ (absβ(π£ β π)) < π ) β (absβ((π’πΉπ£) β (π πΉπ))) < π₯)) |
Ref | Expression |
---|---|
rlimcn2 | β’ (π β (π§ β π΄ β¦ (π΅πΉπΆ)) βπ (π πΉπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimcn2.1a | . 2 β’ ((π β§ π§ β π΄) β π΅ β π) | |
2 | rlimcn2.1b | . 2 β’ ((π β§ π§ β π΄) β πΆ β π) | |
3 | rlimcn2.4 | . . . 4 β’ (π β πΉ:(π Γ π)βΆβ) | |
4 | 3 | adantr 479 | . . 3 β’ ((π β§ π§ β π΄) β πΉ:(π Γ π)βΆβ) |
5 | 4, 1, 2 | fovcdmd 7597 | . 2 β’ ((π β§ π§ β π΄) β (π΅πΉπΆ) β β) |
6 | rlimcn2.2a | . . 3 β’ (π β π β π) | |
7 | rlimcn2.2b | . . 3 β’ (π β π β π) | |
8 | 3, 6, 7 | fovcdmd 7597 | . 2 β’ (π β (π πΉπ) β β) |
9 | rlimcn2.3a | . 2 β’ (π β (π§ β π΄ β¦ π΅) βπ π ) | |
10 | rlimcn2.3b | . 2 β’ (π β (π§ β π΄ β¦ πΆ) βπ π) | |
11 | rlimcn2.5 | . 2 β’ ((π β§ π₯ β β+) β βπ β β+ βπ β β+ βπ’ β π βπ£ β π (((absβ(π’ β π )) < π β§ (absβ(π£ β π)) < π ) β (absβ((π’πΉπ£) β (π πΉπ))) < π₯)) | |
12 | 1, 2, 5, 8, 9, 10, 11 | rlimcn3 15572 | 1 β’ (π β (π§ β π΄ β¦ (π΅πΉπΆ)) βπ (π πΉπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β wcel 2098 βwral 3057 βwrex 3066 class class class wbr 5150 β¦ cmpt 5233 Γ cxp 5678 βΆwf 6547 βcfv 6551 (class class class)co 7424 βcc 11142 < clt 11284 β cmin 11480 β+crp 13012 abscabs 15219 βπ crli 15467 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pow 5367 ax-pr 5431 ax-un 7744 ax-cnex 11200 ax-resscn 11201 ax-pre-lttri 11218 ax-pre-lttrn 11219 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-po 5592 df-so 5593 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-res 5692 df-ima 5693 df-iota 6503 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-fv 6559 df-ov 7427 df-oprab 7428 df-mpo 7429 df-er 8729 df-pm 8852 df-en 8969 df-dom 8970 df-sdom 8971 df-pnf 11286 df-mnf 11287 df-xr 11288 df-ltxr 11289 df-le 11290 df-rlim 15471 |
This theorem is referenced by: rlimaddOLD 15626 rlimsub 15627 rlimmulOLD 15629 |
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