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Mirrors > Home > MPE Home > Th. List > rlimaddOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rlimadd 15619 as of 27-Sep-2024. (Contributed by Mario Carneiro, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
rlimadd.3 | β’ ((π β§ π₯ β π΄) β π΅ β π) |
rlimadd.4 | β’ ((π β§ π₯ β π΄) β πΆ β π) |
rlimadd.5 | β’ (π β (π₯ β π΄ β¦ π΅) βπ π·) |
rlimadd.6 | β’ (π β (π₯ β π΄ β¦ πΆ) βπ πΈ) |
Ref | Expression |
---|---|
rlimaddOLD | β’ (π β (π₯ β π΄ β¦ (π΅ + πΆ)) βπ (π· + πΈ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rlimadd.3 | . . 3 β’ ((π β§ π₯ β π΄) β π΅ β π) | |
2 | rlimadd.5 | . . 3 β’ (π β (π₯ β π΄ β¦ π΅) βπ π·) | |
3 | 1, 2 | rlimmptrcl 15584 | . 2 β’ ((π β§ π₯ β π΄) β π΅ β β) |
4 | rlimadd.4 | . . 3 β’ ((π β§ π₯ β π΄) β πΆ β π) | |
5 | rlimadd.6 | . . 3 β’ (π β (π₯ β π΄ β¦ πΆ) βπ πΈ) | |
6 | 4, 5 | rlimmptrcl 15584 | . 2 β’ ((π β§ π₯ β π΄) β πΆ β β) |
7 | rlimcl 15479 | . . 3 β’ ((π₯ β π΄ β¦ π΅) βπ π· β π· β β) | |
8 | 2, 7 | syl 17 | . 2 β’ (π β π· β β) |
9 | rlimcl 15479 | . . 3 β’ ((π₯ β π΄ β¦ πΆ) βπ πΈ β πΈ β β) | |
10 | 5, 9 | syl 17 | . 2 β’ (π β πΈ β β) |
11 | ax-addf 11217 | . . 3 β’ + :(β Γ β)βΆβ | |
12 | 11 | a1i 11 | . 2 β’ (π β + :(β Γ β)βΆβ) |
13 | simpr 483 | . . 3 β’ ((π β§ π¦ β β+) β π¦ β β+) | |
14 | 8 | adantr 479 | . . 3 β’ ((π β§ π¦ β β+) β π· β β) |
15 | 10 | adantr 479 | . . 3 β’ ((π β§ π¦ β β+) β πΈ β β) |
16 | addcn2 15570 | . . 3 β’ ((π¦ β β+ β§ π· β β β§ πΈ β β) β βπ§ β β+ βπ€ β β+ βπ’ β β βπ£ β β (((absβ(π’ β π·)) < π§ β§ (absβ(π£ β πΈ)) < π€) β (absβ((π’ + π£) β (π· + πΈ))) < π¦)) | |
17 | 13, 14, 15, 16 | syl3anc 1368 | . 2 β’ ((π β§ π¦ β β+) β βπ§ β β+ βπ€ β β+ βπ’ β β βπ£ β β (((absβ(π’ β π·)) < π§ β§ (absβ(π£ β πΈ)) < π€) β (absβ((π’ + π£) β (π· + πΈ))) < π¦)) |
18 | 3, 6, 8, 10, 2, 5, 12, 17 | rlimcn2 15567 | 1 β’ (π β (π₯ β π΄ β¦ (π΅ + πΆ)) βπ (π· + πΈ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β wcel 2098 βwral 3051 βwrex 3060 class class class wbr 5143 β¦ cmpt 5226 Γ cxp 5670 βΆwf 6539 βcfv 6543 (class class class)co 7416 βcc 11136 + caddc 11141 < clt 11278 β cmin 11474 β+crp 13006 abscabs 15213 βπ crli 15461 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-pm 8846 df-en 8963 df-dom 8964 df-sdom 8965 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-rlim 15465 |
This theorem is referenced by: (None) |
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