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Mirrors > Home > MPE Home > Th. List > rlimsub | Structured version Visualization version GIF version |
Description: Limit of the difference of two converging functions. Proposition 12-2.1(b) of [Gleason] p. 168. (Contributed by Mario Carneiro, 22-Sep-2014.) |
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 |
---|---|
rlimsub | β’ (π β (π₯ β π΄ β¦ (π΅ β πΆ)) βπ (π· β 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 15558 | . 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 15558 | . 2 β’ ((π β§ π₯ β π΄) β πΆ β β) |
7 | rlimcl 15453 | . . 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 15453 | . . 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 | subf 11466 | . . 3 β’ β :(β Γ β)βΆβ | |
12 | 11 | a1i 11 | . 2 β’ (π β β :(β Γ β)βΆβ) |
13 | simpr 484 | . . 3 β’ ((π β§ π¦ β β+) β π¦ β β+) | |
14 | 8 | adantr 480 | . . 3 β’ ((π β§ π¦ β β+) β π· β β) |
15 | 10 | adantr 480 | . . 3 β’ ((π β§ π¦ β β+) β πΈ β β) |
16 | subcn2 15545 | . . 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’ β π£) β (π· β 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’ β π£) β (π· β πΈ))) < π¦)) |
18 | 3, 6, 8, 10, 2, 5, 12, 17 | rlimcn2 15541 | 1 β’ (π β (π₯ β π΄ β¦ (π΅ β πΆ)) βπ (π· β πΈ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 β wcel 2098 βwral 3055 βwrex 3064 class class class wbr 5141 β¦ cmpt 5224 Γ cxp 5667 βΆwf 6533 βcfv 6537 (class class class)co 7405 βcc 11110 < clt 11252 β cmin 11448 β+crp 12980 abscabs 15187 βπ crli 15435 |
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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-sup 9439 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-seq 13973 df-exp 14033 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-rlim 15439 |
This theorem is referenced by: rlimneg 15599 rlimle 15600 dvfsumrlim2 25922 logexprlim 27113 |
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