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Mirrors > Home > MPE Home > Th. List > rlimaddOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rlimadd 15531 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 15496 | . 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 15496 | . 2 β’ ((π β§ π₯ β π΄) β πΆ β β) |
7 | rlimcl 15391 | . . 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 15391 | . . 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 11135 | . . 3 β’ + :(β Γ β)βΆβ | |
12 | 11 | a1i 11 | . 2 β’ (π β + :(β Γ β)βΆβ) |
13 | simpr 486 | . . 3 β’ ((π β§ π¦ β β+) β π¦ β β+) | |
14 | 8 | adantr 482 | . . 3 β’ ((π β§ π¦ β β+) β π· β β) |
15 | 10 | adantr 482 | . . 3 β’ ((π β§ π¦ β β+) β πΈ β β) |
16 | addcn2 15482 | . . 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 1372 | . 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 15479 | 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 397 β wcel 2107 βwral 3061 βwrex 3070 class class class wbr 5106 β¦ cmpt 5189 Γ cxp 5632 βΆwf 6493 βcfv 6497 (class class class)co 7358 βcc 11054 + caddc 11059 < clt 11194 β cmin 11390 β+crp 12920 abscabs 15125 βπ crli 15373 |
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 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 ax-pre-sup 11134 ax-addf 11135 |
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 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-pm 8771 df-en 8887 df-dom 8888 df-sdom 8889 df-sup 9383 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-div 11818 df-nn 12159 df-2 12221 df-3 12222 df-n0 12419 df-z 12505 df-uz 12769 df-rp 12921 df-seq 13913 df-exp 13974 df-cj 14990 df-re 14991 df-im 14992 df-sqrt 15126 df-abs 15127 df-rlim 15377 |
This theorem is referenced by: (None) |
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