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Mirrors > Home > ILE Home > Th. List > resubcld | GIF version |
Description: Closure law for subtraction of reals. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
renegcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
resubcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
resubcld | ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | renegcld.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | resubcld.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | resubcl 8252 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℝ) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 (class class class)co 5897 ℝcr 7841 − cmin 8159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-resscn 7934 ax-1cn 7935 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-addcom 7942 ax-addass 7944 ax-distr 7946 ax-i2m1 7947 ax-0id 7950 ax-rnegex 7951 ax-cnre 7953 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-sub 8161 df-neg 8162 |
This theorem is referenced by: ltsubadd 8420 lesubadd 8422 ltaddsub 8424 leaddsub 8426 lesub1 8444 lesub2 8445 ltsub1 8446 ltsub2 8447 lt2sub 8448 le2sub 8449 rereim 8574 ltmul1a 8579 cru 8590 lemul1a 8846 ztri3or 9327 lincmb01cmp 10035 iccf1o 10036 rebtwn2z 10287 qbtwnrelemcalc 10288 qbtwnre 10289 intfracq 10353 modqval 10357 modqlt 10366 modqsubdir 10426 ser3le 10552 expnbnd 10678 crre 10901 remullem 10915 recvguniqlem 11038 resqrexlemover 11054 resqrexlemcalc2 11059 resqrexlemcalc3 11060 resqrexlemnmsq 11061 resqrexlemnm 11062 resqrexlemcvg 11063 resqrexlemglsq 11066 resqrexlemga 11067 fzomaxdiflem 11156 caubnd2 11161 amgm2 11162 icodiamlt 11224 qdenre 11246 maxabslemab 11250 maxabslemlub 11251 maxltsup 11262 bdtrilem 11282 bdtri 11283 mulcn2 11355 reccn2ap 11356 climle 11377 climsqz 11378 climsqz2 11379 climcvg1nlem 11392 fsumle 11506 cvgratnnlembern 11566 cvgratnnlemsumlt 11571 cvgratnnlemfm 11572 cvgratnnlemrate 11573 cvgratnn 11574 efltim 11741 sin01bnd 11800 sin01gt0 11804 cos12dec 11810 uzwodc 12073 pythagtriplem12 12310 pythagtriplem14 12312 4sqlem15 12440 blss2ps 14383 blss2 14384 blssps 14404 blss 14405 ivthinclemlopn 14591 ivthinclemuopn 14593 dvcjbr 14649 reeff1oleme 14670 efltlemlt 14672 sin0pilem1 14679 tangtx 14736 cosordlem 14747 cosq34lt1 14748 cvgcmp2nlemabs 15259 iooref1o 15261 trilpolemlt1 15268 trirec0 15271 apdifflemf 15273 neap0mkv 15296 |
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